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7/29/2019 8. IMAMSS - Shrinkage - Hadeel Salim Alkutubi
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SHRINKAGE ESTIMATOR BETWEEN BAYES AND MAXIMUM
LIKELIHOOD ESTIMATORS
HADEEL SALIM ALKUTUBI
Informatics Center for Research & Rehabilitation, Kufa University, Iraq
ABSTRACT
In this paper the shrinkage estimator between MLE estimator and Bayes estimator. Also the shrinkage
estimator between Bayes estimator and MLE estimator, and shrinkage estimator between and for estimating
the parameter of exponential distribution of life time is presented. Through simulation study the performance of this
estimator was compared to the standard Bayes and MLE estimator with respect to the mean square error (MSE) . We found
the shrinkage estimator between and is the best estimator.
KEYWORDS: Exponential Distribution, Bayes Method, Maximum Likelihood Estimation and Shrinkage Estimator
INTRODUCTION
One of the most useful and widely exploited models is the exponential distribution. Epstein,(1984) remarks that
the exponential distribution plays as important role in life experiments as that played by the normal distribution in
agricultural experiments. Maximum likelihood estimation has been the widely used method to estimate the parameter of an
exponential distribution. Lately Bayes method has begun to get the attention of researchers in the estimation procedure.
The only statistical theory that combines modeling inherent uncertainty and statistical uncertainty is Bayesian statistics.
The theorem of Bayes provides a solution on how to learn from data. Related to survival function and by using Bayes
estimator, Elli and Rao,(1986), estimated the shape and scale parameters of the Weibull distribution by assuming a
weighted squared error loss function. They minimized the corresponding expected loss with respect to a given posterior
distribution.[1]
Consider the one parameter exponential life time distribution1
( ; ) exp( )t
f t
.We find Jeffery prior by
taking g I , where 2
2 2
ln , f t n I nE
. Taking n g
, n
g k
, with k a
constant.
The joint probability density function 1, 2( ,..., , )n f t t t is given
by 11
,..., , ,n
n ii
H t t f t g
1,..., n L t t g , where
11
1
1,..., exp
n
ini
n i ni
t L t t f t
1 1
1 1
1( ,..., , ) exp exp .
n n
i t i i
n n n
t t k n k n H t t
International Journal of Applied Mathematics
& Statistical Sciences (IJAMSS)
ISSN 2319-3972
Vol. 2, Issue 1, Feb 2013, 77-82
© IASET
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78 Hadeel Salim Alkutubi
The marginal probability density function of given the data1 2( , ,..., )nt t t is
1 1,..., ( ,..., , )n n p t t H t t d
1
1
0
1
1 !exp
n
i
i
n nn
i
i
t k n nk nd
t
, where1
0
1 ( 1) !
( )
it
n n
i
ne d
t
The conditional probability density function of given the data1 2
( , ,..., )n
t t t is given by
11
1
( ,..., , ),...,
,...,
nn
n
H t t t t
p t t
1 1
1
1
1
1
exp exp
1 !1 !
n n
i i
i in
n n
i
i
n
nn
i
i
t t k n
t
nk nn
t
By using squared error loss function 2
c , we can obtain the Risk function, such that
1
0
, ,..., n R t t d
21 1
0
2 exp1 !
n n
i ini i
t t
c c d n
,
where
2( )
( ) .( 1) ( 2)
ic t
n n
Let
,0
R
, then the Bayes estimator is
1
1 1 1
0
exp1 ! 1
nn n n
i i ini i i
B
t t t
d n n
[1].
Where and ,then
In this paper, we proposed shrinkage estimator between MLE estimator and Bayes estimator and calculate
the shrinkage estimator between Bayes estimator and MLE estimator, and then shrinkage estimator between and
Shrinkage Estimator between MLE Estimator and Bayes Estimator
let
Then
7/29/2019 8. IMAMSS - Shrinkage - Hadeel Salim Alkutubi
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Shrinkage Estimator between Bayes and Maximum Likelihood Estimators 79
Let , then this implies that the value of which minimizes is
We have
Then
Then the shrinkage estimator between MLE estimator and Bayes estimator is
Shrinkage Estimator between Bayes Estimator and MLE Estimator
Let
Now we calculate , and , then this implies that the value of which
minimizes is
Also , we have
Then
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80 Hadeel Salim Alkutubi
Then the shrinkage estimator between Bayes estimator and MLE estimator is
Shrinkage Estimator between and
To get , we can find , , and such that
Then
Also , we find , that is
Then
And
Now ,we calculate such that
,
where
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Shrinkage Estimator between Bayes and Maximum Likelihood Estimators 81
+n
Simulation Study
In this simulation study, we have chosen n=30,60,90 to represent small, moderate and large sample size, several
values of parameter =0.5, 1.1, 1.5 .The number of replication used was R=1000.
The simulation program was written by using Matlab program. After the parameter was estimated , mean square
error (MSE) is calculated to compare the methods of estimation, where MSE
1000 2
1i
R
The results of the simulation study are summarized and tabulated in Table 1 for the MSE of the five estimators
for all sample sizes and values respectively.
The order of the estimator is from the best (smallest MSE) to the worst (largest MSE). It is obvious from these
tables, shrinkage estimator between and , is the best estimator in most of the cases.
Table 1: The Ordering of the Estimators with Respect to MSE
n
30
0.5 0.0114 0.0102 0.0104 0.0105 0.0102
1.1 0.0113 0.0101 0.0114 0.0119 0.0100
1.5 0.0111 0.0110 0.0115 0.0116 0.0109
60
0.5 0.0121 0.0110 0.0112 0.0115 0.0108
1.1 0.0119 0.0109 0.0111 0.0112 0.01071.5 0.0119 0.0110 0.0110 0.0109 0.0107
90
0.5 0.0117 0.0118 0.0108 0.0108 0.0107
1.1 0.0116 0.0162 0.0108 0.0107 0.0105
1.5 0.0115 0.0155 0.0107 0.0107 0.0105
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82 Hadeel Salim Alkutubi
CONCLUSIONS
The new estimator that is shrinkage estimator between and is the best estimator when compared to
standard Bayes, MLE estimator, and other estimators. We can easily conclude that MSE shrinkage estimator between
and decrease with an increased of sample size.
REFERENCES
1. Alkutubi Hadeel S. And Akma N.2009. Bayes Estimator for Exponential Distribution with Extension of Jeffery
Prior Information. Malaysian Journal of Mathematical Sciences 3(2): 295- 310.
2. Alkutubi Hadeel S. And Akma N.2009. Comparison three shrinkage estimators of parameter exponential
distribution. International journal of applied mathematics, 22(5):785-792.
3. Chiou, P. 1993. Empirical Bayes shrinkage estimation of reliability in the exponential distribution. Comm.
Statistic, 22(5):1483-1494.
4. Epstein, B. and Sobcl, M. 1984. Some theorems to life resting from an exponential distribution, Annals of
Mathematical Statistics, 25
5. Ellis, W. C. and Rao, T. V. 1986. Minimum expected loss estimators of the shape and scale parameters of Weibull
distribution. IEE Transaction on Reliability, 35(2), pp. 212-215.