Chapter IV

SOME DERIVED MODELS

4.1. INTRODUCTION

There are several attempts in literature to

construct bivariate distributions which has specified forms

for its marginal and conditional distributions of which the

systems with specified marginals are reviewed in Johnson and

Kotz (1972). Seshadri and Patil (1964) studied the problem

of determining the joint distribution of X1 and X2 given the

marginal distribution of Xi and conditional distribution of

Xi given X. = xi, i,j = 1,2, i Pf J. They showed that a

sufficient condition for the uniqueness of the joint density

function of X1 and X2 is that the conditional distribution

of Xi given Xi is of the exponential form. The question of

determining the joint distribution using the conditional

distributions has received considerable attention in the

recent times, on the ground that information about the

conditional densities are available in many real life

phenomenon. Some recent papers in this area are of Castillo

92

and Galambos (1987 ) Arnold (1987 ), Arnold and Strauss

(1988 ), etc. Arnold and Press (1989) determined a necessary

and sufficient condition for the existence of joint density

given the conditional densities . Gourieroux and Monfort

(1979). Most of the attempts in these papers were to obtain

joint distributions which have a specified form for their

conditionals , such as bivariate distributions whose

conditionals are normal, Weibull , Pareto etc. In the

following section, we provide a uniform framework in which a

class of bivariate distributions can be generated. This

class contains models whose conditionals are exponential,

Weibull , Pareto I, Pareto II and finite range distributions.

4.2. DERIVATION OF THE FAMILY.

The lack of memory property defined by Nair and

Nair ( 1991) given in equation (1.36) is generalised here as

follows.

P(Xi>G(ti.si)I X11, sip Xj = xj) = P(Xi_ tiIXj = xj) (4.1)

93

for all si t ti, xj in ( u,c) holds , i,j = 1,2 i Pf j where

G(.,.), U. c etc; are all as explained in the beginning of

Chapter II.

Writing the conditional survival function of Xi

given Xj = xj as,

P(Xi> xiIXj = xj) = S(xi.xj)

equation (4.1) becomes,

(4.2)

P(Xi>o(ti,si)IXj=xj)

P(X2>si Xj=xj) = P(X>_ ti^Xj= xj). (4.3)

Using (4.2) and ( 4.3) we have

S(G(ti, si) ,x . ) = S(t i,x j ).S(si,xj). (4.4)

For a fixed , but otherwise arbitrary xj (4.4 ) has the

solution , following the arguments in Muliere and Scarsini

(1987)

S(xi.xj ) = exp(-Xi(xj)g(x1 )], Xi(xj) > 0. (4.5)

Thus the problem of finding the bivariate distributions

94

characterized by (4.1) reduces to find the joint

distribution of (X1, X2), where the conditional distribution

of X. given Xj=x j forms,

P(X1 >_ x1IX2 = x2) = exp[ -X1(x2)g(x1)] (4.6)

and

P(X2 ? x21X1 = x1) = exp [-X2(xl)g(x2)1 (4.7)

The probability density function corresponding to

(4.6) and ( 4.7) are then

and

-X (x )g(x )

f(x1Ix2) = X1( x2) e 1 2 1 g'(x1) (4.8)

-X (x )g(x )

f(x2Ix1 ) = X2(xl) e 2 1 2 g'(x2) (4.9)

respectively.

Representing the marginal densities of X1 and X2 by fl(x1)

and f2( x2) we arrive at the identity,

-X (x )g(x 1 )

X1(x2) e 1 2 g'( x1) f2(x2)

-X (x )g(x2

)

= X2(xl) e 2 1 9 '(x2) fl ( x1) (4.10)

95

or equivalently for all x1,x2 in (u,c).

log X1(x2) - X1(x2)g(x1 ) + log g'(x1) + log f2(x2

= log X2(x1) - ^.2( x1)g(x2) + log g'(x2) + log fl(x1)

(4.11)

with primes indicating differentiation. Differentiating

(4.11) with respect to x2,

Of 2(x2)

0 log .1(x2) 6x1(x2) 6x2

Ox 2 - 6x2 g ( x1) + f2(x2

6g(x2) 6x2

-^2(xl ) 6x2 + g'(x2)

Now differentiating (4.12) with respect to x1, we have

811(x2) 6g(x1> 612(x1) 6g(x2)

6x2 Ox I 6x1 6x2

or

611(x2) 6X2(x1)

6x2 6x1

6g(x2 ) 6g(x1)

(4.12)

(4.13)

0 x2 6x1

96

For equation (4.11) to be true for all xl,x2 it must be true

that

Ox

3g(x j)

Oxj

i, j = 1,2 1 s9 j (4.14). = g,

where 9 is a constant independent of both X1 and X2. Since

this solution is unique , the value of Xi(xj) that satisfy

(4.10) is

Xi(xj) _ (ai + 9 g(xj)). (4.15)

Introducing this value of Xi(xj) in ( 4.10) and simplifying

[g'(x2)]_1 f2(x2) (al+e g( x2)) exp(a2g(x2))

_ 19'(x1)]-1 f1(x1) (a2+e g (x1)) exp (a1g(x1))

for all x1,x2. This however means that for some constant

c>O,

f i ( x i ) = C g'(xi) (aj+9 g(xi))1 exp(ctig (xi)). (4.16)

From ( 4.10),(4 . 14) and ( 4.16) the joint density of

(x1,x2) is

97

f(xl.x2) = C g'(xl )9'(x2) exp (-a19(x1)-a29(x2 )-e9(xl)9(x2))

(4.17)

xl,x2 belonging to (u,c ), ai>0, e)o , i=1,2.

in particular when a=0, we have the case of independence of

x1,x2. The constant C can be obtained as follows.

We have

00 00f f f(xl,x2)dx1dx2 = 1

u u

w -a g(x )

Cfe 1 1 g '(x1) (a2+eg ( xl))-ldx1= 1.u

That is,

or

Ce-1eXp(a1a2e-1) E1(a1a2e-1) = 1

[ 1(a1a2e_1)] ' .C exp (-a1a2e)

Corresponding survival function is obtained as

OD ODR(Xl ► x2 f f f( xl,x2) dxl dx2.

xl x2

ao 0 -a g(x )-a 9 ( x )-eg(x )9(x )= C f f g '(x1) g'(x2 ) e 1 1 2 2 1 2 dx2dxl

xl x2

98

-a2g ( x2)(a +9g( x ))-1 -alg

( xl)-e9(xl)g(x2)00= C f e 2 I))- e g (xl)dxl

xl

-a29(x2) OD (a +9g ( x )-1 -(a1+eg

( x2))9(xI)g'(xl) dxl= C e f 2 1 e-

xl

= E1(a1a291)1 E1 ( a1a2e -1+alg ( xl)+a29(x2)+89(x1)9(x2))•

(4.19)

4.3. PARTICULAR CASKS.

1. Taking G ( xl,x2) = xl+x2

9(x1+x2 ) = 9(x1) + 9(x2

and this reduces to the density function of the form

f(xl,x2 ) = C exp (-a1x1-a2x2 -9x1x2)

x1,x2 > 0, which is the bivariate exponential distribution

obtained in Arnold and Strauss ( 1988 ) and Abrahams and

Thomas (1984).

2. G(xl , x2 ) = xl x2

implies

99

g(xi) = a log xi i = 1,2

and gives

LVFSISY I^gAR(A,

%f,2

21f(xl,x2

) = C xl-(a1+1) x2-(a2+1)x1-91ogx

al,a2 > 0, 9 ?0, x1,x2 > 1.

On,O.ni= COCHIN - 682 022

\ oz

y^..G.

s OF sc'I

This is the bivariate distribution with Pareto I model as

conditionals.

3. G(x1 , x2) = (xi + X13

gives

f(xl,x2) = 02x1-1x2-1 exp(-alxl-a2x2-9xix2)

(l > 0, ai > 0, 9 >_ 0, i = 1,2, x1,x2 > 0.

A bivariate distribution with Weibull conditionals results.

4. 0(xl , x2) = xl+x2+axlx2,

implies

g(xl+x2+ax1x2 ) = g(xl) + g(x2)

and

9(x i ) = log ( 1+axi ) , i = 1,2.

The joint density is

100

-a -a -91oq(1+ax )

f(x1,x2) = C (1+ax1 ) 1 (1+ax2 ) 2 (1+ax1) 2

a1,a2, a > 0, e > 0.

5. G(x1 , x2) = xl+x2 [i+ xlx2 lJP

implies

4Cp2 xl a1 p-x2 a1

f(x1'x2) (p 2 -x1 2 )(P 2 -x22

)p+x

1) G+x

2,

P-x219 log +x

p_x1 l^' 2J

C p+x,1

0 < x1 < p, 0 < x2 < p , a1,a2 > 0, e > 0, p > 0.

In all these cases C is as in equation (4.18). The solution

of the functional equations in the examples are available in

Aczel (1966).

Remarks

The unique bivariate distribution with Pareto II

conditionals obtained in example 4 of above differs from a

similar model derived in Arnold (1987). Arnold chooses the

101

scale parameters to depend on the conditioned variable and

the shape parameter is fixed. in the model described here,

the scale parameter remains unaltered , while the shape

parameter changes with the values of the conditioned

variable.

4.4. RANDOM ENVIRONMENTAL MODELS.

A working system is often affected by the changes

in its surroundings. The environment in which the system is

working need not be the same as the laboratory environment,

under which the system was designed and the prospective

reliability was determined . The working environment

comprises of a number of observable and unobservable factors

whose intensities change over time in a random manner. For

example , the system might have been built on the premise

that the components are structurally independent so that

when they RTork in a common environment , the expectation is

that they fail independently. However the common working

condition may induce certain kind of relationships among the

r

102

components that makes the assumption independent failure

times untenable. Thus the reliability of the system is

often affected sometimes adversely and sometimesfavourably,

when the system operates in places different from the

initial test site. It is important to assess the manner and

extent by which the reliability is affected due to a change

in environment and therefore extensive studies have been

earned out by various researchers on models that can explain

this fact.

Lindley and Singpurwalla ( 1986) have studied

systems sharing common environment and Currit and

Singpurwalla ( 1988 ) analysed the reliability function of

Lindley and Singpurwalla model , in the parallel and series

systems and have obtained a formula for making Bayesian

inferences for the reliability function . Nayak ( 1987),

Cinlar and Ozeckici ( 1987 ), Roy (1989), Bandyopadhyay and

Basu ( 1990 ), Gupta and Gupta ( 1990 ) Lee and Gross ( 1991),

Sankaran and Nair (1993), Singpurwalla and Youngren (1993)

etc; have considered environmental models in detail. It is

103

customary in modelling problems to assume that the failure

rate of the system working in the new environment is given

by yh( xi,x2 ) where h(xi,x2 ) = (h1(x1,x2),h2( xl,x2)) is the

vector failure rate , when the system has worked in the test

environments . In this representation , n stands for the

effect on the failure rate due to the change in environment.

Thus when the environment factor n > 1 (Y) < 1, - = 1) the

new working conditions are assumed to be barsher ( milder,

same as) than the original work site. Since the influence

of the changed environment is seldom known exactly, it is

reasonable to take n as a random variable and to assume a

suitable probability density function for it. One such

choice for the distribution of ''' is the gamma density

pf(nIm,P) = Fp e-m^ np-1, m > 0, p > 0. (4.20)

Consider a two component system, with life lengths

X1 and X2 . Originally, the system is assumed to have a

distribution function specified by (2.8 ). The vector valued

failure rate of the system is given in equation ( 2.58).

While working in an environment with environment factor 71,

104

its failure rate vector get changed to n h(x1,x2), as

77 h(x1,x2) _ (7)(a1+eg( x2))g'(xl ),,(a2+eg(x1))g'(x2) (4.21)

which gives the new survival function of (X1,X2) as

R(x1,x2 ) = exp(-na1g(xl)-77a2g(x2)-T)eg (xl)g(x2 ))• (4.22)

Accounting the uncertainty of », by averaging this over the

distribution of n, given by (3.20)

0

= (1+a1g( x1)+a2g ( x2)+bg (xI)g(x2)] p (4.23)

aiwhere ai = m , and b = e/m, i = 1,2.

The corresponding density function is given by

f(x;,x2) = p[p(a1+bg ( x2)(a2+bg ( x1))+a1a2-b] g'(x1 ) g'(x2)

[1+alg ( x1)+a29(x2)+bg ( xl)g(x2 )]-( p+2)

(4.24)

and the marginal density functions are

105

f i ( x i ) = aiP(1 +aig(xi ))-( P+1)g.(xi ), i=1,2. (4.25)

The conditions on the parameters of the model derives from

R(xl,u ) Z R(xl,x2)

or

(1+a1g(xl)) p ? (1 +aIg(xI)+a2g(x2)+bg(xl ) g(x2))-p

1+aIg ( x1) <_ (1+alg ( xl)+a2g ( x2)+bg ( x1)g(x2))

0 <_ (a2+bg ( xI))g(x2).

since g(.) is monotonic increasing and g ( u) = 0, the above

inequality holds good for all xl,x2 if and only if a2 > 0

and b > 0. Similarly we get

0 <_ (a1+bg(x2))g(x1)

which gives al > 0, b > 0. From the assumption of gamma

density one gets p > 0.

Also f(u,u) >_ 0 leaves the condition,

P(P(aIa2)+ala2-b) > 0

or

(p+1)ala2 >_ b.

106

Thus 0 <_ b _< (p+1)a1a2.

Thus the conditions on the parameters are

ai > 0, i = 1,2 , p > 0, 0 <_ b <_ (p+1)a1a2.

The family of distributions obtained under aforementioned

framework includes a large class of distributions, like

Pareto distributions of Hutchinson (1979), Lindley and

Singpurwalla (1986), Burr distributions of Takahasi (1965),

Durling et al (1970). These distributions are considered in

the forthcoming section.

4.5. PARTICULAR CASES.

1. When g(xi) = xi , i=1,2 , u=0, the form of original

distribution is Gumbel's bivariate exponential

distribution specified by (2.22 ) corresponding

environmental model takes the form,

R7) (x1,x2 ) _ [1+a1x1+a2x2+bx1x2] p (4.26)

which is bivariate Pareto of Hutchinson ( 1979). By

taking b=0 in equation ( 3.26), bivariate Pareto

107

distribution of Lindley and Singpurwalla (1986) is

obtained.

2. g(xi ) = log xi i = 1,2 original distribution is bivariate

Pareto Type I , specified by equation ( 2.23) and

accordingly, equation ( 4.23) changes to

R11 (xl, x2) s [1+a11ogx1 +a21ogx2+blogx11ogx2] p,

x1,x2 > 1. (4.27)

3. g(xi) = xi, the parent distribution becomes bivariate

Weibull given by (2.27), and the environmental model is

R (xl,x2) _ [1+alxi13 +

a2x2+bxix2 ] p' x1,x2>0 (4.28)

which is bivariate Burr distribution of Durling et al

(1970). When b = 0, bivariate Burr distribution of

Takahasi (1965 ) results.

4. When g ( xi) = log(1+axi ), the parent distribution

becomes bivariate Pareto Type II, and the environmental

model arising from ( 4.23) is

R1) (xl, x2) _ [1+a1log (1+cx1)+a2log ( 1+cx2)

+b1og ( 1+cx1 ) log(1+cx2)] p. (4.29)

108

c+x.

5. When g(xi) = log c-x 1, with 0 < xi < c , i=1,2 the form

i

of original distribution is bivariate finite range

distribution specified by (2.32 ). Corresponding

environmental model is

c+xl c+x2R^(xl , x2) = ^1 + al log c-x + a2 log c-x

1 2

c+xl log -p

(4.30)+ b log c-x og c-x I-

1 2,J

ITo be able to analyse the reliability of this

system in a changed environment, we note that, the vector

valued failure rate of the system is

h(xl,x2) (h1(x1,x2),h2( xl,x2))

with

p(a +bg( x )) g'(x )

hi(x1 'x2) - [1+aIg ( xI)+a2g ( x2)+bg(x1)g(x2)](4.31)

Thus for a quantitative assessment of the effect of the new

environment on the system the objects of comparison are the

failure rates in (4.31) and (2.58). Using the superscripts

'e' and 'o' to differentiate the failure rates of changed

i j i

109

and original environments the relative measures that

facilitate comparison are

h1e (xl,x2 )-h1(xl,x2 ) h1*(xl,x2)- 1.

h10 (xl,x2 ) h10 (x1,x2)

(4.32)

Thus when actually the system is operated in a

different set of conditions, the error that would be

committed through the measurement of the failure rate would

be positive or negative according as

e

he(xl,x2) > 1.

hi(xl,x2)

With respect to our model , this happens when

p(ai+bg(xi ))1

(1+aig ( xl)+a2g (x2)+bg-( xl)9(x2))(ai+eg (xj)) < .

For i = 1, the first condition reduces to

P(a1+bg ( x2)) > (1+a19 ( x1)+a2g ( x2)+b9(xI ) g(x2))(a1+e9(x2))

m(1+a1g ( xl)+a2g(x2)+bg(xI)g(x2 ))( aI+bg(x2))

or when

110

p/m > (1+alg(xl)+a2g(x2)+bg(x1)9(x2))•

Since E(-Q) = p/m, we conclude that whenever

E(*q) > (<) (1+alg (x1)+a29(x2)+bg (x1)g(x2)) the

changed environment would cause failures more (less)

frequently than in the test condition, when

E(^) _ [1+ a1 g(xl) + a2 g(x2) + b g ( xl) g(x2 )], the two

environments are identical.

4.6. CHARACTERIZATIONS

As discussed in Section 3.2, measures similar to

AFR, OFR and HFR can be obtained if the concept of failure

rate in them is replaced by mean residual life in the

bivariate case . Accordingly, the arithmetic mean mean

residual life ( AM MRLF ) is defined as the vector,

K(xl,x2 ) _ ( K1(xl , x2),K2 ( xl,x2))

where

xi

Ki(xl,x2) = x f ri(xl ,x2) dxi (4.33)

1 0

111

and

ri(xl,x2) _1 m

R(x x ) f R (xl,x2) dxi, i=1,21 2 xi

is the mean residual life of ith component.

Likewise, the bivariate geometric mean mean residual life

(OM MRLF ) is defined as,

L(x1,x2 ) = ( L1(x1,x2),L2(xl,x2))

where

xi

Li(xl, x2) = exp {_!_ f log ri ( xl,x2) dxi (4.34)i

and the bivariate harmonic mean mean residual life ( HM MRLF)

is defined as

M(x1,x2) = (M1(x1 , x2),M2 ( x1,x2))

with

-1l l

Mi(x1,x2) = Cx;

f r (x,x

) dxi ) (4.35)1 1 2

Using the concepts of AN MRLF , GM MRLF and HM MRLF together

with AFR, GFR and HFR we can characterize some of the models

already considered in the sequel.

112

Theorem 4.1

A random vector X = (x1,x2 ) in R2 with absolutely

continuous distribution satisfies the property

K(x1,x2 ) = r(x1,x2). (4.36)

For every xl,x2 > 0 if and only if the distribution of

(X1,X2) is bivariate exponential of Gumbel (1960).

Proof :

When ( X1,X2 ) is of Gumbel's form,

ri(xlx2 ) (Xi+exj)-l.

so that from (4.33) ri( x1,x2 ) = Ki(x1 , x2), establishing

(4.36). Conversely , if (4.36 ) holds differentiating the

identity ( 4.33) with respect to xi,

xi 6Ki (x1'x2)8x + Ki ( x1'x2) ri(x1'x2)

i

or

xi 0Ki( xl,x2)= 0

giving Ki(x1,x2) = ri ( xl,x2 ) = Pi(xy).

113

Thus ri(xl,x2) = (p 1(x2 ),

p2(xI )) and hence the result

follows from Nair and Nair (1988).

Corollary :

K( x1,x2) ))

if and only if (X1, X2) has Gumbel ' s bivariate exponential

distribution.

Adopting the same logic , but with a little

different algebra , it can be seen that the following

theorems hold.

Theorem 4.2

L(x1,x2 ) = r(x1,x2 ) for every xl,x2 > 0 if and

only if (X1,X2) has Gumbel 's distribution.

Theorem 4.3

M(xl,x2 ) = r(xi,x2 ) for every xl,x2 > 0 if and

only if ( X1,X2) has Gumbel's distribution.

Theorem 4.4

A necessary and sufficient condition for (X1,X2) to be

114

an absolutely continuous random vector in the support of R2

satisfies any one of the following conditions

(1) Ki(xl,x2) Hi(x1x2) = C

(2) Li(xl,x2) ai(xl,x2) = C

(3) Mi(xl,x2) Ai(x1 x2) = C

for i = 1, 2, every xl , x2 > 0 and some positive real c is

that (X1,X2) is distributed either as Gumbel's bivariate

exponential distribution for c = 1 or a bivariate Pareto

type in (4.26) for c>1, or as bivariate finite range with

survival function

P(X1>x1, X2>x2) = (1-p1x1 -p2x2+gx1x2)d(4.37)

P1' p2, d>0, O<x1 <p1k, 0 <x2 < (1-P1x1 ) , 1-d < gP11P21 <_ 1

(p2-qxl

for0<C<1.

Proof :

Suppose (1) holds. Then for i = 1

x1 xl -1

II f r (t , x ) dtl C1 r dtx 1 2 J x1 ,j hl (t , x0 0

115

or

xl r(t,x )dt = C xl dtf 1 2 f h (t x)0 0 1 2

Differentiating with respect to x1,

r1(xl , x2) . h1 ( x1,x2) = C

Similarly for i = 2

r2(xl , x2) . h2(x1,x2) = C

This gives the form of

ri(xl, x2) = Axi + Bi(x i,j = 1,2 i le j

which characterizes the models in the Theorem for the

specified values of C as given in Sankaran and Nair (1992).

When ( 2) holds . for i = 1,

x1 1 xlexp Xl f log r1( t,x2)dt exp x1 f log h1 ( t,x2)dt = C

0 0

which gives the same expression for r1(x1,x2 ) as in the case

of assumption ( 1). The proof for case (3) follows suit and

this establishes the Theorem.

116

Theorem 4.5

An AM MRLF of the form,

Ki(xl, x2) = axi +bi(x.)

characterises the Gumbel ' s bivariate law for a = 0 Pareto II

distribution for a > 1/2 and the finite range distribution

for 0 < a < 1/2.

Proof:

Kl (x1 , x2 ) = ax1 + b1(x2 )

xl

4-+ a f log ri (t,x2)dt = axl + b1(x2)1 0

+-# rI(x1 , x2) = 2ax1 + bl(x2).

4.7 CONCLUSION

The present study has considered three general

families of distributions , each bringing a class of

bivariate distributions under a uniform frame work. They

provide new derivations for some well known distributions as

117

well as certain new bivariate continuous distributions.

Derivation of all the models are based on extensions of

concepts that have found acceptance among a large audience.

We have presented characterization theorems that will enable

identification of the member which will suit the

observations in a practical situation.

In view of the general functional form appearing

in 'the survival function in each family , general

characterization theorems were hard to establish , as in many

cases the assumed properties lead to functional equations

that are difficult to solve , by the existing methods.

However, characterization theorems based on basic

reliability concepts have been established, where the models

are most apt to use. More characteristics of the families

are being investigated and is hopefully expected to be

presented in a future work.