A characterization of a bivariate geometric distribution

STATI~I'I~ i ELSEVIER

Statistics & Probability Letters 23 (1995) 307-311

A characterization of a bivariate geometric distribution* Kai Sun, Asit P. Basu* Department of Statistics, University of Missouri, Columbia, MO 65211. USA

Received June 1992; revised May 1994

Abstract In this paper a characterization of a bivariate geometric distribution is obtained. The results are based on the discrete analogue of Cox's conditional failure rate. Keywords: Bivariate geometric distribution; Failure rate; Characterization

1. Introduction A m o n g m a n y bivariate geometric distributions there is a fundamental one defined as follows. Let M, N be discrete r a n d o m variables taking values in the set {0, 1, 2 .... }. Then (M, N) is said to have a bivariate geometric distribution if P { M > ~ m , N > > , n } = P ~ 1 1 ( P l o + P 1 1 ) m-n

form>~n,

= P'~I(Pol + P l l ) "-m

for n >1 m,

(1.1)

where Poo + Plo + Pol + Pll = 1, P l o , P o l , P l l > O, Poo >>-O. The distribution (1.1) was introduced by Hawkes (1972), and studied by m a n y authors, e.g., Esary and Marshall (1973), Block (1977), L a n g b e r g et al. (1977), and Marshall and Olkin (1985). Since a fundamental characterization of the univariate geometric distribution is constant failure rate, it is natural to hope that a bivariate geometric distribution would also have constant failure rate. In the univariate case, the failure rate of a discrete r a n d o m variable is defined by r(m) = P { M = m } / P { M

>1 m}.

(t.2)

Extension of the above concept to two dimensions is not straightforward and there have been different approaches in defining failure rate in the bivariate case, for example, the bivariate failure rate

'~This research was supported in part by the US Air Force Office of Scientific Research under grant number AFOSR F49620-92-J0371. * Corresponding author. 0167-7152/95/$9.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 1 6 7 - 7 1 5 2 ( 9 4 ) 0 0 1 29-V

Kai Sun, A.P. Basu / Statistics & Probability Letters 23 (1995) 307-311

308

r(m,n) = P { M = m, N = n } / P { M >1 m, N >1 n} discussed by Puri and Rubin (1974), and the vector hazard failure rate h(m,n) = ( P { M = m, N >1 n } / P { M >1 m, N >~ n } , P { M >1 m , N = n } / P { M >>.m, N >>,n}) discussed by Nair and Nair (1990). Unfortunately, the distribution (1.1) does not have either constant bivariate failure rate or constant vector hazard rate. Cox (1972) introduced a concept of conditional failure rate and a failure rate formulation for absolutely continuous bivariate random variables. Cox's failure rate formulation which views the bivariate lifetime model as a point process has been used in many statistical problems. However, little work has been done in characterizing bivariate distributions by this formulation. In this paper we define total failure rate for the discrete bivariate random variable by using Cox's conditional failure rate concept and a characterization of the distribution (1.1), in terms of total failure rate, is obtained. Failure of a two-component parallel system can be considered to consist of two stages: first one of the two components fails, then the remaining component fails. From this point of view, it seems reasonable to represent a system failure rate by using both the first stage failure rate, that is, the failure rate of the minimum lifetime of the two components, and the second stage failure rate which is the conditional failure rate of one component given that the other component fails first. To this end, we define total failure rate of a twocomponent system as follows.

Definition 1.1. Let (M, N) be a bivariate random variable taking values in the set {0, 1, 2 .... } x {0, 1, 2 .... }. The vector (r(t), r l (mln), r2(nl m)) is called the total failure rate of (M, N), where r(t) = P { m i n ( M , N ) = t } / P { M >1 t, N >>.t}, rl(mln) = p ( m , n ) / ~

(1.3)

p(u,n) for m > n,

(1.4)

p(m,u) for n > m,

(1.5)

u=m

r2(nlm) = p ( m , n ) / ~ u:n

with p(m,n) = P { M = m , N = n}, and t e {0, 1,2 .... }. Notice that r(t) is the failure rate of min(M,N) and r~(mln) is the conditional failure rate of M given M > N and N = n (Cox, 1972). The quantity r2(nlm)is defined similarly. According to this definition in the distribution (1.1) has a constant total failure rate (1 - Pl 1, Pol + Poo, Plo + Poo).

2. Results Since total failure rate describes explicitly the two-stage failure risks of a two-component parallel system, a derivation of the distribution (1.1) based on constant total failure rate should be appealing and may shed light on the applicability of the distribution. The following theorem is readily verified. Theorem 2.1. I f (M, N ) has a constant total failure rate (1 - P11, Pol + Poo, Plo + Poo), P { M > N I min (M, N ) = t } = p 1o/(1 - p 11 ), and P {M < N I min (M, N ) = t } = Po 1/(1 - p 11 ), then (M, N ) has the distribution (1.1). To prove our main result, Theorem 2.2, consider the following lemma.

Kai Sun, A.P. Basu / Statistics & Probability Letters 23 (1995) 307-311 2.1. I f

( M , N ) has a constant total failure rate and the geometric marginals, P { M > N l m i n ( M , N ) = t} = 01(0 and P { M < N l m i n ( M , N ) = t} = g2(t) are free oft. Lemma

309

then both

Proof. Let the total failure rate be (q, ql, q2), w h e r e 0 < q, qx,q2 < 1, a n d P { M = N l m i n ( M , N ) = = g3(t). F o r m > n,

t}

P { M = m, N = n} = P { m i n ( M , N ) = n, M > N, M = m} = P{min(M,N)=

n}P{M >Nlmin(M,N)=

n}P{M = mlmin(M,N)=

n, M > N } .

N o t i c e t h a t the d i s t r i b u t i o n of m i n ( M , N ) is a u n i v a r i a t e g e o m e t r i c w i t h a failure rate q, a n d t h a t the c o n d i t i o n a l d i s t r i b u t i o n of M g i v e n b y M > N = n is a u n i v a r i a t e g e o m e t r i c with a failure rate ql a n d a s u p p o r t { n + 1, n + 2 . . . . }. H e n c e

p(m,n) = q(1 - q)"gl(n)ql(1

for m > n.

-- ql) m-n-1

(2.1)

Similarly,

p(m, n) = q(1 - q)'g2(m)q2(1 - q 2 ) " - " - 1 for m < n.

(2.2)

F o r m = n,

p(m, n) = q(1 - q)"ga(n).

(2.3)

Let the m a r g i n a l s be P { M = m} = px(1 - P I ) " a n d P { N = n} = p2(1 - P2)". F o r m > 0, since P { M = m} = P { M = m, N >~ 0}, px(1 - p~)m =

~

p(m,n) + p(m,m) +

n=O

p(m,n).

(2.4)

n=m+l

S u b s t i t u t i n g (2.1)-(2.3) i n t o (2.4), we o b t a i n p~(1 - pl)m/[qq~(1 - q t ) " ] m--1

=

~. [(1 -- q)"g~(n)]/(1 -- qt)"+x + (1 - q)m[1 - gx(m)]/[q~(1 _ q l ) m ] .

(2.5)

n=O

Similarly, pl(1 - p l ) ' + l / [ qqx(1 - ql) m] = ~

[(1 - q)"g~(n)]/(1 -- ql) "+~ + (1 -- q ) , , + l [1 - g~(m + 1 ) ] / [ q a ( 1 - q a ) " + l ] .

(2.6)

n=O

S u b t r a c t i n g (2.5) f r o m (2.6) we o b t a i n a difference e q u a t i o n

g~(m) + (q - 1)91(m + 1) = p~(1 - p~)m(q~ - pl)/[q(1 - q)m] + (q__ q~).

(2.7)

Since (1 -- P l ) " = P { M >1 m) = P { M >1 m, N >1 0} > P { M >~ m, N / > m} = (1 - q ) ' , for large m, we h a v e q > Pl a n d l i m , , ~ o [(1 - p~)/(1 - q ) ] " = ~ . Since 0 ~< g~(t) ~< 1, the l e f t - h a n d side of (2.7) is finite. We thus obtain (2.8)

Pl = ql < q. Therefore, (2.7) b e c o m e s

gl(m) + ( q - - 1 ) g l ( m + 1) = ( q -

ql)

for m > 0.

(2.9)

310

Kai Sun, A.P. Basu / Statistics & Probability Letters 23 (1995) 307-311

Eq. (2.9) is a simple difference equation subject to the condition 0 ~< g~ ~< 1. The unique solution is

gt(rn) = (q - qx)/q

for m > 0.

(2.10)

We can also show yl(O) = (q - ql)/q. Similarly, g2(m) = (q - q2)/q The p r o o f is complete.

for m >~ 0.

(2.11)

[]

We are now ready to prove our main result.

Theorem 2.2. ( M , N ) has the bivariate #eometric distribution (1.1) if and only if ( M , N ) has the geometric marffinals and a constant total failure rate. Proof. The "only if" part is trivial. We prove the "if" part. Let the total failure rate be (q, q~, q2). It follows from L e m m a 2.1 that P{M > N l m i n ( M , N ) = t} = (q -- q~)/q, P { M < N l m i n ( M , N ) = t} = (q -- q2)/q, a n d q > q i , i = 1,2. Noting that P { M > N I min(M, N) = t} + P { M < N I min(M, N) = t} ~< 1, we have q~ + q2 ~> q. Hence p(m,n) = (1 - q)"(1 - q l ) m - n - l q l ( q - -

ql)

for m > n,

=(1--q)m(1--q2)n-m-lq2(q--q2)

form
= (1 -- q)n(qx + q2 -- q)

for m = n.

(2.12)

Using a o n e - o n e parameter transformation, we obtain

p(m,n) = P']I(PIo + P l l ) m - " - l p l o ( P o t + P00),

for m > n,

=P~'l(Pol + P l l ) " - " - l P o l ( P l o + P o o ) ,

forn>m,

= P]lPoo,

for m = n,

(2.13)

where Poo + Plo + P01 + Pll = 1, Pxo, P o l , P l l > 0, Poo ~> 0. This is the probability function of the distribution (1.1). []

Remark. The concept of total failure rate introduced in this paper is extended to the case of continuous bivariate variables and characterizations of a n u m b e r of bivariate exponential distributions are obtained by Sun and Basu (1993).

Acknowledgements The authors are grateful to the referee for his constructive suggestions.

References Block, H.W. (1977), A family of bivariate life distributions, in: C.P. Tsokos and 1. Shimi, eds., The Theory and Applications of Reliability, Vol. I (Academic Press, New York) pp. 349-371. Cox, D.R. (1972), Regression models and life-tables, J. Roy. Statist. Soc. Ser. B 34, 187 220. Esary, J.D. and A.W. Marshall (1973), Multivariate geometric distributions generated by a cumulative damage process, Technical Report 55 #Y73041A, Naval Postgraduate School, Monterey, CA.

Kai Sun, A.P. Basu / Statistics & Probability Letters 23 (1995) 307-311

311

Hawkes, A.G. (1972), A bivariate exponential distribution with applications to reliability, J. Roy. Statist. Soc. Ser. B 34, 129-131. Langberg, N., F. Proschan and A.J. Quinzi (1977), Converting dependent models into independent ones, with applications in reliability, in: C.P. Tsokos and I. Shimi, eds., The Theory and Applications of Reliability, Vol. I (Academic Press, New York) pp. 269-275. Marshall, A.W. and I. Olkin (1985), A family of bivariate distributions generated by the bivariate Bernoulli distribution, J. Amer. Statist. Assoc. 80, 332 338. Nair, N.U. and K.R.M. Nair (1990), Characterizations of a bivariate geometric distribution, Statistica, anno L, no. 2, 247-253. Purl, P.S. and H. Rubin 0974), On a characterization of the family of distributions with constant multivariate failure rates. Ann. Probab. 2, 738-740. Sun, K. and A.P. Basu (1993), Characterizations of a family of bivariate exponential distributions, in: A.P. Basu, ed., Advances in Reliability (Elsevier, Amsterdam) pp. 395-410.