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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
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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)
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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
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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)
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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
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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
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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
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-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
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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
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-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
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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
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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
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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,
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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
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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.
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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
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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)
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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
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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
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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
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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.
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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).
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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
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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
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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.
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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
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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.