a multivariate exponential distribution - Semantic Scholar

a multivariate exponential distribution - Semantic Scholar

BOBINO BCIBIMTIPIC RB8BARCH LABORATORIES DOCUMENT OI-BB-OBOB J i 00 00 50 A Mvltivoriote Exponential Distribution Albert W. Marshall lagran Olki...

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BOBINO BCIBIMTIPIC RB8BARCH LABORATORIES DOCUMENT

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A Mvltivoriote Exponential Distribution

Albert W. Marshall lagran Olkia -

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i MATHEMATICS RESEARCH

MARCH 18SB

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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 GP-3837 by the Department of Statistics, Stanford University, Stanford, California, March 15, 1966.

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Manuscript prepared by Karen Harles

.j-JLi

i *'^WV^* »SMH

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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

r

^

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 - X-t - 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 two-component 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 two-component 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

Non-fatal shock models.

Again consider a two-component 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

fe-0 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(t-S)).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 two-component 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 two-component

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

-Qy-b^x-y) 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 two-dimensional

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

-imevmaamssrr-r^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

expf-Aj* - 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) + (l-a)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 U-i\2+\l2)V

du e

X

+ X2(A1+A12) j

du J

y .XX-X2y-X

y

dv e

-(A1+A12)k-A2y

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

'

1-a

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

(l-e~sx)a-e~ty), 00

e

^

^(O,^) = 0 i G(x,0).

satisfying

of the Laplace transform by

obtaining

00

OO

00

Hs,t) = j J a-e'SX)a-e'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 t-l

t7-l

^=0 T(k+l)y\ V+A:

^=0 r(fc+l)7t2

A'7 ^

Convolutions.

From the "fatal shock" model (§2.1) with

k-l 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 k-l-UP{Z2(s) <_ k-l-i-n}

2.

k-l S^

"A12sn .1 e (X-,25^

,^o

*:

e

-X12U-S)

^:

^"^ e'XlS(xlS)i

,

-1

-

i

Anos

e

00

/ x

^ A.s

.4



(A12s)

k-l-a e -x OJ:3 (fe-l-Ol '

1>12^~S^

-'

-A10(t-e)

e

LA12(t-s;J

r

I ^

A A^t

fe-i-i-we -2^div (fe-i-^-w).'

1. «mmm -16-

s < t.

In particular, if

-(2) -X.s-X^-X FK '(Stt) = e

Of course, 3.4 If for all if

F

(k)

t [(1+X1s)(l+X2t) + A12(t-s)(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_^ -X--max(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-.r-A-iz-A »max^.iv)

fU,i;) - F.Cx)^) = e

is positive for all

x

and

-\

d-e

y,

min(x,zv)

)

so the probability that both items

survive is greatest in case of dependence.

However, it is easily

-i — -.Ä-WUs-ü*^ • .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 ' ^ (1-e

is decreasing in

y,

so that

-(.-,,) - F,MFAy) <_ e-' ""^-^d-e

where

t = "

t = min(x,y). 1

A 12 log(l + ~~r~)

12

' ) = a-U(l-e

The maximum of the right-hand 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 three-component system. Z^itiX.), Z-d^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^ -A-a' -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 n-dimensional 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,

(n-D-

/'=1,2,. .. ,n-l)

In particular, the two-dimensional marginals are BVE.

In the bivariate case (§2.2) the fatal and non-fatal 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

-x-u-öxu -

",

or FC-,:-) = e"^ -[1 + cxd-e'-'Hl-e ^)],

-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 non-negative 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','J-i:',V-l:')

min(A',y) and

and

miniU,V,W).

X - Y

follows from the independence

The latter may be obtained by

applying the two-dimensional 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^x-y),

z l_y*

F2{y-x),

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

(x2-x1) [f2(y1-a:1) - f2(z/2-x1)]e

1 F2(i/1-a:2) - F2(zy2-x1).

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'iv-u) +Qf(v-u)]

OO

00

dv )

X

= e

F (x.,y)

du e'^if'iu-v) + Qf(u-v)]

V

-ex

F(x,x) = exp(-ex),

so that

F (x,x) = [F(x,x)-aF (x,x)]/(l-a) = 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

^^U-i/),

^ 1 ^»

e

FAy-x) ,

^ 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?^)

+ (l-y)^-, ,

satisfies (i) and (ii). Y/O

+

(1-Y)^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-

REFERENCES [1]

Epstein, B. (1958). in life testing.

[2]

Industrial Quality. Control 15 2-7.

Esary, J. D. (1957). and fatality.

The exponential distribution and its role

A stochastic theory of accident survival

Ph.D. dissertation, University of California,

Berkeley. [3]

Ferguson, T. S. (1964). distribution.

[4]

A characterization of the exponential

Ann. Math. Statist. 35 ]199-1207.

Frechet, M. (1951). marges sont donndes.

Sur les tableaux de correlation dont les Ann. Univ. Lyon Seat. A, Series 3, 14

53-77. [5]

Freund, J. E. (1961). distribution.

[6]

A bivariate extension of the exponential

J. Amev. Statist. Assoa. 56 *^ 971-977.

Gaver, D. P., Jr. (1963).

Random hazard in reliability problems,

Teahnometrios 5 211-226. [7]

Gumbel, E. J. (1960).

Bivariate exponential distributions.

J. Amer. Statist. Assoo. 55 698-707. [8]

Morgenstern, D. (1956). Verteilungen.

Einfache Beispiele zweidimensionaler

Mittelungsblatt für Mather.atisehe Statistik %

234-235. [9]

Plackett, R. L. (1965).

A class of bivariate distributions.

■\ Arer. Statist. Assoa. 60 516-522.

* 3Et^t*ta%«^i^B

-36-

[10]

Young, W. H. (1917).

On multiple integration by parts and the

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Pvoc. London Math. Sac, Series 2,