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CLASSICAL AND BAYESIAN INFERENTIAL
PROCEDURES FOR SOME PROBABILISTIC
MODELS USEFUL IN RELIABILITY ANALYSIS
THESIS
SUBMITTED TO THE
KUMAUN UNIVERSITY, NAINITAL
BY
SANJAY KUMAR
FOR THE AWARD OF THE DEGREE OF
DOCTOR OF PHILOSOPHY
IN
STATISTICS
UNDER THE SUPERVISION OF
Dr. NEERAJ TIWARI
SENIOR LECTURER AND CAMPUS HEAD
DEPARTMENT OF STATISTICS
KUMAUN UNIVERSITY, S. S. J. CAMPUS, ALMORA-263601
UTTARAKHAND (INDIA)
2007
CERTIFICATE
This is to certify that the thesis entitled “CLASSICAL AND BAYESIAN
INFERENTIAL PROCEDURES FOR SOME PROBABILISTIC MODELS
USEFUL IN RELIABILITY ANALYSIS” submitted to the Kumaun University,
Nainital for the degree of DOCTOR OF PHILOSOPHY IN STATISTICS is a
record of bonafide work carried out by Mr. Sanjay Kumar, under my guidance and
supervision. I hereby certify that he has completed the research work for the full
period as required in the ordinance 6. He has put in the required attendance in the
Department and signed in the prescribed register during the period. I also certify
that no part of this thesis has been submitted for any other degree or diploma.
(Dr. Neeraj Tiwari)
Department of Statistics
Soban Singh Jeena Campus
Almora (Uttarakhand)
ACKNOWLEDGEMENT
I am very grateful to my supervisor Dr. Neeraj Tiwari, Senior Lecturer and
Campus head, Department of Statistics, Soban Singh Jeena Campus, Almora for
superbly guiding and constantly inspiring me to carry out my research work on
“CLASSICAL AND BAYESIAN INF ERENTIAL PROCEDURES FOR SOME
PROBABILSTIC MODELS USEFUL IN RELIABILITY ANALYSIS”.
I am extremely grateful to Dr. Ajit Chaturvedi, Reader, Department of
Statistics, University of Delhi for his encouragement, constructive suggestions and
help in carrying out the work.
I would like to express my gratitude to all the staff members and research
scholars of Department of Statistics at S. S. J. Campus, Almora and D. S. B.
Campus, Nainital.
I am very fortunate in having constant moral support from my local
guardian Ms.Basanti Kandpal, C.S.Kankpal, Anju Kandpal, Ayush, Suyesh and
Shepher at Almora for their co-operation in carrying out the work.
I am very thankful to my friend Mr. Girish Chandra Kandpal, Assistant
Professor, Central Agricultural University, Gangtok for his constant support in
carrying out the work. I am also thankful to Mr. Lalit Mohan Joshi, Girja Shankar
Pandey, Pankaj Pandey, Hemant Joshi, Rakesh Pant, Negi Thakur , Himanshu
Tiwari, Charuda, Vinay Kumar Pandey, I.S.S., Kanishk Kant Shrivastav, I.S.S. and
Dr. Meholika Sah for their co-operation in completing my research work.
I convey my sincere gratitude to my colleagues Mr. B.N.Joshi, B.S. Burfal,
D.S.Rawat, R.C.Pant, K.D.Pandey, D.C.Upreti, J.N.Mishra, K.S.Kathait,
D.P.Singh, B.K.Singh, D.K.Joshi, N.D.Pandey and Anil Joshi at the Sub Regional
Office of Field Operations Division (FOD) of the National Sample Survey
Organization (NSSO) at Jakhan Devi, Almora and a sincere thank goes to the
probationers of the XXVIII th batch of I.S.S. for their co-operation in completing
the work.
I find no words to express my feeling for constant encouragement, blessing
and inspirations that I received from my father Mr. C.S.Pant, mother Basanti Pant,
uncle H.C.Pant, brother Y.K.Pant and lovely sister Meena Nainwal.
Help received from Central Science Library (CSL) of University of Delhi,
Banaras Hindu University (BHU) and Indian Statistical Institute (ISI), Delhi
Centre is also gratefully acknowledged. The computer assistance received from
Institute of Economic Growth (IEG), Delhi, Institute for Integrated Learning in
Management (IILM), Delhi and Computer Centre of Ministry of Statistics and
Programme Implementation, Govt. of India, New Delhi is also acknowledged.
Last but not least, I am much grateful to the almighty God for giving me the
talent. I express my deepest sense of gratitude to the People of ‘Dev Nagari’
Almora and my hometown Bhatronjkhan for their constant moral support.
Date: (Sanjay Kumar)
CONTENTS
CHAPTERS PAGE NO.
Chapter I: THE GROWTH AND DEVLOPEMENT OF THE CLASSICAL
AND BAYESIAN INFERENTIAL PROCEDURES USEFUL IN
RELIABILITY ANALYSIS. 1-38
1.1 A Brief Historical Development of Reliability Analysis 1
1.2 Some Basic Definitions 4
1.3 Various Probabilistic Models Useful in Reliability Analysis 12
1.4 Classical Inferential Procedures in Reliability Analysis 26
1.5 Bayesian Inferential Procedures in Reliability Analysis 32
1.6 Contents of the Thesis 35
Chapter II: CLASSICAL AND BAYESIAN RELIABILITY
ESTIMATION OF BINOMIAL AND POISSON
DISTRIBUTIONS 39-61
2.1 Introduction 39
2.2 The Hazard-Rates of Binomial and Poisson Distributions and the
Set-up of the Estimation Problems 42
2.3 The UMVUE of the Powers of ?, R (to ) and ‘P’ for Binomial
Distribution 46
2.4 The Bayesian Estimation of the Powers of ?, R (to ) and ‘P’ for
Binomial Distribution 50
2.5 The UMVUE of the Powers of ?, R (to) and ‘P’ for Poisson
Distribution 55
2.6 The Bayesian Estimation of the Powers of ?, R (to) and ‘P’ for
Poisson Distribution 58
Chapter III: SEQUENTIAL POINT ESTIMATION PROCEDURES FOR
THE GENERALIZED LIFE DISTRIBUTIONS 62-79
3.1 Introduction 62
3.2 The Generalized Life Distributions 64
3.3 The Set-Up of the Estimation Problem 66
3.4 The Sequential Estimation Procedure and Second Order
Approximations 68
3.5 Condition for Negative Regret and an Improved Estimator
for ? 75
Chapter IV: BAYESIAN ESTIMATION PROCEDURES FOR A FAMILY
OF LIFE TIME DISTRIBUTIONS UNDER SELF AND GELF
80-115
4.1 Introduction 80
4.2 The Family of Lifetime Distributions 83
4.3 Bayes Estimators of Powers of ?, ?(t) and ‘P’ Under SELF 84
4.4 Bayes Estimators of Powers of ?, ?(t) and ‘P’ Under GELF 101
4.5 Bayes Estimators of the Parameters under SELF and GELF When
Parameters are Unknown 112
Chapter V: TWO STAGE POINT ESTIMATION PROCEDURE FOR THE
MEAN OF A NORMAL POPULATION WITH KNOWN
COEFFICIENT OF VARIATION 116-135
5.1 Introduction 116
5.2 Minimum Risk Point Estimation 120
5.3 Second Order Approximations to E(N) and Rg(c) 124
5.4 Bounded Risk Point Estimation 131
5.5 Second Order Approximations to E(N), E(N2 ) and RN(A) 132
Chapter VI: SHRINKAGE–TYPE BAYES ESTIMATOR OF THE
PARAMETER OF A FAMILY OF LIFETIME
DISTRIBNUTIONS 136-152
6.1 Introduction 136
6.2 The Set-up of the Estimation Problem 139
6.3 Shrinkage Estimator versus the UMVUE 141
6.4 Shrinkage Estimator versus the Minimum Mean Squared Error
Estimator 144
6.5 Lindley Approximation 147
Chapter VII: SUMMARY 153-157
CHAPTER-I
THE GROWTH AND DEVLOPEMENT OF THE CLASSICAL
AND BAYESIAN INFERENTIAL PROCEDURES USEFUL IN
RELIABILITY ANALYSIS
1.1 A BRIEF HISTORICAL DEVELOPMENT OF RELIABILITY
ANALYSIS
The word ‘reliable’ means able to be trusted or to do what is expected. It is
used in various contexts in our daily life such as reliable friend, reliable news,
reliable service centre etc. The concept of reliability is as old as man himself. He
has long been concerned with the questions about the products he used, such as:
‘Will this function satisfactorily?’, ‘Will that last long?’, ‘Will this be more
reliable than other?’ etc.
The growth and development of reliability theory has strong links with
quality control and its development. Shewhartz (1931) and Dodge and Roming
(1929) laid down the theoretical basis for utilizing statistical methods in quality
control of industrial products.
The science of reliability is new and still growing. During the First World
War, reliability was measured as the number of accidents per hour of the flight
time. During World War II, a group headed by rocket engineer Wernher Von
Braun was developing the V-I missiles in Germany. After the war, it was reported
that the first ten missiles were all failed. In spite of providing high quality parts
and careful attention, all these missiles either exploded on the launching pad or
landed in the English Channel. A mathematician named Robert Lausser was called
as a consultant for analyzing the missile system. He quickly derived the law for
the reliability of the components, which are connected in series. According to this
law, the reliability of a system consisting of a large number of components is
equal to the product of the reliabilities of the individual components that made up
the system. If the system comprises a large number of components connected in
series, the system reliability may be rather low. For example if a system has three
components connected in series having reliabilities 0.4, 0.5 and 0.6, then the
reliability of the whole system is 0.012, which is even worse than 0.4 (the lowest
reliability component).
The development continued thereafter throughout the whole world in the field
of engineering and technology. The estimation of reliability has become a demand
in the context of modern technology. With automation, the need for complicated
control and safety of systems became steadily more pressing for researchers. The
growing need of the concepts of reliability in the fields of Statistics, Mathematics
and engineering sciences has made it a prominent topic of research. Investigations
in reliability analysis began in the 1950’s and the growth of the theory began to
gain momentum after a decade. The first major committee on reliability was set up
by the Department of Defence, USA in 1950. Now almost all countries have
started to take keen interest in the applications of reliability principles.
The reliability analysis has a wide scope in the application areas. It is often
used in risk and safety analysis to evaluate the availability and applicability of
safety systems. Reliability analysis is useful fo r environmental protection. Many
industries have realized that most of the pollution caused by their plants is due to
production irregularities, which is most important factor in reducing pollution.
Many industries like aerospace, automobile and aviation have adopted reliability
principles in design process. Reliability has a wide field of application in quality
and reliability management also, since it is considered as a quality characteristic.
Nowadays, the applications of reliability principles have made remarkable
progress in many firms.
1.2 SOME BASIC DEFINITIONS
In this sub-section, we define some important terms used in reliability theory.
(a) Failures: Let us consider a system or a unit under some sort of stress. It may
be a steel beam under a load, a fuse inserted into a circuit or an electric device put
into service. The steel beam may crack or break, the fuse may burn out or the
electronic device may fail to function, all these undesirable states are defined as
“Failure”. A failure is the partial or total loss or change in the property of unit or
system in such a way that its functioning is partially or completely stopped. In the
reliability analysis, failure means that the system is incapable of performing its
required function.
The penalties of failures are paid by people in terms of money, time and
even life itself. A failure in a single unit of a system may cause the complete
breakdown in the industrial plant. A failure in the network of railways may cause
the delay of trains. A failure in the break system of a metro train in Japan was the
cause for its accident with another train and resulted in deaths of hundreds of
people. A failure occurred in the Union Carbide plant at Bhopal, resulted in
leakage of Methyl Iso Cyanide (MIC) and became the cause of the death of
thousands of people. A failure in the Columbia Space Shuttle of NASA caused the
death of seven aeronauts, including Kalpana Chawla of India.
(b) Lifetime/ Failure time/ Survival time: Lifetime is the time until the failure of
the unit occurs i.e., it is the length of the failure free time. Lifetime is often
expressed in hours of operation. Mathematically lifetime is merely ‘nonnegative
valued random variable’. Survival time and failure time are the alternate terms for
lifetime that are frequently used in reliability theory.
(c) Reliability and Reliability Function: The reliability of a unit or system is
expressed as the probability that it will perform satisfactorily at least for a stated
period under the given operational and environmental conditions, such as,
temperature, humidity, vibration etc. The reliability stresses four elements namely
probability, intended function, time and environmental conditions.
Mathematically, if X denotes the random variable (rv) representing the lifetime of
a unit, the reliability function at time t is defined as
R (t) = P(X > t)
= 1- P (X = t)
= 1 - F (t),
where F(t) is distribution function of X at specified time t, known as the failure
distribution and sometimes referred as the unreliability function.
In terms of pdf of X, say f(x), the reliability function at time t is
dxf(x)t
(t)R ∫∞
= .
R is called reliability or survival function and is always a function of time. Also
R(0) = 1 and R(8 ) = 0, hence R(t) is non-increasing function between 0 and 1.
(d) Mean Time to Failure (MTTF): The mean time to failure (MTTF) is the
expected time during which the component will perform successfully. It is given
as the mathematical expectation of the lifetime of the component. Thus
MTTF = E(X).
E(X) is also referred as the expected life.
(e) Mean Between to Failure (MTBF): If the system under consideration is
renewed through maintenance and repaired, E(X) is known as the Mean Between
to Failure (MTBF).
(f) Hazard-rate: Hazard rate is a measure of instantaneous speed of failures. It is
denoted by h(t) and given as
t(t).?SN
t)?(tSN(t)SNlim
0t?(t)h
+−
→=
R(t)f(t)= ,
where (t)SN denotes the number of components surviving after time t. If in any
situation, we have some qualitative information about hazard-rate, we can utilize
this information in selecting a lifetime model. Some frequently used hazard-rates
are constant hazard-rate, monotonic increasing hazard-rate, monotonic decreasing
hazard-rate etc. In actuarial science, it is known as ‘force of mortality’ and in
extreme -value theory, it is called ‘intensity function’. In Economics, the reciprocal
of hazard rate is called ‘Mill’s ratio’.
One of the most popular hazard-rate is ‘bathtub-shaped’ hazard-rate, which
is appropriate when individuals in the population are followed right from actual
birth to death. This pattern is also observed in human populations. Up to the age of
about 10 years, a child has high probability of dying due to birth defects or infant
diseases, called the initial failure. Between the ages of 10 to 30 years, the deaths
are assumed due to accidents. This period exhibits a constant failure rate and such
failure is called chance failure. After the age of 30 years, the death rate increases
with age and such failure is termed as wear out failure.
A mathematical relationship between h(t) and R(t) can easily be drawn
∫−= dth(t)t
0exp(t)R .
One more useful function related to hazard-rate, called the ‘Cumulative
Hazard Function’ is defined as
dxh(x)t
0(t)H ∫= .
The probabilistic models with increasing hazard-rate are very common and
frequently used. Models with constant hazard-rates are important and simple also.
Models with decreasing hazard-rates are less common but are sometimes used.
Non-monotone hazard-rates other than the bath-shaped are less common, but
possible.
(g) Censored sampling: In several situations, it is neither possible nor desirable to
record the failure time of all the items under test, since life testing experiments are
usually destructive in nature. Suppose n items are kept on a test and the
experiment is terminated when a pre-assigned number of items, say r (< n) have
failed. Such a sampling is known as ‘failure-censored sampling’ or ‘type II
censored sampling’. On the other hand, if the experiment is terminated after a pre-
assigned time, say t. Such a sampling is known as ‘time-censored sampling’ or
‘type I censored sampling’.
For type I censored sampling the length of the experiment is fixed, while
the number of observations obtained before time t is a random variable. With type
II censored sampling the number of observations is fixed but the length of the
experiment is random.
(h) Loss Function: Suppose ? is an estimator of a parameter ?, then a loss
function, denoted by ?) ,?( L is a real-valued function such that:
0 ?) ,?( L ≥ , for every ?
0 ?) ,?( L = when ? ? = .
The expected value of loss function is known as risk function.
(i) Squared-Error Loss Function (SELF): If a parameter ? is estimated by ? , the
squared-error loss function (SELF) is given by
2?) -?( E ?) ,?( L = .
(j) Absolute Error Loss Function: If a parameter ? is estimated by ? , the
absolute error loss function is given by
? -? ?) ,?( L = .
(k) LINEX (Linear in Exponential) Loss Function: A symmetric loss function
assumes that overestimation and underestimation are equally serious. However, in
some estimation problems such an assumption may be inappropriate. In estimating
reliability, overestimation is usua lly more serious than the underestimation. For
such situation when overestimation and underestimation are not equally serious,
Varian (1975) has suggested the LINEX loss function. If a parameter ? is
estimated by ? , the LINEX loss function is given by
0.b 0,a 1],?)-?a(?)-?( a [exp b ?) ,?( L >≠−−=
(l) General Entropy Loss Function (GELF): If a parameter ? is estimated by ? ,
the general entropy loss function (GELF) is defined as
0a 1,??
log a -a
??
?) ,?( L ≠−=
.
(m) Bayes Estimator: Bayes Estimator B? of parameter ? is defined as the
estimator, which minimizes the posterior expected loss
[ ] d?)x?(O
p?) ,B
?( L?) ,B
?( L x?
E ∫= ,
where O is the parameter space of parameter? .
(n) Shrinkage Estimator: Thompson (1968, a & b) introduced the concept of
shrinkage estimator. Shrinkage estimation procedure is one of the interesting
procedures in which it is assumed that the prior knowledge about the parameter is
available in the form of a prior point estimate or in the form of interval which
contain parameter in it. Thompson suggested shrinkage estimator of a parameter
by giving suitable weights to the usual estimator and the prior point estimate.
(o) Minimum Mean Square Estimator (MMSE): Minimum mean square
estimator is an estimator that minimizes the mean square error of the estimator.
(p) Lindley Approximation: In many situations, Bayes estimators are obtained
as a ratio of two integral expressions and cannot be expressed in a closed form.
However, these estimators can be numerically approximated using complex
computer programming.
Lindley (1980) suggested an asymptotic approximation to the ratio of two
integrals. The basic idea behind it is to obtain Taylor series expansion of function
involved in the integral about the maximum likelihood estimator.
1.3 VARIOUS PROBABILISTIC MODELS USEFUL IN RELIABILITY
ANALYSIS
Some frequently used probability distributions in reliability analysis are
given below. For detailed description of the properties of these distributions, one
may refers to Johnson and Kotz (1969, 1970).
(a) Exponential Distribution: A fundamental distribution to the reliability
analysis is the Exponential distribution. It is widely used in lifetime models. Davis
(1952) examined that the exponential distribution appears to fit most of the data
related to reliability analysis. Epstein (1958) remarks that the exponential
distribution plays an important role in life testing experiments. The reason behind
the wide applicability of this distribution is the availability of simple statistical
methods and its suitability to represent the lifetime of many items.
The pdf of one-parameter exponential distribution is
0 ? x,;?xexp
?1?) (x;f >
−= .
The reliability function and the hazard-rate for this distribution are given by
−=?texp(t)R
and
0.?where?1h(t) >=
For two-parameter exponential distribution the pdf is given by
otherwise.0,
0 ? ,µ- ;?
µ-xexp?1?) µ, (x;f
=
>∞<<∞
−=
This yields,
µt1,
µt,?µ-t
exp(t)R
≤=
>
−=
and
0.?where?1h(t) >=
Hence, the hazard-rate for exponential distribution is constant. It is more
appropriate for a situation where the failure rate appears to be more or less
constant.
(b) Weibull Distribution: The Weibull Distribution is also a widely used
distribution in reliability analysis. This distribution has been named after the
Swedish scientist Weibull. Weibull (1951) showed that it is useful in describing
wear out failures. It has also been used as a model for vacume tubes, ball bearings,
tumors in human beings etc.
The pdf of two-parameter Weibull distribution is given as
0 ? p, x,;?
pxexp1-p x?pp)?, (x;f >−=
,
where p is referred to as the shape parameter and ? as the scale parameter of the
distribution. It reduces to exponential distribution for 1p = and Rayleigh
distribution for 2p = .
The reliability function and the hazard-rate for this distribution are given by
−=
?
ptexp(t)R
and
0.?p,where,1pt?ph(t) >−=
The Weibull distribution has increasing failure rate (IFR) for p>1,
decreasing failure rate (DFR) for 0 < p< 1 and constant failure rate for p=1.
(c) Gamma Distribution: The gamma distribution is also sometimes used as a
lifetime distribution. Its pdf is given by
0 ß a , x,;ß
xexp 1-a xG(a) aß
1ß) a , (x;f >−=
where a and ß are shape and scale parameter, respectively.
The reliability function and hazard-rate for gamma distribution are
)G(a
)(aß tG)G(a(t)R
−=
and
−
−−=
)(aß tG)G(a aß
)ß texp(1ath(t) ,
where )(abG is the well known standard incomplete gamma function, given by
0a dy,yeb
0
1ay )(ab
G >−∫ −= .
There is no closed expression for R (t) and h (t) for this distribution.
However, R (t) and h (t) have been extensively studied and tabulated. Although it
can be shown that for a >1, it has IFR and DFR for 0 <a <1. For a =1, the gamma
distribution coincides with exponential distribution and yields constant hazard-
rate.
The generalized gamma distribution is a three-parameter distribution with
0.k ß, ?, x,;ßx)(?exp (k)G
1kßx)(?ß?)k ß, ?, (x;f >−
−=
This model includes many lifetime distributions as special cases and was
introduced by Stacy (1962).
(d) Normal Distribution: Davis (1952) has shown that the normal distribution
gives quite a good fit for the failure time data, in the context of life testing and
reliability analysis. The support to the normal distribution is (-8 , 8 ) by taking the
mean µ to be sufficiently large positive valued and standard deviation s to be
sufficiently small relative to µ.
The pdf of normal distribution with location parameter µ (mean) and scale
parameter s (standard deviation) is given by
. 0 s ,µ x,- ;2µ)(x2s2
1exp
21) (2ps
1s ) µ, (x;f >∞<<∞
−−=
The reliability function and hazard-rate for this distribution are given by
−−=s
µtF1(t)R
and
,
sµt
F1s
sµt
fh(t)
−
−
−
=
where f (.) is pdf of standard normal variate (SNV) and ? (z) is the cumulative
distribution function (cdf) of SNV, given by
du2
2uexp 21) (2p
1z(z)F
−∫
∞−= .
Although we can not obtain h(t) for this distribution in closed form, yet it
can be shown that it has IFR .
(e) Log-Normal Distribution: In the contexts of life testing and reliability
problems, the lognormal distribution answers a criticism sometimes raised against
the use of normal distribution (-8, 8), as a model for failure time distribution
which must range over 0 to 8. The lognormal distribution is appropriate when
hazard-rate is decreasing for large value of t. Goldthwaite (1961) justified its use
as a failure time distribution.
The pdf of lognormal distribution is given by
0. s ,µ -,x0 ;2µ) x(log2s2
1exp 21) (2ps x
1s ) µ, (x;f >∞<<∞∞<<−−=
The reliability function and hazard-rate for the lognormal distribution are
given as
−−=
sµ tlogF1(t)R
and
.
sµ tlogF1 ts
sµ tlogf
h(t)
−−
−
=
It should be noted that the hazard-rate initially increases over the time and
then decreases as time increases, thus the lognormal distribution serves as a model
when failure rate is rather high initially.
(f) Inverse Gaussian Distribution: Chhikara and Folks (1974, a) proposed this
distribution as a lifetime model and suggested its applications for studying
reliability aspects where the initial failure rate is high. They also developed
inferential procedure for inverse Gaussian distribution after studying its property.
The pdf of this distribution is given by
, 0 ? µ, x,;2µ)(x x22µ
?exp
21
3 x2p
??) µ, (x;f >
−−
=
where ? is shape parameter.
The reliability function and hazard-rate are given as
+λ−−−λ=
µt1
21
tF µ2?e
µt1
21
tF(t)R
and
+
−−
−
−−=
µt1
21
t?F µ2?e
µt1
21
t?F
t22µ2µ)?(texp213t2p?
h(t) .
Chhikara and Folks (1974, a) showed that this distribution has IFR for mtt < and
DFR for 32?t > , where mt is the mode of the distribution.
(g) The Maxwell and Generalized Maxwell Distribution: Tyagi and
Bhattacharya (1989, a & b) considered the Maxwell distribution as a failure
model. Chaturvedi and Rani (1998) considered the generalized form of Maxwell
distribution and termed it as ‘generalized Maxwell failure distribution’.
The pdf of the generalized Maxwell failure distribution is given by
0k ?, x,; G(k) k?
)?2x-exp(1-2k2x)k ?, (x;f >=
Its reliability function and hazard-rate are given as
(k)?2t
G(t)R =
and
.
1
0dsse
1k
2t
2s1?2th(t)
−
∫∞ −
−
+=
It is observed that this distribution has IFR.
(h) Negative Binomial Distribution: Kumar and Bhattacharya (1989) considered
negative binomial distribution as lifetime model. The negative binomial
distribution has probability mass function (pmf)
........ 1, 0, x0,r 1,?0 ; r?)-(1x?x
1xr) ? r, (x;p =><<
−+=
They studied the behaviour of the hazard-rate and showed that it has IFR
for r >1 and DFR for r <1. For r = 1, it leads to geometric distribution and yields
constant hazard-rate.
Kyriakoussis and Papadopoulos (1993) derived the Bayes estimator of the
reliability function for the zero-truncated negative binomial distribution.
Chaturvedi and Sharma (2007) considered this distribution as reliability model.
The justification behind its use as a reliability model is based on the behaviour of
its hazard-rate.
The zero-truncated negative binomial distribution has the probability mass
function (pmf)
........ 1,2, x0, s 1,?0 ; s?)-(1-1
s?)-(1x?x
1xs
) ? s, (x;p =><<
−+
=
The reliability function and the hazard-rate for this distribution are given as:
s?)-(1-1
s?)-(1x?x
1xs
tx)t(R
−+∞
==∑
and
x?
tx1txs
0x
t1ts
)t(h
+−++∞
=
−+
=
∑
.
This distribution has IFR for s<1, DFR for s>1 and constant failure rate for s=1.
(i) Binomial Distribution: Chaturvedi, Tiwari and Kumar (2007) introduced
binomial distribution as a lifetime model. The binomial distribution has pmf
r.2,...,1,0,x1,?0;xr?)(1x?xr
?)r,(x;p =<<−−
=
They studied the behaviour of the hazard-rate and showed that the binomial
distribution has IFR. [For detailed discussion, one may refer to Chapter II of the
thesis].
(j) Poisson Distribution: Chaturvedi, Tiwari and Kumar (2007) considered
Poisson distribution as a lifetime model. The Poisson distribution has pmf
. 2,...1,0,x;!x
x??e?)(x;p =−
=
They showed that the Poisson distribution can represent lifetime model
when one has IFR. [For detailed discussion, one may refer to Chapter II of the
thesis].
(k) Families of Lifetime Models: Various authors have discussed different
families of lifetime models, which contain several lifetime distributions as
particular cases. We have used the following two families of lifetime distributions
for classical and Bayesian inferential procedures discussed in the thesis.
Let us consider a family of lifetime models originated by Chaturvedi et.al.
(2002, 2003, a) with probability density function (pdf) given by
0d?,g(x),;ax;?
g(x)exp
G(d) d?
(x)'g(x)1dg?)d,(x;f >>
−
−=
where ‘a’ is known and d & ? are parameters. Here, g (x) is real valued, strictly
increasing function of X with g (a) = 0 and (x)'g denotes the first derivative of g
(x).
The above family, known as the generalized life distributions, includes the
following life distributions useful in reliability analysis as particular cases:
(a) For g (x) = x, a = 0 and d = 1, we obtain the one-parameter exponential
distribution.
(b) For g (x) = x, a = 0, we get the gamma distribution and for d taking
integer values, it is known as Erlang distribution.
(c) For g (x) = xp and a = 0, the family gives the generalized gamma
distribution.
(d) For g (x) = xp , a = 0 and d = 1, it leads to Weibull distribution.
(e) For g (x) = x2, a = 0 and d =21 , it represents the half-normal
distribution.
(f) For g (x) = x2 , a = 0 and d = 1, we obtain Rayleigh distribution.
(g) For g (x) = 2x2
, a = 0 and d = 2a , it turns out to be chi-
distribution.
(h) For g (x) = 2x2
, a = 0 and d = 23 , we get the Maxwell distribution and
for g (x) = x2, we obtain the generalized Maxwell distribution.
(i) For g (x) = log (1+ xb), a = 0 and d = 1, we obtain Burr distribution.
(j) For g (x) = log x, a = 1 and d = 1, it represents Pareto distribution.
The reliability function and hazard-rate for this family are given by:
( )?
g(x)y wheredy,?g(t)
yexp)( 1dy)t(R =∫
∞−
δΓ
−=
and
=
−
∫∞ −
+=?(t) g -y s where,
?(t) 'g
1
?g(t)dss-e
1d
(t) gs
?1(t)h .
The behaviour of hazard-rate depends upon (t) 'g . If (t) 'g is constant
(exponential distribution and Weibull distribution for p=1), h (t) is also constant. If
(t) 'g is monotonically increasing in t (Weibull distribution for p>1, Rayleigh
distribution and Chi distribution for a = 2), h (t) is monotonically increasing in t. If
(t) 'g is monotonically decreasing in t (Weibull distribution for p<1, Burr
distribution and Pareto distribution), h (t) is monotonically decreasing in t.
We consider one more family of lifetime distributions proposed by Moore
and Bilikam (1978). Let the random variable X follows the distribution given by
the pdf
−−′
= (x)/?ßg exp (x)1ßg (x)g
?ß
?)ß,f(x; ; 0.?ß,x, >
This family is known as family of lifetime distributions since it includes the
following probabilistic distributions useful in reliability analysis as particular
cases:
(i) For xg(x) = and 1ß = , we get exponential distribution.
(ii) For xg(x) = , it yields Weibull distribution.
(iii) For 0b),bxlog(1g(x) >+= and 1ß = , we obtain Burr distribution.
(iv) For
=
axlogg(x) and 1ß = , it leads to Pareto distribution.
(v) For xg(x) = and 2ß = , it gives Rayleigh distribution.
The reliability function and hazard-rate are given by
θ−= (t)/ßgexp)t(R
and
(t)1ßg(t)g?ß
)t(h −′
= .
The behaviour of hazard-rate depends upon (t) 'g . If (t)g′ is constant
(exponential and Weibull distribution for ß=1), h(t) is also constant, h(t) is
monotonically decreasing in t if (t)g′ is monotonically decreasing in t (Burr,
Pareto and Weibull distribution for ß<1) and h(t) is monotonically increasing in t
if (t)g′ is monotonically increasing in t (Rayleigh and Weibull distribution for
ß>1).
1.4 CLASSICAL INFERENTIAL PROCEDURES IN RELIABILITY
ANALYSIS
Some of the classical inferential procedures commonly used in reliability
analysis are described below.
The classical inferential procedures have been introduced in the field of
reliability analysis for deriving maximum likelihood estimators (MLE’s) and
uniformly minimum variance unbiased estimators (UMVUE’s) of the reliability
and other parametric functions.
In case of censoring from right for one-parameter exponential distribution,
Epstein and Sobel (1953) derived the MLE of scale parameter. Pugh (1963)
obtained the UMVUE of the reliability function of the exponential distribution.
Sinha (1972) discussed the behavior of UMVUE of reliability function of the
exponential distribution when a spurious observation may be present. Epstein and
Sobel (1954) and Epstein (1960) extended these results to the two-parameter
exponential distribution.
The UMUVUE of reliability function of two-parameter exponential
distribution with complete sample was obtained by Tate (1959), Laurent (1963)
and Sathe and Varde (1969). Basu (1964) derived the UMVUE of reliability
function for exponential, gamma and Weibull distributions under type II
censoring. Patil and Wani (1966) derived the UMVUE of reliability function for
the gamma and normal distribution. Harter (1969) obtained the numerical
approximations to the MLE’s for the parameters of generalized gamma
distribution. Feldman and Fox (1968) derived the UMVUES of reliability function
of normal distribution. The reliability function of the inverse Gaussian distribution
was obtained by Roy and Wasan (1968) and Chhikara and Fox (1974, b).
Chaturvedi and Rani (1998) considered the UMVUE of the reliability function for
generalized Maxwell distribution. Chaturvedi and Rani (1997) developed a family
of lifetime distributions and obtained UMVUE of reliability function and
moments. Chaturvedi and Tomer (2002; 2003, a) obtained the UMVUE of
reliability function for negative binomial distribution and generalized life
distributions.
Another measure of reliability under stress-strength set-up is the probability
P=P(X>Y), which represents the reliability of performance of an item of strength
X subject to stress Y. Owen, Craswell and Hanson (1964), Church and Harris
(1970) and Dowton (1973) discussed the estimation of P when X and Y are
normally distributed. Tong (1974) and Kelly, Kelley and Schucany (1976)
considered the case when X and Y are exponentially distributed. Tong (1975) also
considered the case when X and Y follow gamma distribution. Simonoff,
Hochberg and Reiser (1985) discussed various estimation procedures for ‘P’ in
discretized data. The MLE and UMVUE of P when both X and Y follow gamma
distribution with unequal scale and shape parameters were considered by
Constantine, Karson and Tse (1986). Chaturvedi and Surinder (1999) derived
UMVUE of P for exponential case under type I and type II censorings by using a
simpler technique of deriving UMVUEs. Using the same technique Chaturvedi
and Tomer (2002) obtained the MLE and UMVUE of ‘P’ for negative binomial
distribution.
The pioneering work in the field of sequential analysis was due to Wald
(1947), who developed sequential probability ratio test (SPRT) for testing a simple
null hypothesis against a simple alternative hypothesis. He also obtained
expressions for the operating characteristic (OC) and average sample number
(ASN) for the proposed sequential test. Epstein and Sobel (1955) considered
sequential life test in exponential case to test the simple null hypothesis against a
simple alternative hypothesis. They derived approximate formulae for OC and
ASN functions. Several authors contributed to this direction. For a brief review
one may refer to Epstein (1960), Woodall and Kurkjian (1962), Aroian (1976) and
Baryant and Schmee (1979). Phatarfod (1971) proposed sequential test for
composite hypothesis for the shape parameter of gamma distribution. Joshi and
Shah (1990) developed SPRT for inverse Gaussian distribution. Chaturvedi,
Kumar and Kumar (2000) proposed SPRT for testing simple and composite
hypotheses for the parameter of generalized life distributions.
The robustness of SPRT for different distributions have been discussed by
Harter and Moore (1976), Montange and Singpurwala (1985), Chaturvedi, Kumar
and Chauhan (1998), Chaturvedi, Kumar and Kumar (1998) and Chaturvedi,
Tiwari and Tomer (2002).
Dantzing (1940) proved the non-existence of test of student’s hypothesis
having power function independent of variance for normal population.
Consequently, one cannot construct a confidence interval of pre assigned width
and coverage probability for the mean of a normal population when variance is
unknown. To deal with this problem, Stein (1945) proposed a two-stage procedure
determining the sample size as a random variable. Ruben (1961) studied the
properties of Stein’s two-stage procedure. This procedure is easy to apply since it
requires only two stages. However, it has some drawbacks. Firstly it is not
‘asymptotically efficient’ in Chow and Robbins (1965) viewpoint. According to
Chow and Robbins (1965), a sequential procedure is asymptotically efficient, if
the ratio of the average sample size to the optimal fixed sample size converges to
unity. Secondly, this procedure does not utilize the second-stage sample size for
the purpose of the estimation of nuisance parameter. Moreover, the ‘cost of
ignorance’ of the nuisance parameter does not remain asymptotically bounded.
These drawbacks can be removed by sequential upgrading of the
observations. An important contribution to this direction was made by Starr
(1966). Mukhopadhyay (1980) proposed a ‘modified’ two-stage procedure in
order to make Stein’s two-stage procedure ‘asymptotically efficient’. Anscombe
(1949) provided a large sample theory for sequential estimation. Robbins (1959)
considered the problem of minimum risk point estima tion of the mean of normal
population under absolute error loss function and linear cost of sampling. Starr and
Woodroofe (1969) introduced another measure of the optimality of a sequential
point estimation procedure, known as ‘regret’. Regret of a sequential procedure is
defined as the difference between the risks of the sequential procedure and that of
the optimal fixed sample size procedure. A sequential procedure is ‘optimal’ if its
‘regret’ is asymptotically bounded.
Woodroofe (1977) introduced the concept of ‘second order approximations’
in the area of sequential estimation. In this theory, one may be able to study the
behavior of the remainder terms after the optimum position achieved by the fixed
sample size procedure. Chaturvedi (1988) generalized the results of Woodroofe
(1977) by obtaining the second order approximations for the regret of the
sequential procedure for the minimum risk point estimation of the mean of a
normal population by taking a family of loss functions and a general cost function.
Chaturvedi, Tiwari and Pandey
(1992) developed a class of sequential procedures for the point estimation of the
parameters of an absolutely continuous population in the presence of an unknown
scalar nuisance parameter. They also derived second-order approximations for the
expected sample size and the regret of the sequential procedure.
Hall (1981, 1983) proposed three-stage and ‘accelerated’ sequential
procedure. Chaturvedi, Tiwari and Pandey (1993) further analyzed the problem of
constructing a confidence interval of pre-assigned width and coverage probability
considered by Constanza, Hamdy and Son (1986). They utilized several multi-
stage (purely sequential, accelerated sequential, three-stage and two-stage)
estimation procedures to deal with the same estimation problem. Kumar and
Chaturvedi (1993) and Chaturvedi and Rani (1999) proposed the classes of two-
stage procedures to construct fixed width confidence intervals and point
estimation. Chaturvedi and Tiwari (2002) developed a class of three-stage
estimation procedures taking into consideration the common distributional
properties of the estimators of the parameters to be estimated under different
continuous probabilistic models and those of nuisance parameters involved
therein. They also considered the problem of constructing fixed size confidence
regions as well as point estimation. They also presented the asymptotic properties
of the proposed class. Chaturvedi and Tomer (2003, b) considered the three-stage
and accelerated sequential procedures for the mean of a normal population with
known coefficient of variation.
1.5 BAYESIAN INFERENTIAL PROCEDURES IN RELIABILITY
ANALYSIS
The Bayesian ideas in reliability analysis were introduced for the first time
by Bhattacharya (1967), who considered the Bayesian estimation of reliability
function for one-parameter exponential distribution under uniform and beta priors.
Bhattacharya and Kumar (1986) and Bhattacharya and Tyagi (1988) obtained
Bayes estimators of the reliability function with other priors. The Bayes estimators
for the reliability function of exponential and Weibull distributions using uniform
and gamma priors have been obtained by Harris and Singpurwala (1968). Canfield
(1970) considered an asymmetric loss function for the Bayesian estimation of the
reliability function under beta priors. Soland (1969) derived the Bayes estimator of
reliability function for the Weibull distribution using a discrete prior distribution
for the shape parameter. Using Monte-Carlo simulation, Tsokos (1972, b) and
Canvos and Tsokos (1973) showed that the Bayes estimators of reliability function
in case of uniform, exponential and gamma priors have uniformly smaller mean
squared-error than minimum variance unbiased estimators (MVUEs). Lian (1975)
and Martz and Lian (1977) obtained the Bayes estimator of reliability of Weibull
distribution using a piecewise linear prior distribution. Canvos and Tsokos (1971)
obtained Bayes estimator of reliability function for gamma distribution, restricting
the scale parameter as integer valued function. Padgett and Tsokos (1977) studied
the mean squared-error performance of Bayes estimator of reliability function
compared to MLE for lognormal distribution. Tyagi and Bhattacharya (1989, b)
considered the Bayesian estimation of reliability function of Maxwell distribution.
Chaturvedi and Rani (1998) extended the results of Tyagi and Bhattacharya (1989,
a & b) for the generalized Maxwell distribution. Chaturvedi, Tiwari and Kumar
(2007) obtained the Bayes estimator of the reliability function for binomial and
Poisson distributions [For detailed discussion, one may refer to Chapter II of the
thesis].
The pioneering work on the Bayesian estimation of ‘P’ has been done by
Enis and Geisser (1971), who derived Bayes estimator of ‘P’ when X and Y
follow normal distributions. Basu and Tarmast (1987) also considered the problem
of Bayesian estimation of ‘P’. Basu and Ebrahimi (1991) obtained the Bayes
estimator of ‘P’ for the exponential case in complete sample. Chaturvedi and
Tomer (2002) considered Bayes estimation of ‘P’ for negative binomial
distribution. Chaturvedi, Tiwari and Kumar (2007) derived the Bayes estimator of
‘P’ for binomial and Poisson distributions [For detailed discussion, one may refer
to Chapter II of the thesis]. All these authors considered SELF, which is a
symmetrical loss function.
While estimating reliability function, the use of a symmetrical loss function is
inappropriate because of the recognition of the fact that overestimation is usually
more serious than the underestimation [See, Basu and Ebrahimi (1991) and
Calabria and Pulcini (1996) for a detailed discussion]. For the situations when
overestimation and underestimation are not equally serious, Varian (1975)
suggested LINEX (linear in exponential) loss function, which was further used by
Zellner (1986). Basu and Ebrahimi (1991) derived Bayes estimators of the mean
failure time, reliability function and ‘P’ of an exponential distribution for the
complete sample case considering both the SELF and LINEX loss functions. The
LINEX loss function is suitable for the estimation for location parameter but not
for the
estimation of scale parameter and other parametric functions. Calabria and Pulcini
(1994) suggested the use of general entropy loss function (GELF) for estimating
these quantities.
1.6 CONTENTS OF THE THESIS
In Chapter 2 of the thesis, the binomial and Poisson distributions are
introduced as lifetime models. In Section 2.2, we study the behaviour of hazard-
rates of binomial and Poisson distributions and provide the set-up of the estimation
problems. In Sections 2.3 and 2.4, respectively, we obtain the classical and Bayes
estimators of powers of ?, reliability function and ‘P’ for binomial distribution.
The classical and Bayes estimators of powers of ?, reliability function and ‘P’ for
Poisson distribution are obtained in Sections 2.5 and 2.6, respectively. In order to
obtain the estimators of these parametric functions, the basic role is played by the
estimators of the factorial moments of the two distributions.
In Chapter 3 of the thesis, sequential point estimation procedure for the
generalized life distributions, which covers several distributions useful in
reliability analysis, including Weibull and gamma distributions as particular cases,
is considered. The failure of the fixed sample size procedure is established and
minimum risk point estimation for the parameters associated
with the generalized life distributions under SELF is considered. In Section 3.2,
we discuss the generalized life distributions and consider the problem of minimum
risk point estimation. Section 3.3 describes the set-up of the problem. The
sequential estimation procedure and second-order approximations are obtained in
Section 3.4. In Section 3.5, the condition for the negative regret of the sequential
procedure is obtained and an improved estimator is proposed which dominates the
UMVUE.
In Chapter 4 of the thesis, Bayesian estimation procedures for powers of
parameter, reliability function and P(X>Y) for a family of lifetime distributions
under squared-error loss function (SELF) and general entropy loss function
(GELF) is considered. In Section 4.2, the family of life distributions is discussed.
This family includes several probabilistic distributions useful in reliability analysis
as discussed in the Chapter I. In Section 4.3, the Bayes estimators of ?, ?(t)
(reliability function at specified mission time t) and ‘P’ under SELF are discussed.
Bayes estimators of powers of ?, ?(t) and ‘P’ under GELF is considered in Section
4.4. Throughout the chapter, we have assumed that shape parameter is known,
while scale parameter is unknown. Finally, in Section 4.5, we assume that both the
parameters are unknown and the Bayes estimators for both the parameters are
obtained after calculating the marginal posteriors in each case.
In Chapter 5 of the thesis, we develop a two-stage point estimation procedure
for the mean of a normal population when the population CV is known. Both the
minimum risk and the bounded risk estimation problems are considered. Second
order approximations are also considered for the proposed two-stage point
estimation procedure. In Section 5.2, we discuss the minimum risk estimation for
the parameters of normal distribution. In Section 5.3, we obtain second order
approximations for expected sample size, risk corresponding to two-stage point
estimation procedure [ (c)NR ] and the regret of the procedure (c)].g[R In Section
5.4, the bounded risk point estimation case is considered. In Section 5.5 we obtain
second order approximations for expected sample size, E( 2N ) and (A)NR .
In the Chapter 6 of the thesis, we derived the Shrinkage-type Bayes estimator
of the parameter of a family of lifetime distributions. In Section 6.2, the set-up of
the estimation problem is described and the desired shrinkage-type Bayes
estimators are obtained. The optimality in the sense of efficiency of the shrinkage-
type Bayes estimator over the UMVUE and the minimum mean squared error
estimator is established in the Section 6.3 and Section 6.4, respectively. Finally, in
Section 6.5, the Lindley approximation of the reliability function of the family of
lifetime distribution is considered.
In Chapter 7, a brief summary of the thesis is presented.
CHAPTER II
CLASSICAL AND BAYESIAN RELIABILITY ESTIMATION OF
BINOMIAL AND POISSON DISTRIBUTIONS
2.1 INTRODUCTION
A lot of work has been done in the literature for estimating various
parametric functions of several discrete distributions through classical and
Bayesian approaches. Halmos (1946) provided a necessary and sufficient
condition for the existence of an unbiased estimator. Kolmogorov (1950)
investigated for what functions of parameter of success of binomial distribution,
there exist an unbiased estimator. Blyth (1980) studied the expected absolute error
of UMVUE of the probability of success of binomial distribution. For a random
variable (rv) X following binomial distribution, Pulskamp (1990) has shown that
the UMVUE of P(X = x) is admissible under squared-error loss function when x =
0 or n. Cacoullos and Charalambides (1975) obtained MVUE for truncated
binomial and negative binomial distributions. Bayesian estimation of the
parameter of binomial distribution has been considered by Chew (1971). Barton
(1961) and Glasser (1962) obtained UMVUE of P(X = x) for Poisson distribution.
For MVUEs of generalized Poisson and decapitated generalized Poisson
distributions, onemay refer to Patel and Jani (1977). Irony (1992) developed
Bayesian estimation procedures related to Poisson distribution. Guttman (1958)
and Patil (1963) provided UMVUEs of parametric functions of negative binomial
distribution. Patil and Wani (1966) obtained UMVUEs of distribution functions of
various distributions. Roy and Mitra (1957) considered the problem of minimum
variance unbiased estimation of a univariate power series distribution. Patel (1978)
generalized their results to multivariate modified power series distribution. Patil
and Bildikar (1966) derived MVUE for logarithmic series distribution. Extensive
tables concerning UMVUEs of different parametric functions of various
distributions are available in Voinov and Nikulin (1993, 1996).
Discrete distributions have played important role in reliability theory.
Kumar and Bhattacharya (1989) considered negative binomial distribution as the
life-time model and obtained UMVUEs of the mean life and reliability function.
Another measure of reliability under stress-strength set-up is the probability PrX
≤ Y, where X is the stress variable and Y is the strength variable. Maiti (1995)
considered the estimation of PrX ≤ Y under the assumption that X and Y
followed geometric distributions and derived UMVUE and Bayes estimators.
Chaturvedi and Tomer (2002) considered classical and Bayesian estimation
procedures for the reliability function of the negative binomial distribution from
a different approach. Generalizing
the results of Maiti (1995), they dealt with the problem of estimating PrX1 +
…+ Xk ≤ Y, where the rv’s X’s and Y were assumed to follow negative binomial
distribution.
In this chapter the problems of estimating the reliability function and Pr X1
+ … + Xk ≤ Y are considered. The random variables X’s and Y are assumed to
follow binomial and Poisson distributions. Classical as well as Bayes estimators
for these distributions are derived. In order to obtain the estimators of these
parametric functions, the basic role is played by the estimators of factorial
moments of the two distributions.
In order to obtain Bayes estimators of parameters and various parametric
functions of different distributions, the researchers have adopted conventional
technique, i.e. obtaining their posterior means. In the present discussion, we
consider binomial and Poisson distributions and studying the behaviour of their
hazard-rates, we investigate the situations when these distributions can be
recommended as life-time models. We consider the problems of estimating
reliability function and P = PrX1 + …+ Xk ≤ Y from Bayesian viewpoint. It is
worth mentioning here that, in contrary to conventional approach, only estimators
of factorial moments are needed to estimate these parametric functions and no
separate dealing is needed. [see ‘REMARKS’ 1 and 2].
In Section 2.2, we study the behaviour of hazard-rates of binomial and
Poisson distributions and provide the set-up of the estimation problems. In
Sections 2.3 and 2.4, respectively, we obtain the classical and Bayes estimators of
powers of ?, reliability function and ‘P’ for binomial distribution. The classical
and Bayes estimators of powers of ?, reliability function and ‘P’ for Poisson
distribution are obtained in Sections 2.5 and 2.6, respectively.
2.2 THE HAZARD-RATES OF BINOMIAL AND POISSON
DISTRIBUTIONS AND SET-UP OF THE ESTIMATION
PROBLEMS
The rv X is said to follow binomial distribution with parameters (r, ?) if its
probability mass function (pmf) is given by
r.2,...,1,0,x1,?0;xr?)(1x?xr
?)r,(x;p =<<−−
= (2.2.1)
Throughout the remaining part of this discussion, we assume that r is known
but ? is unknown. The reliability function for a specified mission time, say,
)0(ot ≥ cycles, is given by
)ot(XP)o(tR ≥=
.xr?)(1x?r
otx xr −−∑
=
= (2.2.2)
From (2.2.1) and (2.2.2), the hazard-rate is
)o(tR?)r,;o(tp
)o(th =
xr?)(1x?
r
otx x
r
otr?)(1ot?
otr
−−∑=
−−
=
.
1
otr
otr
0x
x
?1?
otxr
−
∑−
=
−
+
= (2.2.3)
Let
,x
?1?
ot
rotx
r
)o(tu
−
+
=
so that
1)ot(x)ot(r
1)(rx1
)o(tu
1)o(tu
++−+
−=+
r.otallfor1, ≤<
Thus, u (to) is monotonically decreasing in to and we conclude from (2.2.3) that
binomial distribution can be taken as reliability model when we encounter
increasing failure rate.
Suppose that Xi, i = 1, 2, …, k be k independent rv’s, where Xi follows
the binomial distribution (2.2.1) with parameters (ri, ? ) and Y be another rv,
independent of Xi’s, following binomial distribution with parameters (s, ß). Using
the additive property of binomial distribution and denoting by ∑=
=k
1i iX *X and
∑=
=k
1i ir*r , we conclude that
YX...XPrP k1 ≤++=
).ßs,(y;p *r
0*x
s
*xy?),*r;*(xp ∑
=∑=
= (2.2.4)
The rv X follows Poisson distribution with parameter ? if its pmf is
. 2,...1,0,x;!x
x??e?)(x;p =
−= (2.2.5)
The reliability function at a specified mission time, say, )0(ot ≥ cycles is
.x!
x??e
otx)o(tR
−∑∞
== (2.2.6)
Denoting by (a) (b) = a (a – 1) … (a – b + 1), from (2.2.5) and (2.2.6), the hazard-
rate is
x!
x??e
otx
!ot
ot??e
)o(th−
∑∞
=
−
=
.
1
0x (x))ot(x
x?
−
∑∞
= += (2.2.7) Let
.(x))otx(
x?)o(tu+
=
Since
1)ot(x
1ot)o(tu
1)o(tu
+++
=+
<1,
we conclude that u (to) is monotonically decreasing in to and, from (2.2.7), Poisson
distribution can represent life-time model when we have increasing failure rate.
Let Xi, i = 1, 2, …, k are k independent rv’s and Xi follows Poisson
distribution (2.2.5) with parameter ?i and Y is another rv, independent of Xi’s,
following Poisson distribution with parameter ß. Denoting by ∑=
=k
1i iX *X and
∑=
=k
1i i? *? , from the additive property of Poisson distribution,
).ß(y;p0*x *xy
)*?;*(xpP ∑∞
=∑∞
== (2.2.8)
Our goal is to estimate R (to) and ‘P’ for binomial and Poisson distributions. In
what follows, we derive classical and Bayes estimators of powers of ?.
2.3 THE UMVUE OF THE POWERS OF ?, R (to ) AND ‘P’ FOR
BINOMIAL DISTRIBUTION
In the following theorem, we obtain the UMVUE of p? (p>0), which comes in
the expression for the pth factorial moment about origin.
Theorem 1: For p>0, the UMVUE of p? is
T.pif,Tnr
pTpnrp
U? ≤
−−
=
otherwise0,=
(2.3.1)
where .n
1i iXT ∑=
=
Proof: Given a random sample )nX...,2X,1(XX = from (2.2.1), it can be seen
that T is complete and sufficient for the family of binomial distributions [see Patel,
Kapadia and Owen (1976, p.157)]. Moreover, T follows binomial distribution with
parameter (nr, ?). Now we choose a function g (T) such that
p? E[g(T)] =
i.e. ( ) p?tnr?1t?Tnr
)n
0tg(T =−−
∑=
or
( ) 1tnr?1p-t?Tnr
)n
ptg(T =−−
∑=
. (2.3.2)
Equation (2.3.2) holds if we choose g (T) = pU? , as given by (2.3.1). Hence the
theorem.
In the following theorem, we provide UMVUE of R (to) and ‘P’, given at
(2.2.2) and (2.2.4), respectively. Given ni observations X ij, i = 1, 2, ..., k; j = 1, 2,
…, ni from Xi, i = 1, 2,..., k and m observationsY j, j = 1, 2,…, m on Y’s, it is
to see that ∑=
∑=
=k
1i jiXin
1j1T and jYm
1j2T ∑=
= are complete and sufficient for
the family of binomial distributions p(x* ; r*, ?) and p(y; s, ß), respectively.
Furthermore, 1T and 2T follow binomial distributions with parameters
)?,ir ink
1i( ∑
= and (ms, ß), respectively.
Theorem 2: The UMVUE of R (to ) and ‘P’ are given, respectively, by )o(tUR
and UP , where
−
∑=
=
Tnr
xTr1)-(nT
otx xr
)o(tUR (2.3.3) and
.
2T
s m
1T
ir ink
1i
1T
0*x
2T
*xy y2T s 1)-(m
*x1
T
ir 1)-i(nk
1iys
*x
*r
UP
∑=
∑=
∑=
−
−
∑=
= (2.3.4)
Proof: we can write the pmf (2.2.1) as
.ix?xr
0i ixri1)(
xr
?)r,(x;p +∑−
=
−−
= (2.3.5)
Using Lemma 1 of Chaturvedi and Tomer (2002) and Theorem 1, it follows from
(2.3.5) that, the UMVUE of p (x; r, ?), at a specified point ‘x’, is
ixU?
xr
0i i
xri1)( x
r?)r,(x;Up +∑
−
=
−−
=
.xr
0i Tnr
i-x-Ti-x-nr
ixri1)(
xr
∑−
=
−−
= (2.3.6)
Using a result of Feller (1960, p.62) that
integers,positivekj,n,;knan
kjn
jaj
j(-1)
−−
=
−
∑
we obtain from (2.3.6) that
T.x;Tnr
xTr 1)(n
xr
?)r,(x;Up ≤
−−
= (2.3.7) Now
from (2.2.2), we get
?)r,(x;Upr
otx)o(tUR ∑
== .
Utilizing (2.3.7), we have from the above equation
−
∑=
=
Tnr
xTr1)-(nT
otx xr
)o(tUR
and (2.3.3) follows.
From arguments similar to those used in obtaining the UMVUE of R (to ),
it can be shown that
)ßs,(y;p *r
0*x
s
*xy?),*r;*(xp UP ∑
=∑=
=
.
2T
s m
1T
ir ink
1i
1T
0*x
2T
*xy y2T s 1)-(m
*x1
T
ir 1)-i(nk
1iys
*x
*r
∑=
∑=
∑=
−
−
∑=
=
Hence the result (2.3.4) follows.
REMARK 1: The method of obtaining UMVUE of the parametric function
discussed in the Section (2.3) is very simple, as one can avoid the calculations for
conditional distributions and then going for Rao-Blackwelliazation. The UMVUEs
of powers of parameter form the basis of whole analysis.
2.4 THE BAYESIAN ESTIMATION OF THE POWERS OF ?, R (to)
AND ‘P’ FOR BINOMIAL DISTRIBUTION
We first consider the estimation of powers of ? under natural conjugate family
of prior densities and SELF.
Given a random sample )nX...,2X,1(XX = from (2.2.1), let ∑=
=n
1i iX T .
Denoting by )x\(?L , the likelihood of observing X , we note that
.tnr?)(1t?t)\(?L −−∝ (2.4.1)
Thus we consider the conjugate prior for ? to be beta with parameters (?, µ), i.e.
).integerspositiveµ ?,(1µ?)(11??)?(g −−−∝ (2.4.2)
Combining (2.4.1) and (2.4.2) via Bayes’ theorem, the posterior density of ?
comes out to be
,1µt rn ?)(11?t ?k t)\(?*g −+−−−+=
where the normalizing constant k can be obtained as
d?t)\(?*g1
0 1-k ∫=
).tµrn?,t(B −++=
Hence, the posterior density of ? is
.)tµrn?,t(B
1tµnr?)(11?t?t)\(?*g−++
−−+−−+= (2.4.3)
In the following theorem, we obtain Bayes estimator of p? (p > 0), which
comes in the expression for the pth factorial moment about origin [see Kendall and
Stuart (1958, p.122)].
Theorem 3: For p > 0, Bayes estimator of p? is given by
.)tµrn?,t(B
)tµrn,p?t(BpB?
−++−+++
=
Proof: We know that, under squared-error loss function, Bayes estimator of any
parametric function is its posterior mean.
d?1tµnr?)(11?pt?1
0)tµrn?,t(B1p
B? −−+−−++−++
= ∫
)tµrn?,t(B
)tµrn,p?t(B−++
−+++=
and the theorem follows.
In the following theorem, we provide Bayes estimators of R (to ) and ‘P’,
given at (2.2.2) and (2.2.4), respectively. Given ni observations Xij, i = 1, 2,..., k;
j = 1, 2,…, ni from Xi, i = 1, 2,..., k and m observationsYj, j = 1, 2,…, m on
Y’s, let us define ∑=
∑=
=k
1i jiXin
1j1T and .jYm
1j2T ∑=
= In order to estimate ‘P’,
we choose independent beta priors for ? and ß with parameters (ν1, µ1 ) and (ν2,
µ2), respectively.
Theorem 4: Bayes estimators of R (to) and ‘P’ are given, respectively, by
)o(tBR and BP , where
)tµrn?,t(B)xtµr)1n(x,?t(Br
otx xr
)o(tBR−++
−−++++∑=
= (2.4.4)
and.
)2t2µms,2?2tB()1tk
1i 1?irin,1?1tB(
*r
0*x
s
*xyy)2t2µs1)(m,y2?2tB()*x
1t*r1µir
k
1i1)i(n,*x1?1tB(
ys
*x
*r
BP−++−∑
=++
∑=
∑=
−−++++−
−++∑=
+++
=
(2.4.5)
Proof: Using Lemma 1 of Chaturvedi and Tomer (2002) (which holds good if we
replace UMVUEs by Bayes estimators) and Theorem 3, from (2.3.5), Bayes
estimator of p(x; r, ?), at a specified point ‘x’, is
ixB?
xr
0i i
xri1)( x
r?)r,(x;Bp +∑
−
=
−−
=
.)tµrn?,t(B
)tµrni,x?t(B
xr
0i i
xri1)( x
r
−++−++++
∑−
=
−−
= (2.4.6)
Utilizing Lemma 2 of Chaturvedi and Tomer (2002), it follows from (2.4.6) that
.)tµrn?,t(B
)xtµr1)(nx,?t(Bxr
?)r,(x;Bp−++
−−++++
= (2.4.7) Now
from the fact that
?).r,(x;Bpr
otx)o(tBR ∑
==
Utilizing (2.4.7), we obtain
)tµrn?,t(B)xtµr)1n(x,?t(Br
otx xr
)o(tBR −++−−++++∑
=
=
and the result (2.4.4) follows.
Similarly we can obtain
)ßs,(y;Bp*r
0*x
s
*xy?),*r;*(xBpBP ∑
=∑=
=
utilizing (2.4.7), we obtain
)2t2µms,2?2tB()1tk
1i 1?irin,1?1tB(
*r
0*x
s
*xyy)2t2µs1)(m,y2?2tB()*x
1t*r1µir
k
1i1)i(n,*x1?1tB(
ys
*x
*r
BP−++−∑
=++
∑=
∑=
−−++++−
−++∑=
+++
=
and the result (2.4.5) follows.
2.5 THE UMVUE OF THE POWERS OF ?, R (to ) AND ‘P’ FOR
POISSON DISTRIBUTION
In the following theorem we derive the UMVUE of p? (p>0) using the method
discussed in the Section (2.3).
Theorem 5: For p>0, the UMVUE of p? is
Tpif,p)(Tpn
TpU? ≤
−=
otherwise0,= (2.5.1)
where
.n
1i iXT ∑=
=
Proof: Given a random sample )nX...,2X,1(XX = from (2.2.5), it can be seen
that T is complete and sufficient for the family of Poisson distributions [see Patel,
Kapadia and Owen (1976, p.158)]. Moreover, T follows Poisson distribution with
parameter n?. Now we choose a function g (T) such that
p? E[g(T)] =
i.e. p?!t
t?tnn?e )
0tg(T =
−∑∞
=
or
1.!t
p-t?tn)ptg(T n?e =∑
∞
=
− (2.5.2)
Equation (2.5.2) holds if we choose g (T) = pU? , as given by (2.5.1). Hence the
theorem.
In the following theorem, we provide UMVUE of R (to) and ‘P’, given at
(2.2.5) and (2.2.8), respectively.
Theorem 6: The UMVUE of R (to) and ‘P’ are given, respectively, by
)o(tUR and UP , where
x-T1)-(nT
otx xT
Tn
1)o(tUR ∑
=
= (2.5.3) and
.y2T
1)-(m
*x1T1in
k
1I
1T
ot*x y 2
T2T
* xy *x1T
2T
m1
T
in
k
1I
1UP
−−
−∑=
∑=
∑=
∑=
=
(2.5.4)
Proof: we can write (2.2.5) as
.0i x!i!
ix?i1)(?)(x;p ∑∞
=
+−= (2.5.5)
Using Lemma 1 of Chaturvedi and Tomer (2002) and Theorem 5, it follows from
(2.5.5) that, the UMVUE of p (x; ?), at a specified point ‘x’, is
∑−
=
+−=
xT
0i x!i!
ix?i1)(?)(x;Up
∑−
= −+−=
xT
0i x)(T x!xin
T!i1)(
.xT
n1
1xnxT −
−−
= (2.5.6)
Now from (2.2.6), we get
?).(x;UpT
otx)o(tUR ∑
==
Utilizing (2.5.6)
x-T1)-(nT
otx xT
Tn
1)o(tUR ∑
=
=
and the result (2.5.3) follows.
From arguments similar to those used in obtaining the UMVUE of R (to), it
can be shown that
).ß (y;p 0*x *xy
?);*(xp UP ∑∞
=∑∞
==
.y2T
1)-(m
*x1T1in
k
1I
1T
ot*x y 2
T2T
* xy *x1T
2T
m1
T
in
k
1I
1 −−
−∑=
∑=
∑=
∑=
=
Hence the theorem.
2.6 THE BAYESIAN ESTIMATION OF THE POWERS OF ?, R (to)
AND ‘P’ FOR POISSON DISTRIBUTION
Denoting by ∑=
=n
1i iXT , the likelihood of observing a random sample
)nX...,2X,1(XX = from (2.5) is
.t?n?et)\(?L −∝ (2.6.1) We
consider the conjugate prior for ? to be gamma with parameters (a, ?), i.e.
integer).positive(a??e1a?)?(g −−∝ (2.6.2)
From (2.6.1) and (2.6.2), the posterior density of ? is
.?) ?n ( e1a t?a)(tG
at?)(nt)\(?*g +−−+
+
++=
In the following theorem, we mention Bayes estimator of p? (p > 0),
which takes place in the expression for the pth factorial moment about origin [see
Johnson and Kotz (1969, p.91)].
Theorem 7: For p > 0, Bayes estimator of p? is
.p ?)(na)(tG
p)a(tGpB? −+
+++=
Proof of the theorem is similar to those of Theorem 3.
In the following theorem, we provide Bayes estimators of R (to ) and ‘P’. In
order to estimate ‘P’, we consider independent priors for ?* and ß to be gamma
with parameters (a1, ?1 ) and (a2, ?2), respectively.
Theorem 8: Bayes estimators of R (to) and ‘P’ are given, respectively, by
)o(tBR and BP , where,
x1)?(notx x
1xat
at
1)?(n?)(n)o(tBR −++∑
∞
=
−+++
+++= (2.6.3)
and
.y2a2t1)2?(m
k
1i
*x1a1t1)1?in(
0*x *xy2a2t)2?.(m
k
1i1a1t)1?in(
y
1y2a2t
*x
1*x1a1t
BP++
++∑=
++++
∑∞
=∑∞
= ++
∑=
++
−++
−++
=
(2.6.4)
Proof: We can write (2.2.5) as
,x!i!
xi?
0ii1)(?)(x;p
+∑∞
=−=
which on using Theorem 3 gives that Bayes estimator of p(x; ?), at a specified
point ‘x’, is
i!
xiB?
0ii1)(
x!1
?)(x;Bp+
∑∞
=−=
xi?)(n1)!a(ti!
1)!xia(t
0ii1)(
x!1
++−+
−+++∑∞
=−=
i?)(n
1i
1xiat
0ii1)(
x1xat
x?)(n
1
+
−+++∑∞
=−
−++
+=
.xat1)?(n
at?)(nx
1xat++++
++
−++= (2.6.5)
Results (2.6.3) and (2.6.4) follow, respectively, from (2.2.6) and (2.2.8), on using
(2.6.5).
REMARK 2: Looking at the proofs of above Theorems, we conclude that, in the
present approach, the classical and the Bayes estimators of R (to ) and ‘P’ can be
obtained simply using the estimators of factorial moments and no separate dealing
is needed to estimate these parametric functions.
CHAPTER III
SEQUENTIAL POINT ESTIMATION PROCEDURES FOR THE
GENERALIZED LIFE DISTRIBUTIONS
3.1 INTRODUCTION
A lot of work has been done in the area of sequential point estimation for
the parameters associated with various probabilistic models useful in reliability
analysis. Robbins (1959) considered the problem of sequential point estimation of
the mean of a normal population under absolute error loss and linear cost. Starr
(1966) generalized these results considering a family of loss functions and cost
function of the general form. Starr and Woodroofe (1969) proved the bounded
nature of ‘regret’ of the sequential procedure of Starr (1966). Later on, Starr and
Woodroofe (1972) proposed sequential procedure for the point estimation of mean
of an exponential distribution, which is very useful in reliability analysis, and
proved the asymptotic bounded nature of ‘regret’. Several authors have obtained
similar results for different estimation problems. For a brief review, one may refer
to Wang (1973, 1980), Nago and Takada (1980), Chaturvedi (1986, a; 1987)
and Chaturvedi and Shukla (1990). Woodroofe (1977) introduced the concept of
‘second-order approximations’ in the area of sequential estimation and obtained
such approximations for the regret of the sequential procedure with the minimum
risk point estimation of the mean of gamma distribution. He considered UMVUE
at both the stopping and estimation stages. Chaturvedi (1986, b) obtained second-
order approximations for sequential procedure to estimate mean vector of a
multinormal population. Isogai and Uno (1995), through the bias-correction of
UMVUE, proposed another sequential estimator and showed its dominance over
the UMVUE in terms of having the smaller risk. Similar results for normal and
exponential distributions have been obtained by Isogai and Uno (1993, 1994) and
Mukhopadhyay (1994).
In the present chapter , we develop sequential estimation procedure for the
generalized distributions considered by Chaturvedi et.al. (2002; 2003, a). In
Section 3.2, we discuss the generalized life distributions and consider the problem
of minimum risk point estimation. Section 3.3 describes the set-up of the problem.
The sequential estimation procedure and second-order approximations are
obtained in Section 3.4. In Section 3.5, the condition for the negative regret of the
sequential procedure is obtained and an improved
estimator is proposed which dominates the UMVUE.
3.2 THE GENERALIZED LIFE DISTRIBUTIONS
Let the random variable (rv) X follows the generalized life distributions
considered by Chaturvedi et.al. (2002; 2003, a) with probability density function
(pdf)
0d?,g(x),;ax;?
g(x)exp
G(d) d?
(x)'g(x)1dg?)d,(x;f >>
−
−= (3.2.1)
where ‘a’ is known and d and ? are parameters. Here, g (x) is real valued, strictly
increasing function of X with g (a) = 0 and (x)'g denotes the first derivative of g
(x).
The model (3.2.1) is called the generalized life distributions since it
includes various life distributions useful in reliability analysis as discussed in the
Chapter I.
The reliability function R(t) for specified mission time t (t > 0) can be
obtained as
t)(XP(t)R >=
= dxt ?
g(x)exp)( d?
(x)'g(x)1dg∫∞
−δΓ
−
( ) dy?g(t)
yexp)(
1dy∫∞
−δΓ
−= (3.2.2)
?g(x)y where = .
The hazard-rate h(t) is defined as
(t)R?)d,(t;f
(t)h =
from (3.2.1) and (3.2.2), the above expression yields
?(t) 'g
1
?g(t)
?(t) g
-y -exp1d
(t) gy
(t)h
−
∫∞ −
=
?(t) 'g
1
?g(t)dss-e
1d
(t) gs
?1
−
∫∞ −
+=
.?
(t) g -y swhere
=
The behaviour of hazard- rate depends upon (t) 'g . If (t) 'g is constant
(exponential distribution and Weibull distribution for p=1), h (t) is also constant. If
(t) 'g is monotonically increasing in t (Weibull distribution for p>1, Rayleigh
distribution and Chi distribution for a = 2), h (t) is monotonically increasing in t. If
(t) 'g is monotonically decreasing in t (Weibull distribution for p<1, Burr
distribution and Pareto distribution), h (t) is monotonically decreasing in t.
The classical and Bayesian inferential procedure for the model (3.2.1) are
considered by Chaturvedi and Tomer (2003, a).
3.3 THE SET-UP OF THE ESTIMATION PROBLEM
Our aim is to estimate parameter ? assuming d to be known. Given
a random sample )nX,...2X,1(XX = of size n, observed from (3.2.1), the
joint pdf of X is
[ ]
.n
1i)ig(x
?1expnG(d)nd?
n
1i)i(x'g)i(x1dg
?);nx,...,2x,1(xf
∑=
−∏=
−
=
(3.3.1)
It can be seen from (3.3.1) that ∑=
n
1i)ig(x = S (say) is complete and
sufficient for the model (3.2.1). Moreover from the additive property of gamma
distribution, S follows gamma distribution with parameters nd and ? [see Johnson
and Kotz (1970, p.170)].
The UMVUE of ? is ndS
n? = with pdf
.?
n?ndexp
G(nd)
1nd)n?(nd
?
nd?),n?(f
−
−=
Also
δ
θ==n
2)n?(Vand? )n?(E .
Let the loss incurred in estimating ? by n? under squared-error loss
function (SELF) and linear cost of sampling be
n,2?)n?(A(A)nL +−= (3.3.2)
where A is known and positive constant. The risk corresponding to the loss
function (3.3.2) is
(A)]nL[E(A)nR =
n2?)n?(EA +−=
ndn
2?A += . (3.3.3)
Our aim is to minimize the risk (3.3.3) while estimating ? by n? . The value
onn = minimizing the risk (3.3.3) is the solution of the equation
0onnn
(A)nR=
=
∂
∂.
Which yields
?21)1(Adon −= (3.3.4)
and the associated minimum risk is
on2(A)
onR =
since δ= 2on2?A .
It is obvious from (3.3.4) that o
n depends upon unknown parameter ?. In
the absence of any knowledge about parameter ?, no fixed sample size procedure
yields solution to the problem. In this situation, we adopt the following sequential
estimation procedure.
3.6 THE SEQUENTIAL ESTIMATION PROCEDURE AND SECOND
ORDER APPROXIMATIONS
Let us start with a sample of size 2m ≥ . Then, motivated by (3.3.4), the
stopping time N = N (A) is defined by
.n?21)1(Adn : m n inf N −≥≥= (3.4.1)
When stop, we estimate ? by N? . The risk associated with the estimator N? is
(A)]N[L E(A)NR =
(N). E2?)N?(AE +−= (3.4.2)
In the following theorem, we derive the second-order approximations for the
expected sample size and risk associated with the sequential procedure.
Theorem: For all md >1, as A ∞→ ,
o(1)1d?on (N)E +−−+= (3.4.3)
and
o(1),13do2n(A)NR +−+= (3.4.4)
where ? is specified.
Proof: Denoting ∑=
==n
1i iZnSandd ?
)ig(xiZ , the stopping rule (3.4.1) can be
written as
∑=
−≥≥= n
1i nd
)ig(x 21)1(Adn :mn inf N
≤∑=
≥= on
2nn
1i d?
)ig(x :mn inf
.on
2nnS:m n inf
≤≥= (3.4.5)
Comparing (3.4.5) with equation (1.1) of Woodroofe (1977), we obtain in his
notations,
1.oLand onßcßµ?
1,1a
1ß ,1d)iV(Z2t1,)iE(Z µ 2,a 1,L(n) ,
on1
c
==−=
=−
=−=======
We have from Theorem 2.4 of Woodroofe (1977), for md > ß,
o(1)2µ2t2ßa21
0Lß?1µß? (N)E +−−−−+=
o(1),1d?on +−−+=
and (3.4.3) follows.
It can be easily seen from (3.4.1) that N is of the form tC given by
Woodroofe (1977) with Xi = Zi.
Now
iZN
1i
iX
N
1i
∑∑
==
= or
NNS
N?θ
= .
Hence
2?
NNS?
A2?)N?(A
−=−
2N)N(S2N
2ond
−=
.1)-2-N2o(n12N)N(Sd
+−= (3.4.6)
Utilizing Theorem 1 of Chow et.al. (1965), we obtain that
(N). E1d2N)NE(S −=− (3.4.7)
On combining (3.4.2), (3.4.6) and (3.4.7),
−+= 2N)N(S 1)-2-N 2
o(n dE(N) 2E(A)NR . (3.4.8)
Now, we can write
( ) ( ) 2NNS 2N2-on-1 d2NNS 1-2-N2
on d −
=−
2-N 2on d+ ( ) 2NNS
22N2-on-1 −
(say). III +=
We estimate I and II separately.
Firstly,
)N 1-o
n(1 )N 1-o
n-(1 )2N 2-o
n-(1 +=
)2N1-on-(N1-N N)1-
on(1+=
utilizing (1.2) of Woodroofe (1977), we get
01-onas) 1-
oO(nN)-N(S 1-o2n )2N 2-
on-(1 →+−= .
Now,
2N)-N(S )1-oO(n2N)-N(S2-
o4n 2on2-dN II +=
( ) )1-oO(n4NNS2N4d +−−=
)1-oO(n
4
21N
N-NS4d +
= .
Also
) ,(~)ig(xandN
1i d ?
)ig(xNS θδΓ∑
== .
Now
.N
)NS(VandN)NS(Eδ
==
Hence
.1
,0NL
21N
N-N
S
→
δ (3.4.9)
The asymptotic distribution of II is that of 4Z4δ , where Z has the normal
distribution with mean 0 and variance 1−δ .
Now, utilizing (3.4.9), we have
∫∞
−=∞−
dz )2z2dexp(4Z
2p21-d
14dE(II)lim
∫∞
−=0
dz )2z2dexp(4Z
2p
23d 8
.112d−= (3.4.10)
Now I can be written as
2N)-N(S2N)1-on-(1)2N1-
on-(N1-o2n d I +=
δ−−= 1on2 3N)-N(S δ−− 1
on2
NS -2N1-on 2N)-N(S
2
N1-on12N)-N(S
−δ+
).say(3
I2
I1I +−−=
To estimate I1 , we use Theorem 8 of Chow et.al. (1965), which asserts that
)NNS(NE1on6)N(E1
on143)NNS(E1on2 −−+−−δ=−δ−
)1(o1onE1
on14)NNS(NE1on6 +−δ−ν+−−δ+−−=
)1(o14)NNS(NE1on6 +−δ+−−= .
Finally
)]NS2N1on(N2N1
on[N1on)NNS(N1
on −−−−−−=−−
)NS2N1on(N1
on)onN()12N2on()onN( −−−−−−−+−=
)NS2N1on(N1
on)1N1on()2
on2N(1on)onN( −−−−−−−−+−=
).say(13
I12
I11
I −+=
Now, on utilizing (3.4.3), we get
o(1).1d? )11(I E +−−=
The estimation of I12 is similar to that of II. Hence
.1d2 )12(I E −=
Using the fact that
,1on
N
Alimas,)NS2N1
on(E =∞→
ν→−−
we get
.? )13
I ( E =
From the Theorem 2.1 and 2.2 of Woodroofe (1977) that
δ→
1,0NL
21on
on-N
and
on
2)on-(N is uniformly integrable for all .1m >δ
Utilizing above fact and (3.4.9), we get
.01onas1-3 )3(I E limand2 )2(I E lim →−δ=ν=
Collecting the terms, we get
.)1(o21-7 (I) E lim +ν−δ−= (3.4.11)
Finally utilizing (3.4.10) and (3.4.11), we get
o(1) 2?-1d 52N)-N1)(S2-N( d E 2on +−=
− . (3.4.12)
Utilizing (3.4.3) and (3.4.12), we obtain from (3.4.8) that
o(1)215d)1don 2( (A)NR +ν−−+−−ν+=
o(1),13do2n +−+=
and the result (3.4.4) follows.
3.7 CONDITION FOR NEGATIVE REGRET AND AN
IMPROVED ESTIMATOR FOR ?
Following Starr and Woodroofe (1969), we define the ‘regret’ of the sequential
procedure (3.1) by
o(1).13d
(A)onR - (A)NR (A)gR
+−=
=
Woodroofe (1977) concluded theoretically that, ,on 2 (A)NR > for all sufficiently
large value of on .
We give below a theoretical justification of numerical finding of Starr and
Woodroofe (1972) by giving the condition under which the regret may be negative
i.e. o2n (A)NR < .
Let us consider a stropping rule ‘N’ such that N? overestimates ?. Under
such condition, from (3.4.1)
??N21
)1(Ad N? −≥−−−
and (3.4.2) yields
(N). EN
2N)N(S Eond(A)NR +
−≤
(3.5.1)
Utilizing (3.4.9), we obtain from (3.5.1) that
E(N)on(A)NR +≤
utilizing (3.4.3), we get
o(1) 1d?o2n(A)NR +−−+≤ .
Hence
o(1).1d?o2n(A)NR +−−≤−
For the distributions having 1d? −< , the regret will be negative. The
generalized life distributions (3.3.1) include exponential, Weibull, Rayeigh and
Burr distribution, which have negative regret.
In what follows, we propose an improved estimator of ?. An improved
estimator of ? having smaller risk as compared to N? for stopping rule (3.4.1) is
( ) 21Adk N? NT −+= (3.5.2)
where k is any scalar.
Now we find the value of k for which the dominance of N?overNT can
be established.
Let (A)oNR is risk associated with the improved estimator (3.5.2). Then,
(N). E2?)N(TAE(A)oNR +−=
1d2k?)N?(E21)1-(Ad2k (A)NR −+−+=
.1d2kN
NNSE
21)1-(Ad 2k?(A)NR −+
−+=
(3.5.3)
Using Wald’s lemma, it can be shown that
o(1)1dN
NNSE?21)1(Ad +−−=
−− .
From (3.5.3),
1d2k1d2k(A)NR (A)oNR −+−−= .
From the above expression it is clear that
2][0,k provided (A),NR (A)oNR ∈≤
and
2.kor 0keither when (A),NR (A)oNR ===
Using principle of minima and maxima, the optimum value of k for which
NT has minimum risk can be obtained as k = 1 and such an optimum estimator of
? is given as ( ) 21Ad N? NT −+= .
REMARK: The sequential procedure for the generalized life distributions
considered in this chapter provides a better solution, where the fixed sample size
procedure fails to provide solution if the parameter ? is unknown. The second
order approximations for the ASN and the risk associated with the proposed
sequential procedure are derived. The condition for the negative regret of the
sequential procedure is achieved and it is found that the generalized life
distribution considered in this chapter contains many distributions that have
negative regret. An improved estimator of ? is also proposed and it is found that it
has smaller risk as compared to the traditional UMVUE.
CHAPTER-IV
BAYESIAN ESTIMATION PROCEDURES FOR A FAMILY OF LIFE
TIME DISTRIBUTIONS UNDER SELF AND GELF
4.1 INTRODUCTION
Bhattacharya (1967) introduced the Bayesian ideas in the reliability
analysis. He considered the problem of estimating the parameter and reliability
function of one-parameter exponential distribution under type II censoring and
SELF. Bhattacharya and Kumar (1986) and Bhattacharya and Tyagi (1988)
obtained Bayes estimators of the reliability function with other priors. The Bayes
estimators for the reliability function of exponential and Weibull distributions
using uniform and gamma priors have been obtained by Harris and Singpurwala
(1968). Chaturvedi, Tiwari and Kumar (2007) obtained the Bayes estimator of the
reliability function for binomial and Poisson distributions. Since then a lot of work
has been done in this direction. For a brief review, one may refer to Martz and
Waller (1982) and Sinha (1986).
Another measure of reliability under stress-strength set-up is the probability
P = P (X > Y), where the random variable X and Y represent strength and stress
respectively. For the case when X and Y both were assumed to follow normal
distributions, the Bayes estimator of ‘P’ under SELF was obtained by Enis and
Geisser (1971). Chaturvedi and Tomer (2002) considered the Bayes estimation of
‘P’ under SELF for negative binomial distribution. Chaturvedi, Tiwari and Kumar
(2007) derived the Bayes estimator of ‘P’ for binomial and Poisson distributions.
The use of symmetrical loss function is inappropriate while estimating
reliability function because the overestimation is usually more serious than the
underestimation. For the situation when overestimation and underestimation are
not equally serious, Varian (1975) has suggested LINEX (linear in exponential)
loss function. Zellner (1986) used LINEX loss function in Bayesian estimation and
prediction. Basu and Ebrahimi (1991) obtained Bayes estimators of reliability
function and ‘P’ for exponential distribution under both the SELF and LINEX loss
functions. The LINEX loss function is suitable for the estimation of location
parameter but not for the estimation of scale parameter and other parametric
functions. Calabria and Pulcini (1994) suggested the general entropy loss function
(GELF) for estimating these quantities.
In the present chapter, we consider a family of lifetime distributions as
considered by Moore and Bilikam (1978). They obtained Bayesian estimation
procedures for the parameter and reliability function under SELF and type II
censoring. We consider the Bayes estimators for the powers of the parameter,
reliability function and P = P (X > Y) both under SELF and GELF with complete
sample. Bayes estimators of powers of parameter are utilized to obtain Bayes
estimator of the pdf at a specified point. This estimator is now used to obtain the
Bayes estimators for the reliability function and P = P (X > Y).
In Section 4.2, the family of life distributions is discussed. This family
includes several probabilistic distributions useful in reliability analysis as
discussed in the Chapter I. In Section 4.3, the Bayes estimators of ?, ?(t)
(reliability function at specified mission time t) and ‘P’ under SELF are discussed.
Bayes estimators of powers of ?, ?(t) and ‘P’ under GELF is considered in Section
4.4. Throughout the above discussion, we have assumed that shape parameter is
known, while scale parameter is unknown. Finally in Section 4.5, we assume that
both the parameters are unknown and the Bayes estimators for both the parameters
are obtained after calculating the marginal posteriors in each case.
4.2 THE FAMILY OF LIFETIME DISTRIBUTIONS
Let the random variable X follows the family of lifetime distributions
considered Moore and Bilikam (1978) given by
−−′
= (x)/?ßg(x)exp1ßg (x)g
?ß?)ß,f(x; ; 0.?ß,x, > (4.2.1)
We assume that shape parameter ß is known but the scale parameter ? is
unknown. Here g(x) is a real-valued strictly-increasing function of x with
0)g(0 =+ and ∞=∞)g( . The family (4.2.1) is known as family of lifetime
distributions since it includes various probabilistic distributions useful in
reliability analysis as particular cases as discussed in Chapter I.
For the family (4.2.1), the reliability function at a specified mission time ‘t’
is
t)P(X?(t) >=
.(t)/?ßgexp
−= (4.2.2)
Now, the hazard- rate is given as
?(t)?)ß,f(t;
h(t) =
utilizing (4.2.1) and (4.2.2), we have
(t).1ß
g(t)g?
ß)t(h
−′=
(4.2.3)
The behaviour of hazard-rate depends upon (t) 'g . If (t)g′ is constant
(exponential and Weibull distribution for ß=1), h(t) is also constant, h(t) is
monotonically decreasing in t if (t)g′ is monotonically decreasing in t (Burr,
Pareto and Weibull distribution for ß<1) and h(t) is monotonically increasing in t
if (t)g′ is monotonically increasing in t (Rayleigh and Weibull distribution for
ß>1).
4.3 BAYES ESTIMATORS OF POWERS OF ?, R(t) AND ‘P’ UNDER
SELF
Let n items are put on a test and we obtain the random sample
)nX...,2X,1(XX = . The likelihood of observing X is
∑
=−∏
=
−′= /?n
1i)
i(xßgexp
n
1i)
i(x1ß)g
i(xg
n
?
ß)x\L(?
)/nS(expn?)x\L(? θ−−∝ (4.3.1)
.n
1i)i(x gnSwhere ∑
=
β=
Looking at (4.3.1), we consider the natural conjugate prior (NCP) for ? to
be inverted gamma as
µ/?)exp( 1?? G(?)
?µp(?) −+= ; 0µ ?, > and ν positive integer.
Combining the likelihood and the prior via Bayes’ theorem, the posterior density
of ? is
µ)/?)n(Sexp( 1)?(n?
k)x|h(? +−++= (4.3.2)
where k is constant of proportionality and after calculation
.
1
?)G(n ?n-
µn
Sk
−
+−
+=
Now from (4.3.2),
( )
µ)/?).n(Sexp( 1)?(n? ?)G(n
?nµnS
)x|h(? +−+++
++
= (4.3.3)
In the following theorem we obtain Bayes estimators of powers (positive as
well as negative) of ?.
Theorem 1: For p>0, the Bayes estimators of p? and p?− under SELF, are pBS?
and pBS?− , respectively, where
( ) ν+<++
−+=
np;p
µnS?)G(n
p)?G(npBS? (4.3.4)
and
( ) .pµnS?)G(n
p)?G(npBS? −+
+++=−
(4.3.5)
Proof: Since under SELF, Bayes estimator is posterior mean, we have
( )
d? µ)/?)n(Sexp( 0
p1-?n??)G(n
?)(nµnSpBS? +−∫
∞ +−−+
++= .
Solving it, (4.3.4) follows. Similarly (4.3.5) can be obtained.
In the following theorem, we obtain expressions for the risks, posterior
risks and Bayes risks of Bayes estimators of pBS? and p
BS?− under SELF. In what
follows, we use
)pBS?(SR : Risk of p
BS? under SELF
)pBS?(PSR : Posterior risk of p
BS? under SELF and
)pBS?(BSR : Bayes risk of p
BS? under SELF.
Theorem 2: The risk, posterior risk and Bayes risk for Bayes estimator of p? and
-p? , under SELF are respectively given by
∑=
+−
+−+=
2p
0ii?
G(n)n)G(ii2pµ
i2p2
?)G(np)?G(n)p
BS?(SR
;p
0i2p?i?
G(n)n)G(iipµ
ip
?)G(np)?G(np2? ∑
=+
+−
+−+
− ?np +< ,
(4.3.6)
( ) ;2pµnS2
?)G(np)?G(n
?)G(n2p)?G(n
)pBS?(PSR +
+−+
−
+−+
=
?n2p +< , (4.3.7)
;2pµG(?)
2p)G(?2p)??)G(nG(n
p)?(n2G1)pBS?(BSR −
−++−+−=
2?p < , (4.3.8)
∫∞
+
−−
+++−=−
0 2pµ/?)(z
dzze1nzG(n)
12
?)G(np)?G(n2p?)p
BS?(SR
+∫
∞
+
−−
+++− 1
0 pµ/?)(z
dzze1nzG(n)
1?)G(n
p)?G(n2 (4.3.9)
( ) 2pµnS2
?)G(np)?G(n
?)G(n2p)?G(n
)pBS?(PSR −+
+++
−
+++
=−
(4.3.10)
and
2pµG(?)
2p)G(?2p)??)G(nG(n
p)?(n2G1)pBS?(BSR −+
+++++−=− .
(4.3.11)
Proof : The risk corresponding to pBS? is
2p?pBS?/?nSE)p
BS?(SR
−=
p2pBS?/?nSEp2
2pBS?/?nSE θ+
θ−
=
2pµ)n(S/?nSE2
?)G(np)?G(n
+
+−+
=
.2p?pµ)n(S/?nSE?)G(n
p)?G(np2? ++
+−+
−
(4.3.12)
Let us make transformation (X)ßgU = . It is easy to see that U follows the
exponential distribution with pdf
)u/?exp(?1?)h(u; −= ; 0u > .
Since )i(xn
1ißg nS ∑
== , from the additive property of exponential distribution, the
pdf of nS is
?)/nSexp( G(n) n?
1nnS
?);nh(S −−
= ; 0nS > . (4.3.13) Hence
for q>0,
ndS )0
/?nS( expG(n)n?
1qnnS
)qnE(S ∫
∞−
−+
=
.q? G(n)
n)G(q
+
= (4.3.14)
On using (4.3.14) in (4.3.12), the result (4.3.6) follows.
Now
2p?pBS?x?/E)p
BS?(PSR
−=
2pBS?p?x?/E p
BS?22p?x?/E +
−
=
on using (4.3.4)
( )
( )2pµnS2
?)G(np)?G(n
)p?(x?/E ?)G(n
p)?G(npµnS 2)2p?(x?/E)pBS?(PSR
+
+−++
+−+
+−=
utilizing (4.3.3)
( )
( )
( ) 2pµnS2
?)G(np)?G(n
0d? µ)/?)
n(S( exp1-?-n-p?
2?)G(n
p)?G(n?npµnS 2
d? µ)/?)n(S( exp1-?-n-2p?0 ?)G(n
?nµnS)p
BS?(PSR
+
+−++
∫∞
+−
+
−++++−
+−∫∞
+
++=
solution of above expression yields (4.3.7).
By the definition
.2pµ)n(SnSE
2
?)G(np)?G(n
?)G(n2p)?G(n
)pBS?(BSR +
+−+
−
+−+
=
(4.3.15)
Now, first of all we find the marginal density of nS , from the NCP for ? and
(4.3.13) we have
( )∫∞
+−++
−=
0d? µ)/?n(Sexp 1?n?
1G(?) G(n)
1nnS ?µ
)nf(S
0.nS;?nµ)n(S ?)B(n,
1nnS ?µ
>++
−= (4.3.16)
Now utilizing (4.3.16) we have from (4.3.15)
nSd?n - 2pµ)n(S1nnS
0.
G(?) G(n) ?)G(n ?µ
2
?)G(np)?G(n
?)G(n2p)?G(n)p
BS?(BSR
−+−∞
+
+−+−
+−+=
∫
2pµ G(?)
2p)G(?2p)??)G(nG(n
p)?(n2G1
−
−++−+
−= ;2?p <
hence result (4.3.8) follows.
Now we will find these risks for the negative powers of BS? .
2p-?p-BS?/?nSE)p-
BS?(SR
−=
2p?)pBS?(/?nSEp2?2)p
BS?(/?nSE −+−−−−= .
Utilizing (4.3.5), we have
( )
( ) 2p?pµnS/?nSE
.?)G(n
p)?G(np2?2
?)G(np)?G(np2µnS/?nSE)p
BS?(SR
−+−+
+++−−
+++−+=−
(4.3.17)
Using (4.3.13), for q > 0
∫∞ −−+=−+0
ndS/?nSe 1-nnSqµ)n(S
G(n) n?1qµ)n(S/?nSE
∫∞
+
−−=
0dz qµ/?)(z
ze1nzG(n)q?1 (4.3.18)
where z = /?nS .
From (4.3.17) and (4.3.18)
∫∞
+
−−
+++
=−
0dz 2pµ/?)(z
ze1nz
G(n)2p?
12
?)G(np)?G(n
)pBS?(SR
∫∞
++
−−
+++−
0
2p-? dzpµ/?)(z
ze1nzG(n)p?1
?)G(np)?G(np-? 2
∫∞
+
−−
+++−=
0 2pµ/?)(z
dz ze1nzG(n)
12
?)G(np)?G(n2p?
+∫
∞
+
−−
+++− 1
0 pµ/?)(z
dzze1nzG(n)
1?)G(n
p)?G(n2
hence result (4.3.9) follows.
Proceeding in a similar manner the results (4.3.10) and (4.3.11) can be
obtained.
Hence the Theorem 2 follows.
Theorem 3: The Bayes estimator of pdf (4.2.1)
1)?(n
µ)n(S(x)ßg
1µ)n(S
(x)1ßg (x)g ?)ß(n?)ß,(x;BSf
++−
++
+
−′+= .
Proof: The pdf (4.2.1) can be written as
1)(i? (x)ßig 0i i!
i1)( (x)1ßg (x)gß?)ß,(x; f +−∑∞
=−−′= .
Here we will utilize the Bayes estimators of power of ?. Utilizing
Lemma 1 of Chaturvedi and Tomer (2002), we have
1)(iBS? (x)ßig
0i i!
i1)( (x)1ßg (x)g ß?)ß,(x;BSf +−∑∞
=
−−′=
using result (4.3.5)
( ) 1)(iµnS
.?)G(n
1)i?G(n (x)ßig
0i i!
i1)( (x)1ßg (x)g ß?)ß,(x;BSf
+−+
++++
∑∞
=
−−′=
1)?(n
µ)n(S(x)ßg1
µ)n(S(x)1ßg (x)g ?)ß(n
++−
++
+
−′+=
and the theorem follows.
Theorem 4: The Bayes estimator of reliability function ?(t) defined at (4.2.2) is
given as
?)(n
µ)n(S(t)ßg1(t)BS?
+−
++=
.
Proof : We have
∫∞
=t
?)dxß,(x;BSf(t)BS?
∫∞
++−
++−′
++=
tdx
1)?(n
µ)n(S(x)ßg1(x)1ß(x)gg
µ)n(S?)(n ß
++=∫
∞
++++
+=µ)n(S
(x)ßg1u wheredu,µ)n(t)/(Sßg
1?nu)(1
1?)(n
After solving, we get
?)(n
µ)n(S(t)ßg1(t)BS?
+−
++=
and the theorem follows.
Theorem 5: The risk, posterior risk and Bayes risk of (t)BS? , as defined in
Theorem 4, are respectively,
G(n)
(t)/?ßg2e
0 ?)2(n?(t))/ßg(µz
dzze1nz?)2(nµ/?)(zG(n)
1(t))BS?(SR−
−∫∞
+
++
−−++=
(t)/ ?ß2ge0 ?)(n
(t))/ ?ßg(µz
dz ze1nz?nµ/?)(z. −+∫∞
+
++
−−++ (4.3.19)
+−
++−
+−
++=
?)2(n
µ)n(S(t)ßg1
?)(n
µ)n(S(t)ßg 21(t))BS?(PSR (4.3.20)
and
∑ν+
=
−+
+
+−
+=
n
0i
i2?n
(t)ßgµ
µi
?n
?)?(n,1
?
(t)ß2gµ
µ(t))BS?(BSR
. ( )i2?ni,n ? −++ . (4.3.21)
Proof: By definition, we have
( ) 2)t((t)BS? /?nSE(t))BS?(SR ρ−=
2))t(((t))BS?(/?nSE (t) 22(t))BS?(/?nSE ρ+ρ−=
/?nSE (t)/?ßg2e
?)2(n
µ)n(S(t)ßg1/?nSE −−
+−
++=
(t)/?ßg 2e
?)(n
µ)n(S(t)ßg1 −+
+−
++
/?nSE (t)/?ßg2e
?)2(n
(t))ßgµn(S
µ)n(S/?nSE −−
+
++
+=
(t)/?ßg 2e
?)(n
(t))ßgµn(S
µ)n(S −+
+
++
+
Now, from (4.3.13) for q > 0.
∫∞
++
−+
=
++
+
0ndS q(t))ßgµn(S
/?nSe1-n
nS qµ)n(S
G(n) n?1
q
(t))ßgµn(S
µ)n(S /?nSE
∫∞
++
−−+=0 q?)(t))/ ßg(µ(z
dz ze1nz qµ/?)(zG(n)
1 . (4.3.22)
Utilizing (4.3.22), we get
G(n)
(t)/?ßg2e
0 ?)2(n?(t))/ßg(µz
dzze1nz?)2(nµ/?)(zG(n)
1(t))BS?(SR
−−∫
∞+
++
−−++=
(t)/ ?ß2ge0 ?)(n
(t))/ ?ßg(µz
dzze1nz?nµ/?)(z. −+∫∞
+
++
−−++ and
result (4.3.19) follows.
Now
( ) 2)t((t)BS
? x?/E(t))BS?(PSR ρ−=
[ ] [ ]2)t(x?/E))t((x?/E (t)BS?22(t)BS? ρ+ρ−=
?)(n
µ)n(S(t)ßg1 2
?)2(n
µ)n(S(t)ßg1
+−
++−
+−
++=
−+
− (t)/?ß2gex?/E(t)/?ßgex?/.E
utilizing (4.3.3) , we have
.
?)(n
µ)n(S(t)ßg1 2
?)2(n
µ)n(S(t)ßg1(t))BS?(PSR
+−
++−
+−
++=
∫∞
+++
++−++
0d?
1)?(n? ?)G(n
(t))/?ßgµn(Se ?nµ)n(S
∫∞
+++
++−+++
0d?
1)?(n? ?)G(n
(t))/?ßg2µn(Se ?)(nµ)n(S
?)(n2
µ)n(S(t)ßg1 2
?)2(n
µ)n(S(t)ßg1
+−
++−
+−
++=
.
?)(n
µ)n(S(t)ß2g1
+−
+++
Result (4.3.20) follows after solving above expression.
Now
+−
++−
+−
++=
?)2(n
µ)n(S(t)ßg1
?)(n
µ)n(S(t)ß2g1
nSE(t))BS?(SBR .
(4.3.23)
Utilizing (4.3.16)
∫∞
+
++
−=
+−
++
0 ?)(n(t)ß2gµnS
ndS1nnS
?)B(n,
?µ?)(n
µ)n(S(t)ß2g
1nSE
?
(t)ß2gµ
µ
+=
and
i?nµ?n
0i i?n
?)?(n,
?µ?)2(n
µ)n(S(t)ßg
1nSE −+∑
+
=
+=
+−
++
∫∞
ν+++
−+
0 )2(n(t))ßgµn(S
ndS 1innS
.
i)2?ni,?(n
i2?n
(t)ßgµ
µ?n
0i i?n
?)?(n,1
−++
−+
+∑+
=
+= .
Using the above two results in (4.3.23), the result (4.3.21) follows. Hence the
Theorem 5 follows.
In the following theorem we obtain Bayes estimator of P=P(X>Y), where
‘X’ and ‘Y’ have the pdf’s )1?,1ß(x; f and )2?,2ß(y; f , respectively. Let n items
on ‘X’ and m items on ‘Y’ are put on a life test. Let us denote by ∑=
=n
1i)i(x1ß
gnS
and ∑=
=m
1j)j(y2ß
gmT . Here we assume that 1ß and 2ß are known whereas 1? and
2? are unknown. We consider the conjugate priors for 1? and 2? with parameters
)1?,1(µ and )2?,2(µ , respectively.
Theorem 6: For ß2ß1ß == (say) , the Bayes estimator of ‘P’ under SELF
−≤−
++++++++
+−−
+++
+
<+++++++++
+−
+++
+
=
1,c if ),c1
c1;2?m1?n;1?n,2?m1?(n1F2.
)1?(nc)(1
)2?m1?(n
)2?(m
1|c| if c),1;2?m1?n1;2?m,2?m1?(n1F2.
2?mc)(1
)2?m1?(n
)2?(m
BSP
where )1µn(S)2µm(T1c ++−=
(4.3.24)
For 2ß1ß ≠ ,
( ) .1
0dz
12?nz
)1?(n
2/ß1ß11z
)1µn(S
2/ß1ß)2µm(T
1)2?n(SBSP ∫−+
+−
−−+
+++=
(4.3.25)
Proof: Using similar arguments as in Theorem 4, we have
dydx)2?ß,(y;BSf0y yx
)1?ß,;(xBSfBSP ∫∞
=∫∞
==
∫∞
=0
dy)2?ß,(y;BSf)1?ß,(y;BS?
du
1)2?(m
)2µm(Tu
1
.0
)1?(n
)1µn(Su1
)2µm(T
)2?(m
++−
++
∫∞
+−
++
+
+=
where u = (y)ßg .
After solving, we obtain
dz1
0
1)2?(mcz)(1
12?n1?mz 2?m
c)(1)2?m(BSP ∫++−
−−++++
−+=
where )1µn(S)2µm(T1c ++−= .
The result (4.3.24) follows on using a result of Gradshteyn and Ryzhik (1980, p.
286).
For 2ß1ß ≠
dydx )2?,2ß(y;BSf 0y y x
)1?,1ß(x;BSfBSP ∫∞
=∫∞
==
∫∞
=0
dy )2?,2ß(y;BSf )1?,1ß(y; BS?
( ) du
)1?(n
2/ß1ßu
)1µn(S
2/ß1ß)2µm(T
10
1)2?(mu1)2?m(
+−
+
++∫
∞ ++−++= .
Result (4.3.25) follows after solving above expression.
4.4 BAYES ESTIMATORS OF POWERS OF ?, ?(t) AND ‘P’
UNDER GELF
The general loss entropy function (GELF), when the parameter ? is
estimated by ? is given by
( ) .0a;1?
logaa
? ,? L ≠−
θ−
θ=θ (4.4.1)
The Bayes estimator of ? Under GELF is given by
1/a
)a(?x?/EBe?−
−= . (4.4.2)
In the following theorem we obtain Bayes estimators of powers of ?.
Theorem 7: For p>0, the Bayes estimators of p? and -p? under GELF are given
by pBe? and p
Be?− respectively, where
p)µnS(1/a
ap)?n(G?)n(Gp
Be? +++
+=
(4.4.3)
and
p-)µnS(1/a
ap)?n(G?)n(Gp-
Be? +
−++= ; ?rap +< . (4.4.4)
Proof: We have
d? µ)/?)n(Sexp( 0
1)ap?(n??)G(n
?nµ)n(S)ap(?x?/ E +−∫
∞ +++−+
++=−
ap-)µnS(?)n(Gap)?n(G +
+++= (4.4.5)
From (4.4.5), on utilizing (4.4.2), we obtain
p)µnS(1/a
ap)?n(G?)n(Gp
Be? +++
+=
In the similar manner, we can obtain the result (4.4.4). Hence the theorem follows.
Now we derive expressions for the risks, posterior risks and Bayes risks of
Bayes estimators of powers of ?.
Theorem 8: The risk, posterior risk and Bayes risk for Bayes estimator of p? and
-p? , under GELF are respectively given by
( )( ) ( )
dzze1nzap
0 ?µ
znG1
ap?nG?nG
)pBe?(eR −−
∫∞
+
+++
=
( )
( ) ( ) 1dz ze1nz0 ?
µzlog
nGp a
ap?nG?nG
log −−−∫∞
+−
+++
− (4.4.6)
( )( )
++
+++
−= ?)(n ap?ap?nG
?nG log)p
Be?(PeR (4.4.7)
( )( )
++
+++−= ?)(n ap?
ap?nG?nGlog)p
Be?(BeR (4.4.8)
( )( ) ( )
dzze1nz-ap
0 ?µ
znG
1ap?nG
?nG)p-
Be?(eR −−∫∞
+
−++
=
1dzze1nz 0 ?
µz log
G(n)p a
ap)?G(n?)n(G
log −−−∫∞
++
−++
− (4.4.9)
( )
( )
−++
+=ap?nG
?nG log-?)(n ap?)p-
Be?(PeR (4.4.10)
and
( )( )
−++
+=ap?nG
?nGlog-?)(n ap?)p-
Be?(BeR . (4.4.11)
Where (x) logdxd?(x) Γ= .
Proof: From (4.4.1), we have
−
θ−
θ= 1
p
pBe?
logaap
apBe?
/?nSE)pBe?(eR
utilizing (4.4.3), we get
−
+−
+++−
+
+++
=
1?
µnSaplog
ap)?G(n?)G(nlog
ap
?µnS
ap)?G(n?)G(n
/?nSE)pBe?(eR
on using (4.3.13)
∫∞ −−
+
+++=
0ndS/?nSe1n
nSap
?µnS
G(n)n?1
ap)?G(n?)G(n
1ndS/?nS
e1nnS
0 ?µnS
logG(n)n?ap
ap)?G(n?)G(n
log −−−
∫∞
+−
+++
− .
Result (4.4.6) follows on substituting z = /?ns .
Now,
( )
−++−
+++−
+
+++
=1aplog?µnSaplog
ap)?G(n?)G(nlog
ap
?µnS
ap)?G(n?)G(n
x?/EnSE)p
Be?(PeR
?) (logx?/EnSE p aµ)nlog(S
nSE p aap)?G(n
?)G(nlog ++−
+++
−=
(4.4.12)
also from (4.3.3),
( ) µ)/?)d?n(Sexp(0
1)?(n? )1? (log?)G(n
?nµ)n(Slog x?/E +−∫
∞ ++−−+
++=θ−
µ)u)dun(Sexp(0
1?nu u) (log?)G(n
?nµ)n(S+−∫
∞ −++
++= .
From a result of Gradshteyn and Ryzhik (1980, p. 576), we have.
[ ]µlog?(?)?µG(?)
0dxlogx µxe1?x −=∫
∞ −−
utilizing it, we get
( ) µ)n(Slog?)(n?log? x /?E +−+=− . (4.4.13)
Utilizing (4.4.13) in (4.4.12), we get
( )( )
++++
+−= ?)(n ap?ap?nG
?nGlog)pBe?(PeR
and result (4.4.7) follows.
Now, by definition
= )p
Be?(PeRnSE)p
Be?(BeR
Since this expectation is independent of nS , hence
)pBe?(PeR)p
Be?(BeR =
and (4.4.8) follows. Results (4.4.9), (4.4.10) and (4.4.11) can be obtained in the
similar way. Hence the theorem.
The following theorem provides the Bayes estimator of reliability function.
Theorem 9: The Bayes estimator of reliability function ?(t) defined at (4.2.2),
under GELF is given as
0afor 0,(t)ßagµnS;
?)/a(n
µ)n(S(t)ßag
1(t)Be? ≠>−+
+
+−=
Proof: By definition, we have from (4.4.2)
a1
) / (t)ßg a(expx?/E(t)Be?−
θ−=
Now using (4.4.3), we get
θ
θ++−∫
∞+++
++= d(t))/ßagµnS(exp
0 1?n? ?)G(n
?nµ)n(S)(t)/?ßag(ex?/E
?n(t)]ßagµn[S
?nµ)n(S+−+
++= .
Hence
0afor ,
?)/a(n
µ)n(S(t)ßag
1(t)Be? ≠
+
+−=
and the theorem follows.
In the following theorem we derive risk, posterior risk and Bayes risk of
(t)Be? under GELF.
Theorem 10: The risk, posterior risk and Bayes risk of (t)BS? , under GELF are
respectively,
ndS/?nS
e 1-nnS
?n
L µ)n(S(t)ßga
1G(n)n?
(t)/?ßage(t))Be?(eR
−+
∫∞
+−=
1?
(t)ßagndS
/?nSe 1-n
nSL µ)n(S
(t)ßag1log G(n)n?
?)(n −−∫∞
+−+− (4.4.14)
++
+−+−=
µ)n(S(t)ßg a
µ)n(S(t)ßg a1log ?)(n(t))Be? (PeR (4.4.15) and
∫∞
+++
+++∫∞
+−
+−=
L 1?nµ)n(S
ndS1-nnS
.
(t)ßag?nµ)n(S
ndS1-nnS
L µ)n(S(t)ßag
1log
?)B(n,
??)µ(n(t))Be?(eBR
(4.4.16)
where
0a , 0 0a ,µ - (t)ßga L
<>= (4.4.17)
Proof: By definition, we have
−
ρ−
ρ= 1
)t(
)t(Be? loga
a
)t(
)t(Be?
/?nSE)(t)Be?(eR
( ) ( )
−−
+−+−
+
+−
=
1?
(t)ßg a
µnS(t)ßg a1 ?)log(n(t)/?ßg ae
?n
µnS(t)ßg a1
?/ nSE .
Utilizing (4.3.13), we get
L
ndS
?/ nS-e 1-n
nS
?n
)µnS((t)ßag-1
G(n)n ?
? / (t)ßage(t))Be?(eR ∫∞
+
+=
1?
(t)ßag
Ln
dS?/ nS-
e 1-nn
S
?n
)µnS((t)ßag-1log
G(n)n ?
?)(n- −−∫∞
+
++
and result (4.4.14) follows. Where L has been defined in (4.4.17).
Now
−
ρ−
ρ= 1
)t(
)t(Be
? loga
a
)t(
)t(Be
? x /?E)(t)Be?(PeR
( ) ( )
−−
+−+−
+
+−
=
1?
(t)ßg a
µnS(t)ßg a1?)log(n?(t)/ ßg ae
?n
µnS(t)ßg a1
x /?E
after solving the above expression result (4.4.15) follows.
Finally,
++
+−+=
µ)n(S(t)ßag
µ)n(S(t)ßag1 log ?)(n-
nSE(t))Be?(eBR
utilizing (4.3.16), the result (4.4.16) follows.
In the following theorem we obtain the Bayes estimator of ‘P’ under GELF.
Theorem 11: For )1µn(S)2µm(T1c ++−= and 2ß1ß = , the Bayes estimator
of ‘P’ under GELF is given by
)/a2?(mc)(1
1/a
)1?n, a2??(m
)2?m,1??(n1/a
a2?m
a2?1?nmBeP
+−−
+−+
++
−+
−+++=
[ ] 1/ac)1;a2?1?nm1;a2?ma,2?1?n(m1F2. −+−++++−+−+++
Proof: For ß2ß1ß == (say)
−−′∫
∞
=∫∞
=−′
=
1?(x)ßgexp (y)1ßg (y)g
0y yx (x)1ßg(x)g
2?1?
2ßP
dydx 2?(y)ßg.exp
−
dydu 2?(y)ßg exp (y)1ßg (y)g
0
1?(y)ßg
ue2?ß
−−′∫
∞∫∞ −=
.2?1?
1?
+=
The joint posterior density of 1? and 2? can be written with the help of (4.3.3) as
1)2?(m2?1)1?(n
1?)2?G(m)1?G(n
2?m)2µmT(1?n
)1µnS()2?,1?(h ++++++
++
++
=
( ) ( )2/?)2µm(T exp 1/?)1µn(S .exp +−+− .
Let us consider the transformations )2?1/(?1?P += and 2?1?u += , so
that, uP1? = and P)u(12? −= . The Jacobian of transformation is u. Hence the
joint posterior density of P and u is
1)2?1?n(mu
)2?m(G )1?G(n
2?m)2µmT(1?n
)1µnS(u)P,(h
++++−
++
++
++
=
( )
−
++
+−++−−
++−
P1
)2µmT(
P
)1µnS(
u1exp1)2?(mP1
1)1?(n.P .
Hence the marginal posterior density of P is
11?n
P)(1 12?m
P )2?m,1?n( ?
2?m)1µnS()2µmT(
g(P)−+
−−+
++
+++
=
)2?1?n(m
)cP1.(+++−
−
Now
1/a)a(P xPE BeP
−
−= .
After solving it and utilizing the result of Gradshteyn and Ryzhik (1980), the
Theorem 11 follows.
4.5 BAYES ESTIMATORS OF THE PARAMETERS UNDER SELF AND
GELF WHEN BOTH THE PARAMETERS ARE UNKNOWN
Now we consider the case when the shape parameter ß is also unknown.
We assume the prior distribution of is ), g(ßp(?)?)f(ß = , where p(?) is given by
µ/?)exp(1?? G(?)
?µp(?) −+= ; 0µ?, > and ν positive integer.
and
aß0,a1 g(ß <<∝)
i.e.
( ) µ/?).exp(1?? G(?) a
?µ,f −+∝θβ (4.5.1)
Now the likelihood of observing ß and ? from (4.2.1) with a sample
)nX...,2X,1(XX = is
∑=
−∏=
−′
= /?
n
1i)i(xßg exp
n
1i )i(x1ßg ) i(xg
n
?ß
)x|ß , L(?
−−
= /?nSexp1ß?
n
?ß (4.5.2)
.n
1i)i(xßgnSand
n
1i)ig(x?where ∑
==∏
==
With the prior (4.5.1) and likelihood (4.5.2), the posterior density of (?, ß) is
.?µ)n(S exp1ß?1?n?
nßK )x|ß , h(? +−++
= (4.5.3)
After calculating the normalizing constant, we have
?µ)n(S e 1?n ? ?)G(n
1ß? nß 1
dß?n µ)n(S
1ß? nß a
0)x|ß , h(?
+−+++
−−
++
−∫= .
(4.5.4)
Integrating out ? in (4.5.4), we obtain the marginal posterior of ß as
d?0
)x|ß ,h(?)x|ß g( ∫∞
=
a.ß0;
dß ?n µ)
n(S
1ß? nßa
0
?n µ)n(S1ß? nß<<
++
−∫
++−
= (4.5.5)
Similarly, we obtain the marginal posterior of ? as
dß0
)x|ß ,h(?)x| p( ∫∞
=θ
.?0;
dß ?n µ)n(S
1ß? nßa
0 ?)G(n
dß / µ)n(S-
ea
0
1ß?nß 1?n ?
1
∞<<
++
−∫+
θ+∫
−++
= (4.5.6)
From (4.5.5), the Bayes estimator of ß, under SELF is
.
dß a
0
?nµ)n(S1ß? nß
dß a
0
?nµ)n(S1ß?1nß*BSß
∫
++−
∫
++−+
=
Similarly from (4.5.6) the Bayes estimator of ?, under SELF is
dß a
0 ?n µ)n(S
1ß? nß ?)G(n
dßa
0d?
/ µ)n(S-e
0 ?n ?
11ß? nß
*BS?
∫
++
−+
∫
θ+∫∞
+−
=
.
dß a
0
?nµ)n(S1ß? nß
dß a
0
1-?nµ)n(S1ß?nß
1-?n1
∫
++−
∫
++−
+=
The Bayes estimator for ? and ß under GELF can be easily calculated as
1/a)a(ßx |ßE*
Beß
−=
a1
dß a
0
?nµ)n(S1ß? nß
dß a
0
?nµ)n(S1ß?a-nß
∫
++−
∫
++−
=
and
1/a)a(?x |?E*
Be?
−=
.
a1
dß a
0
?nµ)n(S1ß? nß
dß a
0
1-a?nµ)n(S1ß?nß
1-a?n1
∫
++−
∫
+++−
++=
Hence the Bayes estimators of the parameters even both the parameters are
unknown can be obtained with the help of above expressions.
CHAPTER-V
TWO STAGE POINT ESTIMATION PROCEDURE FOR THE MEAN OF
A NORMAL POPULATION WITH KNOWN COEFFICIENT OF
VARIATION
5.1 INTRODUCTION
English mathematician De-Moivre first discovered the normal distribution
in 1773 as a limiting case of the binomial distribution. In 1809, Gauss used the
normal distribution for the distribution of errors in Astronomy. Laplace also
contributed to this distribution. The normal distribution is of special significance
in inferential Statistics since it describes probabilistically the link between a
statistic and a parameter. It has wide applicability in statistical analysis since most
of the distributions occurring in practice can be approximated by the normal
distribution and many of the distributions of sample statistics tend to the normal
distribution for large samples. The normal distribution has also importance in
reliability analysis. Davis (1952) has shown that the normal distribution gives
quite a good fit for the failure time data, in the context of life testing and reliability
analysis. The support to the normal distribution is (-8, 8) by taking the mean µ to
be sufficiently large positive valued and standard deviation s to be sufficiently
small relative to µ.
The pdf of normal distribution with location parameter µ (mean) and scale
parameter s (standard deviation) is given by
The reliability function and hazard-rate for this distribution are
−−=
sµtF1(t)R
and
−−
−
=
sµtF1s
sµtf
h(t)
where f (.) is pdf of standard normal variate (SNV) and ? (z) is the cumulative
distribution function (cdf) of SNV, given by
Although we can not obtain the hazard-rate in closed form, yet it can be
shown that the hazard-rate for this distribution is IFR .
There arise many situations in which the mean and variance of a population
0. s , µ x, - ; 2 µ) (x 2 s 2
1 exp 2 1 ) (2 p s
1 s ) µ, (x; f > ∞ < < ∞ − − =
du. 2
2 u exp
2 1 ) (2 p
1 z (z) F
− ∫
∞ − =
are unknown but the population coefficient of variation (CV) is known. For a brief
review, one may refer to Snedecor (1946), Hald (1952), Davies and Goldsmith
(1976) and Chaturvedi and Tomer (2003, b). An estimator for the mean of a
normal population when CV is known is proposed by Searls (1964). He showed
that it is more efficient than the sample mean in terms of having lesser mean sum
of squares due to error (MSE).
Wald (1947) developed sequential probability ratio test (SPRT) for testing
simple versus simple hypotheses for normal distribution. Stein (1945) developed a
two-stage point estimation procedure to construct fixed-width confidence interval
for the mean of a normal population. A lot of work has been dome in the literature
for two-stage point estimation procedure for estimating parameters under different
models. For details, one may refer to Chatterjee (1959, 1960), Ruben (1961),
Mukhopadhyay (1980; 1982, a & b), Mukhopadhyay and Abid (1986, a & b),
Costanza et.al. (1986) and Kumar and Chaturvedi (1993). Several authors have
further generalized the concept of two-stage point estimation procedure by
introducing three-stage procedure, sequential procedure, and accelerated
sequential procedures. For some citations, one may refer to Robbins (1959), Starr
(1966), Hall (1981), Hamdy and Son (1991), Chaturvedi and Tomer (2003, b).
Chaturvedi, Tiwari and Pandey (1993) further analyzed the problem of
constructing a confidence interval of pre-assigned width and coverage probability
considered by Constanza, Hamdy and Son (1986). They utilized several multi-
stage (purely sequential, accelerated sequential, three-stage and two-stage)
estimation procedures to deal with the same estimation problem.
Many authors have considered the testing and estimation procedure for the
mean of a population when the population CV is known. Joshi and Shah (1990)
proposed SPRT for testing simple versus simple hypotheses for the mean of an
inverse Gaussian distribution with known CV. Singh (1998) considered the
problem of minimum risk point estimation of the mean of a normal population
under SELF when CV was known. Using Searls’ estimator, he proposed a
sequential procedure and proved it to be ‘asymptotically risk-efficient’ in the sense
of Starr (1966). Chaturvedi and Tomer (2003, b) considered three-stage and
accelerated sequential procedures for the mean of a normal population with known
CV.
In the present Chapter, we develop a two-stage point estimation procedure
for the mean of a normal population when the population CV is known. Both the
minimum risk and the bounded risk estimation problems are considered. Second
order approximations are also considered for the proposed two-stage point
estimation procedure. In Section 5.2, we discuss the minimum risk estimation for
the parameters of normal distribution. In Section 5.3, we obtain second order
approximations for expected sample size (N), risk corresponding to two-stage
point estimation procedure [ (c)NR ] and the regret of the procedure (c)].g[R In
Section 5.4, the case of bounded risk point estimation is considered. In Section 5.5
we obtain second order approximations for expected sample size (N), )2E(N and
(A)NR .
5.2 MINIMUM RISK POINT ESTIMATION
Let us consider that a random variable (rv) X follows the normal
distribution having pdf
(5.2.1)
where ),(µ ∞−∞∈ and )(0,s ∞∈ and are unknown mean and unknown standard
deviation, respectively. The population CV i.e. kµs = (say) is assumed to be
known.
Given a random sample nX...,2X ,1X of size n (= 2) from (5.2.1), let us
define
.2)nXn
1i iX (1-n
12nsand
n
1i iXn1
nX −∑=
=∑=
=
For fixed n, Searls (1964) has proposed the estimator
nX1
n
2k1nµ~−
+=
(5.2.2)
for estimating µ and showed that, under SELF, it has smaller risk than the usual
estimator nX .
. 0 s , µ x, - ; 2 µ) (x 2 s 2
1exp
2 1 ) (2 p s
1 s ) µ, (x; f > ∞ < < ∞ − − =
Let the loss incurred in estimating µ by nµ~ under squared-error loss
function (SELF) be
nc2µ) -nµ~(A)nµ~,(µL += (5.2.3)
where A and c are known and positive constants.
The risk corresponding to the loss function (5.2.3) is
nc2µ) -nµ~(EA(c)nR +=
nc2µ
2
11
n
2k12µ) -nX(E2
n
2k1A +
−
−
++
−
+=
nc2kn
2sA ++
= , where .µs
k = (5.2.4)
The value *n of n minimizing the risk (5.2.4) can be obtained by solving
the following expression
0*nnn
(c)nR ==∂
∂
.
After solving above expression, we get
c22k*n
2sA =
+
which yields
2k - s21
cA*n
= . (5.2.5)
If k = 0 then
s21
cA
on
= .
Now the associated minimum risk is
*nc2k*n
2sA(c)*nR +
+=
*nc2k*nc ++=
2kc*nc2 += .
It is obvious from the above expression that the optimal fixed sample size
*n depends upon unknown parameter s. In the absence of any knowledge about
parameter s, no fixed sample size procedure meets the goals. In what follows, we
propose a two-stage point estimation procedure.
At the first stage, we start with a sample X1 , X2…, Xm of size m (= 2) in
such a manner that )21-(c o m = as 0c → and 1.*n
mSupc
lim <∞→
Now compute
.2)mXm
1iiX (
1-m12
ms −∑=
=
The second stage sample size is given as
+
−
= 12kms 2
1
cA
m,max N (5.2.6)
where [y] denotes the largest positive integer less than y.
By definitions
m2kms21
cAN2kms2
1
cA +−≤≤−
.
After stopping, we estimate µ by .Nµ~
The risk corresponding to the estimator Nµ~ is
E(N).c2µ) -Nµ~(EA(c)NR +=
Following Starr and Woodrofe (1969), we define the regret of the procedure
(5.2.6), as
(c).*nR - (c)NR (c)gR = (5.2.7)
5.3 SECOND ORDER APPROXIMATIONS TO E(N), (c)NR AND (c)gR
In the following theorem, we derive the second-order approximations for
the expected sample size, risk associated with Nµ~ i.e. [ (c)NR ] and the regret of
the two-stage procedure.
Theorem: For two-stage point estimation procedure as c 0→ ,
o(1) 21*n (N)E ++= (5.3.1)
)21(coo
n3c(c)*n
R(c)NR ++= (5.3.2)
and
).21(coo
n3c(c)gR += (5.3.3)
Proof: Let us consider
−
−
−
−= 2kms2
1
cA2kms2
1
cA1mT .
Following Hall (1981), we have
.mas1)U(0,LmT ∞→→
Now we can write
)mE(T2k)m(s E21
cAE(N) +−=
212k)m(s E2
1
cA
+−
= (5.3.4)
since mT follows U(0, 1).
First we calculate )m(s E , as
2)mXm
1iiX (
1)(m1
2s
1-m2ms2s
1)(m −∑=−
=−
2m
1i
mXiX∑= σ
−=
jZ1-m
1i∑=
=
where jZ follows Chi-square distribution with one degree of freedom.
Hence
21)(m?
1)(m
2s2ms −−
=
2121)(m?
211)(m
smsor
−−= .
Now
)21(yE211)(m
s)msE(−
=
21)(m?ywhere −= .
dy
12
1-m
y2y
e
21-m2
1-m
2
121y0211)(m
s)msE(
−
−
Γ
∞
−= ∫
du2.ue
121
21-m
u0
21-m
2
121
21-m
2
21-m211)(m
s −
−+
∞
−+
Γ−
= ∫
2yuwhere = .
Solution of above expression yields
.
21-m
G
21
21-m
G
21
21m
s)mE(s
+
−
=
Using a well known result of Neill and Rohatgi (1973) that
( )( )
.aas)1(aO1aG
baGba ∞→−+=+−
We get
∞→−
−
−+σ=
21mas
1
21mO1)mE(s
0cas21cO1 →
=
+σ
utilizing above result in (5.3.4)
+−
+
=
212k21cOs
21
cA(N)E
utilizing (5.2.5) , we get
o(1) 21*n (N)E ++= .
Hence result (5.3.1) follows.
Furthermore,
)m(TE2k2 )m(SE21
cA
2k2-4k)2m(TE)2
m(sEcA
)2E(N −
++
=
)mTmE(S21
cA
2
+ .
Using Cauchy-Schwartz inequality,
)mV(T)mV(s )mTm(s2cov ≤
=
2)m(sE-)2
m(sE121 since )1,0(UmT →
0cas)1(o →= .
This implies ms and mT are asymptotically uncorrelated. Now utilizing this result,
we obtain
212k2 )21O(c
21
cA
2k2-4k41
1212
cA
)2E(N −
+σ
+
++σ
=
+σσ
+ )21O(c
2121
cA
2
)21-O(cs21
cA2k
31 s
21
cA 2k2-4k2s
cA +
+−+
+
=
)21-o(c 31*n
2*n +++= . (5.3.5)
Now, we can write
1xf(x)where(N),Ec2k*n
2kNfE2k*nc (c)NR −=+
+
+
+=
.2k*nonand2kN'Nwhere(N),Eco
n
'NfEo
nc +=+=+
=
(5.3.6)
Expanding f(x) around ‘x=1’, by second order Taylor’s Series, we have for
,1x1U −≤−
(U)''f! 2
21)-(x (1)'f 1)-(x f(1) f(x) ++= .
Here
3x
2(x)''fand2x
1(x)'f =−= .
Hence for 1on
'N1U −≤− , we get the Taylor series expansion as,
(U)''f
2
1on
'N21(1)'f1
on
'N(1)f on
'Nf
−+
−+=
3U2
2
1on
'N21(-1)1
on
'N1 −
−+
−+= (5.3.7)
utilizing (5.3.7) in (5.3.6), we get
)N(Ec3U
2
1on
'N1on
'N1Eonc)c(NR +
−
−+
−−= (5.3.8)
now utilizing (5.3.1) and (5.3.4), we have
)N(Ec
)'N(Eon2
)2on
2'N(E2o
n
12kon-o(1) 21*n
on1
1onc)c(NR
+
−++
+++−=
+
++++++++−= 2
ono(1)
21*n22k4k)21-o(c
31*n
2*n2on
1
on211
onc
+++
+++− o(1)
21*nc2k o(1)
21*nc2
)21o(con 3
c2k 2c- )on2k (2c)2ko(nc ++++−=
)21o(con 3
c2kc*nc2 +++=
)21(coo
n3c(c)*n
R ++= . (5.3.9)
Hence (5.3.2) follows.
Now utilizing (5.2.7) and (5.3.9), we get
)21(coo
n3c(c)gR +=
Hence the theorem follows.
5.4 BOUNDED RISK POINT ESTIMATION
Now we consider the bounded risk point estimation of µ. Let the loss incurred
in estimating µ by Nµ~ be
( ) 2µ) -nµ~(Anµ~ µ,L = (5.4.1)
where A is known positive constant.
The risk corresponding to the loss function (5.4.1) is
2µ) -nµ~(EA(A)nR =
2kn
2sA
+= ; where
µs
k = . (5.4.2)
In order to achieve the condition that the risk (5.4.2) should not exceed W
i.e. W(A)nR ≤ for a pre-specified W>0, the sample size required is the smallest
positive integer **nn ≥ , where
.2k -W
2sA**n
= (5.4.3)
If k = 0, W
2sAon = i.e.
on2k**n =+ .
The associated minimum risk is
.W2k**n
2sA(A)**nR =
+=
We start with a sample X1, X2…, Xm of size m (= 2) in such a manner that
)A( o m = as ∞→A and 1.**n
mSuplimA
<∞→
At the second stage, we collect
N - m more observations, where
.12k2ms
WA m,max N
+
−
=
After stopping we estimate µ by .Nµ~
The risk corresponding to the estimator Nµ~ is
.2µ) -Nµ~(EA(A)NR =
5.5 SECOND ORDER APPROXIMATIONS TO E(N), E( 2N ) AND
(A)NR
In the following theorem, we derive the second-order approximations to
E(N), E( 2N ) and (A)NR .
Theorem: For two-stage point estimation procedure for A ∞→
o(1) 21**n (N)E ++= (5.5.1)
o(A) 31*n
2**n )2(NE +++= (5.5.2)
and
(A).o2o
n3
1
on211W(A)NR +
+−= (5.5.3)
Proof: Let us consider
−
−
−
−= 2k2
msWA2k2
msWA1mT .
Following Hall (1981), we have
.mas1)U(0,LmT ∞→→
Hence
)m(TE2k)2m(s EAE(N)
W+−=
; 212k)2
m(s EWA
+−
= since mT follows U(0, 1).
o(1) 21**n ++=
hence (5.5.1) follows.
Result (5.5.2) can be similarly derived.
Now
)N(Ec2k*n
2kNfE2k*nc (c)NR ++
++=
on2k**nand2kN'Nwhere'N
on
EW =++=
=
=
on
'NfW ; .1x)x(fwhere −= (5.5.4)
Expanding f (x) around ‘x = 1’ by second order Taylor’s series, we have for
,1x1U −≤−
( ) (U)''f!2
21-x(1) 'f1)-(x(1) f(x) f ++=
Also
3x
2(x)''fand2x
1(x)'f =−= .
Hence, for ,1on
'N1U −≤− applying Taylor Series expression, we get from
(5.5.4)
−
−+
−−= 3U
2
1on
'N1
on
'N1EW)A(NR
−
−+
−−= 3U
2
on'N
2on
1o
n'Non 11EW
−++
−−= 'Non22
on2'NE
2on
1on)'(N E
on 1
1EW
+++−= 2k
on- o(1)
21**n
on11W
+++−+
+++++++
+2ko(1)
21**non22
on
o(1) 21**n2k24k(A)o
31*n
2**n
2on
1W
o(A)2on3
1
on2 11W +
+−=
and (5.5.3) follows. Hence the theorem.
CHAPTER-VI
SHRINKAGE-TYPE BAYES ESTIMATOR OF THE PARAMETER OF A
FAMILY OF LIFETIME DISTRIBNUTIONS
6.1 INTRODUCTION
In the estimation of unknown parameter there often exists some form of
prior knowledge about the parameter which one would like to utilize in order to
get a better estimate. The Bayesian approach is a well known example. There
exists another kind of procedure for estima ting unknown parameter with the help
of prior estimate, which is known as shrinkage estimation. In this estimation
procedure prior knowledge about the parameter is assumed to be available in the
form of prior point estimate or in the form of interval which contain parameter in
it.
According to Thompson (1968, a) there is sometimes a natural origin O?
such that one would like to take the minimum variance unbiased linear estimator
(MVULE) ? for ? and move it closer to O? . Thus obtaining an estimator for ?
which is better than ? near O? , though possibly worse farther away, measured in
terms of MSE. Such a procedure of modifying an estimator i.e., shrinkage of the
MVULE towards a natural origin is called shrinkage estimation. Thompson
suggested shrinkage estimator of a parameter by giving suitable weights to the
usual estimator and the prior point estimate. The shrinkage estimator s? (say) for a
parameter ? is
o? k)-(1 ?ks? +=
where k is any scalar between 0 and 1, such that it minimizes the MSE of s? .
Similarly, the MVULE can be shrunken towards an interval. For detailed
discussion one may refer to Thompson (1968, b).
Singh and Bhatkulikar (1977) considered the shrinkage estimation in Weibull
distribution. They proposed some preliminary test shrinkage estimators of the
shape parameter of the Weibull distribution under censored sampling and proved
that these estimators are more efficient than unbiased estimator. The problem of
shrinking the MLE of mean of various populations toward a natural origin is
studied by Mehta and Srinivasan (1971). Lemmer (1981) discussed a variety of
shrinkage methods for the estimation of some unknown parameter by considering
estimators based on a priori guess value of the parameter or an interval consisting
the parameter. He proposed a simple new estimator and compared a variety of
shrinkage estimator for the parameter of the binomial distribution. Pandey (1983)
considered the shrinkage estimation of the exponential scale parameter. He
derived some shrinkage estimators for the scale parameter of an exponential
distribution and compared them with the minimum squared error (MSE) estimator.
The shrunken estimators have smaller mse than the mse estimator when the prior
estimator is good. Pandey and Singh (1984) considered the estimation of the shape
parameter of the Weibull distribution by shrinking toward an interval. Pandey and
Upadhyay (1985) considered weather it would be more realistic to postulate a
prior distribution for the two parameter of the Weibull distribution around the
prior values and use ordinary Bayes estimator instead of prior value in the
shrinkage estimator. They obtained Bayes shrinkage estimators and proved that
these are better than the unbiased estimator. Jani (1991) suggested a class of
shrinkage estimator for the scale parameter of the exponential distribution. Singh
and Singh (1997) discussed a class of shrinkage estimators for the variance of a
normal distribution. Singh and Shukla (2002) considered the problem of
estimating the square of population mean in normal distribution when a prior
estimate or guessed value of the population variance is available. They have
suggested a family of shrinkage estimators for square of population mean with its
mean squared error formula.
In the present chapter, we derived the Shrinkage-type Bayes estimator of the
parameter of a family of lifetime distributions. In Section 6.2, the set-up of the
estimation problem is described and the desired shrinkage-type Bayes estimators
are obtained. The optimality in the sense of efficiency of the shrinkage-type Bayes
estimator over the UMVUE and the minimum mean squared error estimator is
established in the Section 6.3 and Section 6.4, respectively. Finally, in Section 6.5,
the Lindley approximation of the reliability function of the family of lifetime
distribution is considered.
6.2 THE SET-UP OF THE ESTIMATION PROBLEM
Let the random variable X follows the family of lifetime distributions as
considered in the Chapter IV, i. e.
θ−−′= (x)ßg exp (x)1ßg (x)g
?ß?)ß,f(x; ; 0.?ß,x, > (6.2.1)
We assume that shape parameter ß is known but the scale parameter ? is unknown.
We obtain a random sample )nX,...2X,1(XX = from (6.2.1) and let
∑=
β=n
1i).i(x gnS Denoting by )x|L(? , the likelihood of observing X is
θ
−∏=
−′= nSexp
n
1i)i(x1ß)gi(xg
n
?ß)x|L(? (6.2.2)
.nS expn?)x|L(?
θ−−∝ (6.2.3)
We consider inverted gamma distribution as prior distribution given by
∑=+
= )n
1i i(x g ?
o? 1)-(a- exp
1??
kp(?)
2a, ?
o? 1)-(a- exp
1a? 21a
a0? a1)-(a
>+
=
(6.2.4)
combining (6.2.3) and (6.2.4) via Bayes theorem the posterior density is
−+−
+++
+−+=
?
)o? 1)(an(s exp
a)G(n 1an?
an ]o? 1)(an[S)x|h(? .
In the following theorem we derive the Bayes estimator of ?, under SELF.
Theorem 1: The Bayes estimators of ? , under SELF is BS? , where
1-an
o? 1)-(anSBS?
+
+= . (6.2.5)
Proof: Since under SELF, Bayes estimator is posterior mean, we have
( ) ( )
−+−∫
∞ −−+
+−+=
?o1)?(anS
exp 0
an? a)G(n
ano? 1)(anS
BS?
( )o? 1)-(anSa)G(n
1)aG(n+
+−+
=
solving it, (6.2.5) follows.
Now, let )i(x giY β= , it gives nnS
Y = .
Hence we can express (6.2.5) as
Y1-an
n o?
1-an1)-(a
BS?+
++
=
or
Yk o? k)(1BS? +−= (6.2.6)
where
1)a(nnk
−+= . (6.2.7)
The estimator given by (6.2.6) is shrinkage estimator with shrinkage factor
k, lying between 0 and 1, and is given by (6.2.7).
Thus YBS? 1,k →→ and o.?BS? 0,k →→
6.3 SHRINKAGE ESTIMATOR VERSUS THE UMVUE
In this section we find the condition under which the shrinkage estimator is
more efficient as compared with UMVUE in terms of having lower mean square
error. An expression for efficiency of the shrinkage estimator has also been
derived.
First we calculate the following values
.n
2?)Y( Vand 2? n
1n2)Y( E,22?)2iY ( E ?,)Yi( E =+===
It is easy to see that Y is UMVUE of ?.
In the following theorem the shrinkage estimator is compared with
UMVUE.
Theorem 2: The shrinkage estimator is more efficient than UMVUE if
1a
2n12t
−+< , where 1.
?o?
t −=
Proof: The mean square error of shrinkage estimator is
2?)BS?( E)BS? ( mse −=
[ ] 2 ?Yko?k)(1 E −+−=
2
1)a(n
? 1)a(nYno? 1)(aE
−+
−+−+−=
21)a(n
2? n2
o21)(a
−+
+θ
θ−θ−
=
21)a(n
2? n]2t21)[(a
−+
+−= , where 1.?o
?t −=
The shrinkage estima tor is more efficient than UMVUE if
)Y( mse)BS? ( mse <
Hence
n
2
2)1n(
2n2t21)-( θ<−α+
θ
+α
?21)a(n2t21)(an 2n <−+−−+
1a2
n12t
−+< .
Hence the theorem.
Also as .1
22tif)Y(mse)BS? ( mse ,n −α
<<∞→
Theorem 3: For t = 0, the efficiency of shrinkage estimator with respect to
UMVUE, say sue is given by
.2,2
n1
1sue >α−α
+=
(6.3.1)
Proof: If oθ is the true value of ? i. e. t = 0, we get
21)a(n
2?n )BS? ( mse−+
= . (6.3.2)
In such a situation the efficiency of the shrinkage estimator with respect to
the UMVUE is
)BS? mse(
)Ymse(sue =
2
n1
1
−α+=
and (6.3.1) follows.
It is obvious from (6.3.1), the shrinkage estimator is always efficient than
the UMVUE since sue is always greater than 1, as a >2.
6.4 SHRINKAGE ESTIMATOR VERSUS MINIMUM MEAN
SQUARED ERROR ESTIMATOR
In this section we find minimum mean squared error estimator (MMSE) of
the scale parameter of the family of lifetime distributions as a function of UMVUE
for which the mean square error is minimum. We show that the shrinkage
estimator is more efficient as compared with MMSE in terms of having lower
mean square error.
Firstly, we write
MSE= 2?)Y(k E − , (6.4.1)
where k takes that value which minimizes (6.4.1) and can be calculated as
1nnk +
= .
Hence the minimum mean squared error estimator (MMSE), say msY is
Y1n
nmsY
+= .
In the following theorem the shrinkage estimator is compared with MMSE.
Theorem 4: The shrinkage estimator is more efficient than minimum mean
squared error estimator if
)1n(21)(a
)32(1n
12t+−
−α++
< , where 1.?o
?t −=
Proof: The mean square error of MMSE is
2)msY( E)msY( mse θ−=
)YE(1n
2n22)YE(2
1nn θ
+−θ+
+=
1n
2
+θ= (6.4.2)
The shrinkage estimator is more efficient than minimum mean squared
error estimator if
)msYmse()BS? mse( <
Hence
1n
2
2)1n(
2n2t21)-(
+θ
<−α+
θ
+α
n 3)(2a21)-(a2 t21)-(a1)(n −+<+
)1n(21)(a
)32(1n
12t+−
−α++
< .
Also as .2)1(
322tif)msY(mse)BS? ( mse ,n −α
−α<<∞→
Theorem 5: For t = 0, the efficiency of shrinkage estimator with respect to
minimum mean squared error estimator, say sme is given by
.2,1n21
n11sme >α
+−α+−α+=
(6.4.3)
Proof: If oθ is the true value of ?, the efficiency of shrinkage estimator with
respect to MMSE can be obtained with the help of (6.3.2) and (6.4.3) as
)BS? mse(
)msYmse(sme =
+−α+
−α+=
1n21
n11 .
Hence the theorem.
It is obvious from (6.4.3), the shrinkage estimator is always efficient than
the minimum mean squared error estimator since sme is always greater than 1, as
a >2.
Hence we have shown the dominance of the shrinkage estimator over the
UMVVUE and the minimum squared error estimator.
In the following section, we find Lindley approximation for the reliability
function of the family of lifetime distributions (6.2.1).
6.5 LINDLEY APPROXIMATION
In many situations, Bayes estimators are obtained as a ratio of two integral
expressions and cannot be expressed in a closed form. However, these estimators
can be numerically approximated using complex computer programming.
Lindley (1980) suggested an asymptotic approximation to the ratio of two
integrals.
( )
( )∫
∫
Ωθθθ
Ωθθθ
=d)(Lexp)(v
d)(Lexp)(w
I (6.5.1)
)(w;functionlikelihoodtheofarithmlogtheis)(L);...,,(where m21 θθθθθ=θ
)(vand θ are arbitrary functions of θ .
The basic idea behind it is to obtain Taylor series expansion of function
involved in (6.5.1) about the maximum likelihood estimator.
The reliability function for the family (6.2.1) at a specified mission time t is
t)P(XR(t) >=
θ−= (t)ßgexp (6.5.2)
and the hazard-rate is given as
(t)1ßg(t)g?ßh(t) −′=
Now we consider Jeffery’s prior for (ß, θ) as
( ) .?ß
1?ß, p ∝ (6.5.3)
The likelihood of observing a random sample )nX,...2X,1(XX = from
(6.2.1) when ß is also unknown is
θ−∏
=
−′
=β nS
exp n
1i)i(x1ß)gi(xg
n
?ß
)x|? ,L( (6.5.4)
Combining the prior distribution (6.5.3) and the likelihood function (6.5.4)
via Bayes theorem, the posterior density is
( )?ß, p)x|? ,L(k) x?/ ß, h( β=
where k is the normalizing constant.
Under SELF, the Bayes estimator of R (t) say, )t(BSR is
)x/)t(R(E)t(BSR =
( )
( )∫∞
∫∞
βθβ
∫∞
∫∞
βθβ=
0 0d d ?ß, p)x|? ,L(
0 0d d ?ß, p)x|? ,L()t(R
.
Now, on utilizing (6.5.2) (6.5.3) and (6.5.4), we get
∫∞
β∫∞
θ
θ−+θ
∏=
−′
∫∞
β∫∞
θ
θ
+−+θ
∏=
−′
=
0d
0 dnS
exp 1n1n
1i)i(x1ßg )i(xg1-nß
0d
0 d
(t)ßgnS exp 1n
1n
1i)i(x1ßg )i(xg1-nß
)t(BSR
∫∞ ∏
=
−′
∫∞
+
∏=
−′
=
β
β
0 n]nS[
n
1i)i(x1ßg )i(xg1-nß
0 n(t)ßgnS
n
1i)i(x1ßg )i(xg1-nß
d
d
(6.5.5)
It is obvious from (6.5.5), we can not express )t(BSR in closed form.
Hence we look for Bayes estimator of reliability function using Lindley
approximation say, )t(LSR as
[ ]x/),(uE)t(LSR θβ=
( ) 22s2?2u11s1?1u22
s22
u11
s11
u 21u ++++=
++ 2
22s2u03L211s1u30L
21 (6.5.6)
where
u = R (t), ?u
1u∂∂
= , β∂
∂=
u2u ,
2?
u2
11u∂
∂= ,
2u2
22uβ∂
∂= , ( )[ ]?ß, p log=ρ , ?1 ∂ρ∂=ρ ,
β∂ρ∂=ρ2 , L = )x|L(? ,
2L2
20Lθ∂
∂= , 20L1
11 −=σ , 2L2
02Lβ∂
∂= , 02L1
22 −=σ , 3L3
30Lθ∂
∂=
and 3L3
03Lβ∂
∂= .
On deriving these expressions, we have
θ−= (t)ßgexpu , 2?
(t)ßgu 1u = ,
?(t) g log (t)ßgu
2u −= ,
−
θθ= 2(t)ßg
3(t)ßgu
11u , [ ]
−
θθ= 1(t)ßg2(t) g log (t)ßgu
22u ,
θβ=ρ log - log- , ?1
1 −=ρ , β
−=ρ1
2 ,
θ−−′∑
=+θ−β=
nS
)i(x1ß)gi(xglogn
1ilognlognL ,
2n
20Lθ
−= , n
2
11θ=σ ,
θ
β
−β
−=
2
)ix(glognS
2n
02L ’
β
θ+β=σ)ix(glognSn
2
22 , 3n4
30Lθ
= ,
θ
β
−β
−=
3
)ix(glognS
3n2
03L
all are calculated at θ=θ ˆ , where nnSˆ =θ is the mle of ?.
Substituting these values in the expression (6.5.6), we get the Lindley
approximation to the reliability function.
CHAPTER-VII
SUMMARY
The word ‘reliable’ means able to be trusted or to do what is expected. It is
used in various contexts in our daily life such as reliable friend, reliable news,
reliable service center etc. The concept of reliability is as old as man himself. The
growth and development of reliability theory has strong links with quality control
and its development. The science of reliability is new and still growing.
The classical inferential procedures have been introduced in the field of
reliability analysis for deriving maximum likelihood estimators (MLE’s) and
uniformly minimum variance unbiased estimators (UMVUE’s) of the reliability
and other parametric functions. Davis (1952) examined that the exponential
distribution appears to fit most of the data related to reliability analysis. Epstein
(1958) remarks that the exponential distribution plays an important role in life
testing experiments. In case of censoring from right for one-parameter exponential
distribution, Epstein and Sobel (1953) derived the MLE of scale parameter.
Another measure of reliability under stress-strength set-up is the probability
P=P(X>Y), which represents the reliability of performance of an item of strength
X subject to stress Y. Owen, Craswell and Hanson (1964), Church and Harris
(1970) and Dowton (1973) discussed the estimation of P when X and Y are
normally distributed.
The pioneering work in the field of sequential analysis was due to Wald
(1947), who developed sequential probability ratio test (SPRT) for testing a simple
hypothesis against a simple alternative hypothesis. He also obtained expressions
for the operating characteristic (OC) and average sample number (ASN) for the
proposed sequential test. Epstein and Sobel (1955) considered sequential life test
in exponential case to test the simple null hypothesis against a simple alternative
hypothesis. They derived approximate formulae for OC and ASN functions.
Dantzing (1940) proved the non-existence of test of student’s hypothesis
having power function independent of variance for normal population.
Consequently, one cannot construct a confidence interval of pre assigned width
and coverage probability for the mean of a normal population when variance is
unknown. To deal with this problem, Stein (1945) proposed a two-stage procedure
determining the sample size as a random variable. Woodroofe (1977) introduced
the concept of ‘second order approximations’ in the area of sequential estimation.
In this theory, one may be able to study the behavior of the remainder terms after
the optimum position achieved by the fixed sample size procedure
The Bayesian ideas in reliability analysis were introduced for the first time
by Bhattacharya (1967), who considered the Bayesian estimation of reliability
function for one-parameter exponential distribution under uniform and beta priors.
Bhattacharya and Kumar (1986) and Bhattacharya and Tyagi (1988) obtained
Bayes estimators of the reliability function with other priors. The pioneering work
on the Bayesian estimation of ‘P’ has been done by Enis and Geisser (1971), who
derived Bayes estimator of ‘P’ when X and Y follow normal distributions.
Thompson (1968, a & b) introduced the concept of shrinkage estimator.
Shrinkage estimation procedure is one of the interesting procedures in which prior
knowledge about the parameter is assumed to be available in the form of prior
point estimate or in the form of interval which contains parameter in it.
In many situations, Bayes estimators are obtained as a ratio of two integral
expressions and cannot be expressed in a closed form. However, these estimators
can be numerically approximated using complex computer programming. Lindley
(1980) suggested an asymptotic approximation to the ratio of two integrals.
In Chapter 1 of the thesis, the brief review of the literature including brief
historical development, basic definitions, concepts and different distributions
useful in the reliability analysis is given.
In Chapter 2 of the thesis, the binomial and Poisson distributions are
introduced as lifetime models. We obtain the UMVUES and Bayes estimators of
the powers of parameter, reliability function and Y)KX...2X1X(P ≤+++ ,
where X’s and Y are assumed to follow Binomial and Poisson distributions. In
order to obtain the estimators of these parametric functions, the basic role is
played by the estimators of the factorial moments of the two distributions.
In Chapter 3 of the thesis, sequential point estimation procedure for the
generalized life distributions, which covers several distributions useful in
reliability analysis, including Weibull and gamma distributions as particular cases,
is considered. The failure of the fixed sample size procedure is established and
minimum risk point estimation for the parameters associated with the generalized
life distributions under SELF is considered. The second order approximations are
made and the ‘regret’ of the sequential procedure is obtained.
In Chapter 4 of the thesis, Bayesian estimation procedures for powers of
parameter, reliability function and P(X>Y) for a family of lifetime distributions
under squared-error loss function (SELF) and general entropy loss function
(GELF) is considered. Bayes estimators of these parameters are obtained by using
the technique of Chaturvedi et. al. (2002 & 2003,a). The Bayes estimators of the
parameters for the family of lifetime distributions when both the parameters are
unknown are also obtained after calculating the marginal posteriors.
In Chapter 5 of the thesis, we develop a two-stage point estimation
procedure for the mean of a normal population when the population CV is known.
Both the minimum risk and the bounded risk estimation problems are considered.
Second order approximations are also considered for the proposed two-stage point
estimation procedure. The second order approximations for expected sample size,
risk corresponding to two-stage point estimation procedure and the regret of the
procedure have also been obtained.
In Chapter 6 of the thesis, we derive the shrinkage-type Bayes estimator of
the parameter for a family for lifetime distributions. The optimality in the sense of
efficiency of the shrinkage-type Bayes estimator over the UMVUE and the
minimum mean squared error estimator is established. The Lindley approximation
of the reliability function of the family of lifetime distribution is also considered
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