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A Mvltivoriote Exponential Distribution
Albert W. Marshall lagran Olkia 
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A MULTIVARIATE EXPONENTIAL DISTRIBUTION
by Albert W. Marshall and Ingram Olkin* Stanford University Stanford, California
Mathematical Note No. 450 Mathematics Research Laboratory BOEING SCIENTIFIC RESEARCH LABORATORIES March 1966 *Tiio research of this author was partially supported by the National Science Foundation. This report is also being issued as Technical Report No. 23, National Science Foundation Grant GP3837 by the Department of Statistics, Stanford University, Stanford, California, March 15, 1966.
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Manuscript prepared by Karen Harles
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ABSTRACT
A number of multivariate exponential distributions are known, but they have not been obtained by methods that shed light on their applicability.
This paper presents some meaningful derivations of a
multivariate exponential distribution that serve to indicate conditions under which the distribution is appropriate.
Two of these derivations
are based on "shock models", and one is based on the requirement that residual life is independent of age.
It is significant that the
derivations all lead to the same distribution. For this distribution, the moment generating function is obtained, comparison is made with the case of independence, the distribution of the minimum is discussed, and various other properties are investigated. A multivariate gamma distribution is obtained by convolution, and a multivariate Weibull distribution is obtained through a change of variables.
.J;—
BLANK PAGE
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^
L.
Introduction. Exponential distributions play a central role in life testing,
reliability and other fields of application.
Though the assumption of
independence can often Le used to obtain joint distributions, sometimes such an assumption is questionable or clearly false.
Thus, an under
standing of multivariate distributions with exponential marginals is desirable. A number of such distributions have been obtained by methods that do not shed much light on their applicability.
The purpose of this paper
is to present some meaningful derivations of a multivariate exponential distribution.
These derivations serve to indicate conditions under which
the distribution is appropriate. In considering the general problem of constructing bivariate distributions with given marginals
F
and
G,
Frdchet (1951) obtained
the condition
(1.1)
max[F(.r) + Giy)  1,0] <_äix,y) <_ miv[: (x) tGiy)].
These upper and lower bounds are themselves bivariate distributions with the given marginals, and so constitute solutions to the problem.
Recently
Plackett (1965) constructed a one parameter family of bivariate distributions which includes these solutions as well as the solution /:(.;•,.•) = F{x)Jiij)',
^e
a so
l
surveyed previous work on the problem.
2
The family of solutions
(1.2)
Hixty) = Fix)Giy){l + a[l  Fix)][l  G(y)]}t
M 1 1,
due to Morgenstern (1956) has been studied by Gumbel (1960) when and
G
are exponential.
F
Gumbel also studied the bivariate distribution
H(x,y)  1  e^  e'y + e"^"6^.
which has exponential marginals.
0 <_6 <_l,
However, we know of no model or other
basis for determining how these distributions might arise in oractice. An interesting model based on the exponential distribution has been used by Freund (1961) for deriving a bivariate distribution.
However,
the distribution obtained does not have exponential marginals. The models and characterization investigated in this paper lead to the multivariate distribution with exponential marginals, which in the bivariate case is given by
(1.3)
PiX > s,Y > t} = exp[X s  Xt  A
max(s,t)],
s,t > 0.
Each approach used to derive this distribution was chosen for its intuitive appeal, and it is significant that each leads to the same distribution.
We believe that this distribution is often a natural one.
3
For convenience we say that
X
and
Y
are
BVE(A »X.X.»)
if
(1.3) holds and refer to the distribution of (1.3) as the bivariate exponential,
BVE(X ,A,X..~).
We begin by considering the bivariate case (§2) and its properties (§3) before investigating the multivariate case (§A).
Various properties
concerning the minimum of exponential random variables are also investigated (§5) and ramifications of the condition that residual life is independent of age are explored (§6).
2.
Derivation of the Bivariate Exponential Distribution. The first defining properties (§2.1,§2.2) are motivated by reliability
considerations and are based on models in which a twocomponent system survives or dies according to the occurrences of "shocks" to each or both of the components.
(Shock models in one dimension have been utilized
by several authors; see, e.g., Epstein (1958), Esary (1957), Gaver (1963).) The defining property of §2.3 is based on a bivariate extension of a central property of the exponential distribution that the distribution of P{survival to time t + s \
residual life is independent of age, i.e., survival to time t)  P{survival to time s}. 2.1
A "fatal shock" model.
Suppose that the components of a twocomponent system die after receiving a shock which is always fatal.
Independent Poisson processes
w. ....A^J
4
ZAt',^,)* Z2(t;X2), Z12(t;A12)
govern the occurrence of shocks:
events in the process
Z.(t',X.)
coincide with shocks to component
1,
events in the process
Z2(t;X2)
coincide with shocks to component
2,
Z.j(t\\.y)
and events in the process components.
Thus if
A'
and
7
coincide with shocks to both
denote the life of the first and second
components,
F{s,t)
E
P{X > s,y > t}
' P{Z1(s;A1) = 0, Z2(t;A2) = 0, Z12(max(s,t);X12) = 0} = exp[A..s  X2* ~ V2max(e,t)l.
2.2
Nonfatal shock models.
Again consider a twocomponent system and three independent Poisson processes
Z..(£;6..), Z2(t;62), Z.~(t',6,7)
governing the occurrence of
shocks, with the modification that shocks need not be fatal. Describe the state of the system by the ordered pairs (1,0), (1,1),
where a
1
in the first (second) place indicates that the
first (second) component is operating and a Suppose that events in process
Z.(t\6.)
first component which cause a transition from probability
p.,
(0,0), (0,1),
and from
Similarly, events in process
(1,1)
to
Z (t;62)
(1,1)
0
indicates that it is not. coincide with shocks to the (1,1)
to
(0,1)
with probability
with 1  p .
coincide with transitions from
«Sri
I
BSWM «»«•■—.»m«
5
(1,1)
to
1  : y,
(1,0)
or
(1,1)
respectively.
which occur with probability
Events in process
" (t^,)
;„
coincide with
shocks to both components which cause a transition from state to states
(0,0), (0,1), (1,0), (1,1)
^rin' Pm * Pin*
and
(1,1)
with respective probabilities
Furthermore, assume that each shock to a component
PIT'
represents an independent opportunity for failure. Let
X
and
components.
Y
Since
dencte the life length of the first and second Z.it;6,)t Z:?(t;6«), 2,^(^6,«)
have independent increments, we have for
(2.1)
t >_ s ^_ 0,
P{X > sj > t) 00
1
e
P kl
fe0 00
CO
1
n=0
12
By symmetry, for
Ä=0
M
1
^=0 
&2t U^)1 — (1  p2)
(i Pi)
'
12 m!
= exp{f?[61p1 + '512Poi] "
(2.2)
are independent and
m ~ pll
t[6
2P2
512(W) (612(tS)).1 e ; (p, 1 + pn1) 11 ^01' n.
+ 6
12(1 ~
;
P
ll " Poi^1,
s > t > 0,
P{A' > 3,Y > t} = exp{^[61p1 + 6^(1  p^  r 10)]  t(&2r2 + ^2^10^
Consequently, by combining (2.1) and (2.2), it follows that
6
P{A' > s,y > t} = exp[A s  X^  X12inax(s ,t)],
(2.3) where
X1 = 61p1 + 612p01,
When :
1
p. = p9 = 1, pnf. = 1, =
P?
=
^»
we
X2 = 62p2 + 612p10,
we have the specialized fatal model.
When
effectively eliminate the first two processes; but the
joint distribution obtained from the process 2.3
A12 = 612p00.
Z,»
is of the same form.
Residual life independent of age.
The univariate exponential distribution is chiracterized by
F(s + t) = F(s)FU),
(2.4)
for all
s >_ 0, t >_ 0.
Of course, this is equivalent to
P{/Y > s + t \ X > s] = P{X > t}, time
s + t
i.e., the probability of surviving to
given survival to time
probability of survival to time
s
is exactly the unconditional
t.
Because this characterization is so fundamental in the univariate case, it is important to investigate its multivariate extensions. F(r,t) = P{X > s,y > t},
(2.5)
one obvious extension of (2.4) is
Fio1 + t1,s2 + t2) = f(s1,s2)F(t1,t2)
If
M&i
I
7
for all
i?,, s, f , t2 > 0.
To solve this equation, set
a7 ~ t7 = 0
in (2.5) to obtain
Fl(s1 + t^ = F(s1 + tj^.O) = F1(e1)F1(t1).
Similarly,
F2is2 + t2) E F(0,s2 + t2) = F2(s2)F2(t2),
so that the marginal distributions are exponential. Sy  t, = 0,
(2.6)
By choosing
we obtain
F(s1,t2) = F(s1,0)F(0,t2) = F1(s1)F2(t2),
so that the joint distribution factors into the product of the marginal distributions.
Thus the functional equation (2.5) is too strong to
yield an interesting multivariate exponential distribution,
but it may be a
convenient way to justify the assumption of independence. Let us examine the functional equation (2.5) more critically. Consider again a twocomponent system and suppose both components have survived to time
t.
A physically meaningful extension of (2.4) is
obtained if the conditional probability of both components surviving an additional time
s  (s,,s9)
of surviving to time
(s,,3)
is set equal to the unconditional probability starting at the origin, i.e..
*♦ —
8
?{.V > s1 + t, Y > s2 + t \ X > tt Y > t} * P{X > sv Y > s2},
(2.7)
or F(s1 + t,s2 + t) = F(s1,s2)F(t,t),
(2.8)
for all
s. >_ 0, s~ >_ 0, t >_ 0.
This represents a weakening of (2.5)
Since (2.7) can also be written
P{X > sl + t, Y > s2+ t \ X > sl, 7 > s2} = P{X > t, Y > t},
(2.9)
we may also check the physical meaning of (2.9).
In the univariate case
if we suppose a functioning component is of age it will function to time component were new.
t
s,
the probability that
units from now is the same as if the
With the same interpretation for a twocomponent
system with functioning components of ages
s.
and
s,.,
equation (2.9)
asserts that the probability that both components are functioning t
time
units from now is the same as if both components were new. To solve the functional equation (2.8), first set in (2.8), so that
Fis + t,s + t) = J(s,s)F(t,t),
and hence
(.? ,ß) = (s,c)
«*«■
'AM*»
SÜSd
f
9
fCj,^) = e
where
6 >_ 0.
Then, with
e
= 0
,
in (2.8),
F(s1 + ttt) = F(s1,0)F(t,t)
= F(s1,0)e"et.
Consequently,
e
^FAx  y),
% tlf*
e
0x— F0(y  x),
z ±y.
Fixty) = «
(2.10)
where the marginal distributions F^t)
and
F(t,0)
and F(ö,t)
are denoted by
F2(t).
The requirement of exponential marginal distributions yields
f
Qyb^xy) e " ,
z ^y.
Fix,y) = '
(2.11)
Qx&2^y~z) x < y, where £
1
+
0 >^ ^i»*') •? 
e,
^n order that
then
A
i
= e
" 62,
F
be monotone. A
2
are all positive and the substitution
:=
e
" 61'
6
= A
If, in addition, and
+ A
A
12
=
"1
, 5™ = A
+
'2 ' ''
+ \
,
*•—^7
10
e
=
+
^i
S ^ X12
in
^2,11^ yields
later (§6) that the condition
+
^i
the BVE
*„ ^_ 6
given by (1.3). is necessary for
We show F
given
by (2.11) to be a distribution function.
3.
Properties of the Bivariate Exponential Distribution. 3.1
The distribution function.
An interesting facet of the B^E is that it has both an absolutely continuous and a singular part.
Though distributions in one dimension
with this property are usually pathological and of no practical importance, they do arise naturally in higher dimensions. In the case of the BVE, the presence of a singular part is a reflection of the fact that if X and positive probability, whereas the line Lebesgue measure zero.
If
X
and
Y
I
are BVE, then
x  y
X = Y
with
has twodimensional
are lifetimes, the event
X = Y
can occur when failure is caused by a shock simultaneously felt by both items, as indicated in §2.1 and §2.2.
Simultaneous failure also occurs
with the failure of an essential input, common to both items. X = Y
Sometimes
because one component (say, a jet engine) explodes and the other
component (an adjacent engine) is destroyed by the explosion. Another example where that
X
and
Y
X = Y with positive probability is the case
are waiting times for the registration of an event by
two adjacent geiger counters.
•*~^i JJW
imevmaamssrrr^g .■rri1'» 
Counters are sometimes placed
VTT^SBSB^B^^'
■p
!*■  ^ÄB—new
ir«"1' '■
11
in a specific orientation, say one above the other, so that a simultaneous event in each counter records particles with nearly perpendicular paths. Theorem 3.1.
If
Ffx^y)
is
and
BVEO^,A2,A12)
A = A, + A2 + A,,
then
X +X X l 2 — 12 Fix.y) = ^ Fa{x3y) + ^ F^x^j),
'neve
F
Ax>y}
=
exp[A maxCxjZ/)]
is a singular distribution and
— \ Fa(x3u) =  +x
expfAj*  X2y  X12max{x,y)]  
12 +x
exp[A max(x,y)]
is absolutely continuous. Proof. part
F
of
a
we compute of
To find the singular part F
subtraction.
With
i
a
mp m
a
and
F
s
is then obtained as the integral
determined,
a
0 <_ a <^ 1,
F
s
can be obtained by
We compute
J2F(x,:')
v
and absolutely continuous
Fixty) = aF (x,y) + (la)F (x,y),
from
2— af {x,y) = 9 F(x,y)/dxdy. a
if ix,u). 'a ^
F
A2(A1+A12)F(x,7/),
x > y.
A1(A2+A12)F(x,iv),
x < y.
*fa te*y) ="
*m TV
■*
t*"?**?,?***:^. ...•.:
.J^ 12
and hence, for
OLF
x < y.
(■£,£/)"/ / x y
ot/ (u,v)du dv r
= A v(X +X l
2
oo
) j
12'
r
dv J
2/
X Ui\2+\l2)V
du e
X
+ X2(A1+A12) j
du J
y .XXX2yX
y
dv e
(A1+A12)kA2y
y
V
A12
= e
r e ^
A
with a symmetric expression when
^ ^_ */.
Combining both cases.
Fa(x,y) =nx,y) _ ^ e'^ax^,^)
It follows from the condition j = (A
+ A )/A.
With
a
and
1 / F (0,0) =  2 a a \ F
A+X
12\ —J = 1 A /
that
known, the singular part
F
can
be obtained by subtraction:
s
3.2
'
1a
Moment generating function.
Since we are considering positive random variables, the Laplace transform (moment generating function) exists and is natural to compute in place of the characteristic function.
n,aaf»j
iW* *^^fotUnM»mmt —
13
Bec ause of the fact that the BVE has a singular part, direct computation of the Laplace transform
J
/e
dFiXyu)
is somewhat
tedious; the integral must be computed for the absolutely continuous and singular parts separately. considerable simplification; if
Integration by parts affords a G(0,y) E 0 E G(X,0)
and G
is of
bounded variation on finite intervals, it follows from results of W. H. Young (1917) that 00
(3.3)
j j 0 0
00
00
Gix,y)dF(x,y) = j j 0 0
F
F(x,y)dG(x,y).
Gix,y)
This change is of particular use when and
00
is absolutely continuous
is easy to compute. G
To utilize (3.3) we need a kernel Thus we replace the kernel
(le~sx)ae~ty), 00
e
^
^(O,^) = 0 i G(x,0).
satisfying
of the Laplace transform by
obtaining
00
OO
00
Hs,t) = j J ae'SX)ae'ty)dFix,y) = / / F(x,y)st e'^+^dx dy 0
0
0
0000
0
.
0000
yvr
r r xiX1+X12+s)yi\2+t) r r xil^yiX^X^+t) L = j J st e " ""*' ' dx dy + J j st e ' ^ '"" dy dx 0 y Ox st(x+X12+s+t)
(x+sn)ix1+x12+s)(x2+x12+t) * where
> = X
+ A? + X..
Although the transform
$
is directly useful as
a moment generating function, its powers are not transforms of convolutions.
14
i>is,t)
However, the Laplace transform
can be obtained from the
relation
ipis,t) = (s.t) " <)(",t)  $(s,") + 1
(X+s+t) (A1+X12) (X2+X12)stX12
(x+s+t)(x1n12+s)ix2+xn+t)
'
To obtain the moments of the BVE we compute
EX =
£T =
Var X =
A1+X12
(A1+A12) Var Y =
x2+x12
(A2+X12)
2 '
2 '
from the marginal distributions and
/• .v y
9 $ 3s3t
1/
s=t=0
s=t=0
1
X
\A1+A12
1
+
A +A
2
12
Hence the covariance is given by
CovU,7) =
and the correlation is
12 A(A1+A12)(A2+X12) *
piX^Y) = X.JX.
Note that
0 < o(X,Y) <_ 1.
Higher moments are not difficult to compute directly from the equality ;.r'v i?U'»?) = fi;izl~ly3~lF{z,u)dz du
(t,j > 0)
which follows from (3.3).
►••»««•UBW
r' •■■
15
If
{
and J
2=1,2,
3.3
are positive integers, we obtain with
y. = A, + A10, t
t
12.
that tl
t7l
^=0 T(k+l)y\ V+A:
^=0 r(fc+l)7t2
A'7 ^
Convolutions.
From the "fatal shock" model (§2.1) with
kl F (k)
component, we obtain the fe fold convolution
spares for each of the BVE.
With
s < t,
F(fc)(s,t) =
P{Z12(s)«)i,Z12(*)=)l+m}P{Z1(e) i klUP{Z2(s) <_ klin}
2.
kl S^
"A12sn .1 e (X,25^
,^o
*:
e
X12US)
^:
^"^ e'XlS(xlS)i
,
1

i
Anos
e
00
/ x
^ A.s
.4
„
(A12s)
kla e x OJ:3 (felOl '
1>12^~S^
'
A10(te)
e
LA12(ts;J
r
I ^
A A^t
feiiwe 2^div (fei^w).'
1. «mmm 16
s < t.
In particular, if
(2) X.sX^X FK '(Stt) = e
Of course, 3.4 If for all if
F
(k)
t [(1+X1s)(l+X2t) + A12(ts)(l+X1s) + X12s].
is a bivariate gamma distribution.
Distributions obtained by a change of variables. A'
is an exponential random variable, then
a > 0.
{X,Y)
However, if
a = b > 0.
The distribution of
aX
is exponential
is BVE, then
(aXtbY)
is BVE only
(aX,bY)
a,b > 0
is easily
for
seen to be of the form
Fiz,y) = e
(3.4)
i
4
^
This distribution has exponential marginals and includes the BVE as the special case when
.r,
and
^T
r0
=
^A
* ^IO*
^t also includes the upper bound of (1.1)
are exponential
(A. = A0 = 0).
Other changes of variables in the BVE may be of interest, in particular, the distribution of
(A"
,Y
)
is a bivariate Weibull
distribution, namely. _ X.z BY A_^ Xmax(x BY ,zv ) F(x,y) ~ e
•*SB
l•t■w<'■,
17
3.5
Representation In terms of Independent random variables.
Theorem 3.2.
(A',1)
ie BVE if and only if there exist indepenJenl
e.cicner.tial random variables
U, V
and
W
such that
X  min('',I.'),
}' = rain (7,;/). This theorem is an immediate consequence of the fatal shock model discussed in §2.1.
It can be an aid in reducing questions concerning
dependent exponential random variables to questions concerning independent exponential random variables.
An illustration of its usefulness is given
in §5.
3.6
Comparison of the bivariate exponential with the case of independence.
It is interesting to compare the survival probabilities (with marginals fixed) of the dependent and independent cases. The BVE has marginals
F^x) = e
,
F2(i/) = e
Clearly, the difference
A.rAizA »max^.iv)
fU,i;)  F.Cx)^) = e
is positive for all
x
and
\
de
y,
min(x,zv)
)
so the probability that both items
survive is greatest in case of dependence.
However, it is easily
i — .ÄWUsü*^ • .Mi**«
18
verified that for any bivariate distribution and
F
with marginals
F,
F2,
F(x,y)  F^F^y) = F(x,y)  F^F^y),
so the probability that both items fail is also greatest in the case of dependence.
This means that in the case of a series system (which
functions only when both items function), system survival probability is greatest in the case of dependence.
On the other hand, in the case of a
parallel system (which fails only when both items fail), system survival probability is greatest in the case of independence. To determine the greatest discrepancy,
max[F(x,y)  F^{x)F„(y)]
note
t
z3y that if
x < y.
X x(X„+\ )y ^IOA 1 ; F(x,y)  F1(x)F2(^) = e ' ^ (1e
is decreasing in
y,
so that
(.,,)  F,MFAy) <_ e' ""^^de
where
t = "
t = min(x,y). 1
A 12 log(l + ~~r~)
12
' ) = aU(le
The maximum of the righthand side occurs at
and is equal to
12 max[F(.r,zv)  F^F^u)] = d6/(1+6)1+6 , ■*■*
i J
•A, t i*
12
).
19
where
6 = \/X
.
In terms of the correlation
maximum discrepancy is
4.
[p /(1+p)
pU',y) = A
9/\,
the
]
The Multivarlate Exponential Distribution. 4.1
Derivations.
To fix ideas, we consider first an extension of the fatal shock model to a threecomponent system. Z^itiX.), Zd^A ), Z„(t;A )
Let the independent Poisson processes
govern the occurrence of shocks to
components 1, 2, 3, respectively;
Z^itiX^) * Z.d^A
), Z (i;;A23)
govern the occurrence of shocks to the component pairs 1 and 2, 1 and 3, 2 and 3, respectively; and
Z, „„ (t; A.. __)
governs the occurrence of
simultaneous shocks to components 1, 2, 3.
If
X, , X*, X~
denote the
life length of the first, second, and third components, then
F{xl,x2,x ) = P{A'
> a:, , X2 > x, X~ > x~}
= P{Z1(x1) = 0, Z2(x2) = 0, Z3(a'3) = 0, Z12(raax(^1,a:2)) = 0,
(4.1)
Z.CmaxCr, ,.r)) = 0, Z23(max(a:2,j'3)) = 0, Z „Amaxix.tXy,^^)) = 0} = exp [A a^ Aa' A a^A.^axCx, ,cr2)A „max^ ,a:3)A23max(a*2 x~)
A
2„max(a
,;r2,a: ) ].
It is clear that similar arguments yield the ndimensional exponential distribution given by
mmmm* 20
n
Fix, tX^,... ,x ) = exp[ l \ .x . 
(4.2)
Z
X . .max(x .,x .) 
t<.i
1
— • • • ■" A i o
l
max ^x, »x 0 j • •»».r ) \
To obtain a more compact notation for this distribution, let st
of vectors
(s, ,....s 7)
In
(^.,...,8 ) f (0,...,0). X.
where each
For any vector
s. = 0
j
or
1
Is
a'.'s
for which £. = 1. t
S
denote the
but
s c S, max(a:.s.)
It.
of the
\ . .7ma.x(x.,x .tx,)
t<.}
'
is the maximum
Is
Thus,
Fix,,... ,x ) = exp [  lc X max(x.s .) 1. in szb s t t
(4.3)
For example, for
n  3
X010 = A2, X001 = Xy
the correspondence with (4.1) is
^mn
=
^i '
X110 = A12, A101 = X13, A011  A23, A111  A123.
We call the distribution given by (4.2) or (4.3) the multivariate exponential distribution (abbreviated MVE). dimensional marginals (hence are MVE.
Note that the
^dimensional marginals,
(nD
/'=1,2,. .. ,nl)
In particular, the twodimensional marginals are BVE.
In the bivariate case (§2.2) the fatal and nonfatal shock models both yield the BVE.
Indeed, if we assume in the multivarlate case that
shocks need not be fatal but instead cause transitions with varying probabilities, then by a tedious but direct calculation we again obtain the MVE. Consider now the requirement that the residual life is independent of age, i.e.,
i *^',' '^KW^W
24
Thus,
v(^1.....sn) = ri ^^^io^.o+s^" ^11o...o+si+ö2)~ •••(^i...i"nv)~ '
where
I'
is the summation over all permutations of the indices.
For
example,
iB
V&2^3)
^lll+,c?l+s2+s3 7^
1 =
(
^110+sl+s2)
1
1 +
(
+ ^lOl^l^S^^^lOO^P'1
+
(
+ ^011+S2+S3)'1[(^010+ß2)"1
+
(
f^lOO^^
^010+ö2)
]
^001+;3)'1]
^001+ö3rl]
A final but very important property of the MVE that we mention is its representation in terms of independent exponentials.
As in the bivariate
case (Theorem 3.2), we obtain from the fatal shock model that if are MVE, there exist independent exponential random variables such that X. = min Z . s. = l
A',,...,/ 1 n
Z , st S S
T\
5.
Minima of Exponential Random Variables. An important property of independent exponential random variables
and
Y
is that
min(,V,y)
is exponential.
that if the independent random variables distribution functions, then if
Ä
and
Y
min(Ä,y)
It is also known [Ferguson (1964)] X
and Y
have absolutely continuous
is independent of
X  Y
if and only
are exponentially distributed (with the same location parameter,
possibly not zero).
Thus, minima play an important role in the case of
independent random variables and we examine their role in the dependent case. If
(.V,7)
is BVE, then
P{min(A',y) >_ x} = P{X >_x$ Y >_ x} * e
where
\ = \
+ \
+ A
9,
so that the minimum of
— \T
X
,
and
Y
is exponential
(This fact is also an immediate consequence of Theorem 3.2.) On the other hand, the minimum of dependent exponential random variables X
need not be exponential; e.g., if
and
Y
have one of the bivariate
distributions studied by Gumbel (1960),
—,
.
: ix,y) * e
xuöxu 
",
or FC,:) = e"^ [1 + cxde''Hle ^)],
4 
26
then (;
min(.V,j') or
A
= 0).
is exponential only in the case of independence Similarly, if F is the lower bound of (1.1), i.e.,
FCr,:;) = max [^ Or) +?2(,y)  1, 0],
then
min(A',y)
is not exponential, even though
F,
and
F?
are
exponential. But there are many bivariate distributions with exponential marginals such that
min(A,,y)
is exponential.
Such distributions must satisfy the
functional equation
(5.1)
F(^,^)  FOr.aOFQ/.z/).
In addition to the BVE, the distribution given by (3.4) has this property. Since there are many solutions of (5.1) with exponential marginals, it may be of interest to consider those bivariate distributions for which mintoA'.Zr.i')
is exponential for all
a, b > Q,
An investigation of this
stronger condition shows that such a distribution must have the form
(5.2)
where
Hx.y) = e"^^*,
is a nonnegative function.
Again, there are many such
distributions, including the distribution given by (3.4).
Of course,
defined bv (5.2) is not a distribution function for all functions
/.
F
■^ ■^■^^"^^^fl^P^^W
27
X
and
Y
min(A',y)
and
A  Y
As previously mentioned, it is well known that if independent exponential random variables, then are independent.
This conclusion is also true if
X
and
dependent but have the bivariate exponential distribution.
are
Y are It is some
what tedious to prove this fact direcly, so we make use of Theorem 3.2 and our knowledge of the independent case. Let
X = min(U,W), Y = min(V,W),
independent exponential random variables.
' U  V,
and
X  Y =
0
W  V,
0 < w  V < u  I/,
0,
0 < U  W < V  W
and
W
are
min(XtY) = min(U,V,W)
Then
U  W,
or
0 < U  V < W  V,
or
o < v  w < u  w.
<
The independence of of
U,V
where
('.'l','Ji:',Vl:')
min(A',y) and
and
miniU,V,W).
X  Y
follows from the independence
The latter may be obtained by
applying the twodimensional result to conditional distributions. It would be of theoretical interest to know the class of bivariate distributions for which
min(A',y)
and
A'  7
are independent.
, v.4—■■
^ ^*^^WP^^^B"
28
6.
Further Results for the Functional Equation
^(c^+t.s^+t) = F(s, ,s~)F(t,t)
In Section 2.3 we introduced and motivated the functional equation (2.8) and found the general solution to be
e ÜF^xy),
z l_y*
F2{yx),
z tV'
F(xty) =
(6.1)
e
F., F~
When
are exponential distributions with parameters 6 , &2 
satisfying
Q
— 6i
+
^l*
^{x^y)
is the BVE.
6 , 6„
specified by (6.1) is a distribution only for certain marginals
In order that any two points
(6.2)
Fix,y)
(x*,y,)
f.
and
be a distribution function, it is necessary, for and
that
Cr^'^o^'
Fix^y^ + F(x2,y2)  FC^,^)  F(x2,y1) >_ 0.
(6.2) Is equivalent to conditions on (x1,y1)
Fix^y)
However,
and
(J:2,Z/2);
e.g., if
F,
and
F9
which depend upon
x^ ^^^ —^\ 1^2'
^6*2^
becomes
(x2x1) [f2(y1a:1)  f2(z/2x1)]e
1 F2(i/1a:2)  F2(zy2x1).
Such conditions are not easily verified; we obtain some alternative conditions when the marginal distributions have densities satisfying certain regularity conditions.
f, • 1
and
r„ • 2
—t ■
30
so that
(6.5)
a E / / 0 0
afAx,y)dxdy = 2 ^ [/ (0) +/2(0)]
Thus, the absolutely continuous part f ixfu)
given by (6.4) and (6.5). 00
F ix%x) = a"1 /
+ a"1 /
But
of
Fix,y)
has density
We compute 00
du j
dv e~Qu[f'ivu) +Qf(vu)]
OO
00
dv )
X
= e
F (x.,y)
du e'^if'iuv) + Qf(uv)]
V
ex
F(x,x) = exp(ex),
so that
F (x,x) = [F(x,x)aF (x,x)]/(la) = exp(ex).
Since
F (x,v)
(6.6)
Thus,
is concentrated on x = v,
we conclude that
F (x,y) = exp[6 max(a:,^)].
F
is a valid distribution function if:
mm
r
31
(i)
F
is a convex mixture of
F
F , s
and
a
i.e., '
0 < ct < 1 — —
From (6.5) this condition is 9
(ii)
F
l/y0)
+
/2(0) 
2fl

is a valid distribution function, i.e.,
From (6.4) this is
/'.(e) + e/.(s) ^0, is
/ {x,u)
i=l,2,
>
0,
or
L'
equivalently,
d log f^/dz i e.  We now check the conditions of the theorem for the important Weibull and gamma distributions.
The respective density functions
^(s) E w(2;B,6) = ß6r;ß":lexp(6^ß),
2 > 0, ß > 0, 5 > 0,
g{z) = 7(s;B,5) = p^y 6ß2ß
s > 0, ß > 0, 6 > 0,
1
satisfy the regularity conditions.
exp(63),
It is easilv checked that
.. . d logr u(a) , , d log" "flOj) x lim ^^  lim '^ ^ oo as az lim ::i—07fi ^^ = oo s>0
if
a < 1.
.r if
, a > 1, '
andj
i • (i log~— jjCsj i lim —^ = az n
Because of condition (tt),
FT
or
F0
can be Weibull or gamma distributions only in the special case that they are exponential.
^BÄ
•32
In the exponential case, condition (t) becomes where
^
requires
and
6
are the parameters of
5. f_ 6, 6„ ^_ 6.
/.
and
6 1 ^ /„.
+
*o i. ^0
Condition {{.{)
Thus,
( e
^^Ui/),
^ 1 ^»
e
FAyx) ,
^ 1 i/.
F(ar,i/) = <
is a bivariate distribution with exponential marginals and only if Remark. F(J:,^)
suppose (>;'
f,
and
/"„
if
5. f_ 6, 6? <_ 8, ^ + 6„ >_ 6. Suppose
(/■,»/«)
satisfies conditions (i) and (tz) so that
defined by (6.1) is a valid bivariate distribution, and similarly (^i .i?^)
+ (ly)^, ,
satisfies (i) and (ii). Y/O
+
(1Y)^7)
Then the mixture
satisfies the conditions and yields
another solution to the functional equation (2.8).
In particular,
marginal distributions which are mixtures of certain exponential distribu yield solutions.
33
ACKNOWLEDGEMENTS We would like to thank J. D. Esary for a number of very helpful comments and suggestions and also Max Woods for suggesting the investigation in §3.4.
)utions
— ^
~
~i~
« i
V
I «
BLANK PAGE
k. J »
!*«• f
' 1
*£.„
■.■,

■*■
 r^—^—^(f>—*
^
35
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[2]
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* 3Et^t*ta%«^i^B
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