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A covariance inequality for coherent structures
Statistics & Probability North-Holland
March
Letters 9 (1990) 215-216
A COVARIANCE
INEQUALITY
Kumar
*
JOAG-DEV
Department
Frank
of Statistics,
STRUCTURES
of Illinois, Champaign, IL 61820, USA
University
PROSCHAN
FOR COHERENT
1990
**
Florida State University,
Tallahassee,
FL, USA
Received July 1988 Revised December 1988
Abstract: We generalize a covariance inequality for coherent structures. We replace the earlier assumption of independence among components by association, a weaker assumption applicable to many situations encountered in practice. The proof of the more general result turns out to be simpler than that of the earlier, more restricted case. AMS
1980 Subject Classification:
Keywords:
Reliability,
reliability
60N05. function,
S-shapedness,
1. Introduction An inequality important for proving that the reliability function is S-shaped is developed in Barlow and Proschan (1981, Chapter 2, Section 5). To describe it briefly, we review the following basic concepts. Let X,, i = 1,. . . , n, be performance indicating binary (values 0 or 1) random variables. X, = 1 indicates that the ith component is functioning while X, = 0 indicates that it has failed. Write X for (X,, X,, . . . , X,,), and let the binary function \k(X) denote the performance indicator of a system of n components. The function \k is said to be coherent if (i) \k is coordinatewise nondecreasing, that is x
*
q(x)
< ‘k(v);
(ii) all n components * Research
supported by Research under Grant of Illinois. * * Research supported by Research under Grant State University.
0167-7152/90/$3.50
are relevant. the Air Force Office of Scientific AFOSR 84-0208 to the University the Air Force Office of Scientific AFOSR 88-0040 to the Florida
0 1990, Elsevier Science Publishers
associated
random
variables.
coherent
structure
function.
The ith component is said to be relevant if there exists at least one configuration of the other (n - 1) components such that the system is in the failed state if the ith component is in the failed state and in the functioning state otherwise. Below we assume that X is associated; that is, for every pair of coordinatewise nondecreasing functions ( f, g) defined on [wn + Iw, cov[f(X),
SW)]
ao.
(1)
It is wellknown that if the X, are mutually independent, then X is associated. The main inequality obtained in this note is cov[ xx,
- q(x),
‘k(x)]
> 0.
(2)
Note that the nonstrict version of (2) follows easily from the facts that EX, - \k(X) is coordinatewise nondecreasing in X, and X associated implies (1). The approach in Barlow and Proschan (1981) assumes and uses independence of the components in a crucial manner. As seen from the discussion above, association seems to be the natural condition; we prove ineauality (2) in this more general setting. It should be noted that the covari-
B.V. (North-Holland)
215
Volume
9, Number
STATISTICS
3
& PROBABILITY
ante inequality (2) by itself is not enough for proving the S-shaped property of the reliability function when the components are dependent. However, (2) is an important step towards obtaining such a result. Our approach is based on a result which states that if a bivariate dustribution exhibits positive quadrant dependence, then the uncrrelatedness implies independence. The result is an immediate consequence of a covariance formula due to Hoeffding (see Lehmann, 1966, Lemma 2). In general, the present approach relies directly on the notions of association, independence, and coherence. It is hoped that it will provide a better insight.
LEmERS
March
Due to assumptions CX,=n
1990
on \k made in the theorem,
j
‘k(X)=1
*
@x)=0.
and xx,=0 so that P[Cx,-P(X)=01 >P[CX;=O]
>, I!!JP[X,=O] 1
>o,
(6)
where the second inequality in (6) follows the association of X. Similarly,
from
2. Results P[CX,-‘k(X)=n-1]
>P[CX,=n]
Lemma. Suppose the pair (U, V) satisfies the positive quadrant dependence (PQD) condition, that is P[U>u,
V~u]>,P[u>u]P[V/u],
(3)
for every pair of reals (u, v). Then cov[U,
V] =o
*
U, V are independent.
0
Remark. Suppose the pair (U, V) is associated. Then from (1) it is clear that (U, V) satisfies PQD condition (3). Theorem. Let \k correspond to a system with n (>, 2) associated components X,, X,, . . . , X,, such that for every i, 0 < P[ X, = l] < 1. Further, suppose that q is coordinatewise nondecreasing and 0 < P[q(X) = 0] -C 1. Then cov[xX, - q(X), 9(X)] = 0 implies that only one out about of the n components is relevant for \k. Proof. In view of the lemma and remark above, CX, - \k(X) and 9(X) are independent. Thus P[‘k(X)=O,
From (6) and (7), it follows that the left sides of (4) and (5) are positive. Hence, P[CX,=n-1,
*(X)=0]
>O
(8)
and P[CX,=l,
\k(X)=lj
>O.
(9)
Inequality (9) implies that there is a component j such that the system works even if component j is the only working component. On the other hand, (8) implies the existance of a component whose failure causes system failure even if all other components are funtioning. Due to the nondecreasing character of 9, the latter component has to be j. Thus j is the only relevant component, as claimed. 0
References (4
xx,-*(x)=0]
=P[9(X)=l]P[CX,-*(x)=0]. 216
(7)
CT-*(X)=n-l]
=P[\k(X)=O]P[CX,--*(X)=n--I],
P[*(x)=l,
>O.
(5)
Barlow, R.E. and F. Proschan (1981), Statistical Theory of Reliability and Life Testing (To Begin With, Silver Spring, MD). Lehmann, E.L. (1966), Some concepts of dependence, Ann. Math. Statist. 37, 1137-1153.
March
Letters 9 (1990) 215-216
A COVARIANCE
INEQUALITY
Kumar
*
JOAG-DEV
Department
Frank
of Statistics,
STRUCTURES
of Illinois, Champaign, IL 61820, USA
University
PROSCHAN
FOR COHERENT
1990
**
Florida State University,
Tallahassee,
FL, USA
Received July 1988 Revised December 1988
Abstract: We generalize a covariance inequality for coherent structures. We replace the earlier assumption of independence among components by association, a weaker assumption applicable to many situations encountered in practice. The proof of the more general result turns out to be simpler than that of the earlier, more restricted case. AMS
1980 Subject Classification:
Keywords:
Reliability,
reliability
60N05. function,
S-shapedness,
1. Introduction An inequality important for proving that the reliability function is S-shaped is developed in Barlow and Proschan (1981, Chapter 2, Section 5). To describe it briefly, we review the following basic concepts. Let X,, i = 1,. . . , n, be performance indicating binary (values 0 or 1) random variables. X, = 1 indicates that the ith component is functioning while X, = 0 indicates that it has failed. Write X for (X,, X,, . . . , X,,), and let the binary function \k(X) denote the performance indicator of a system of n components. The function \k is said to be coherent if (i) \k is coordinatewise nondecreasing, that is x
*
q(x)
< ‘k(v);
(ii) all n components * Research
supported by Research under Grant of Illinois. * * Research supported by Research under Grant State University.
0167-7152/90/$3.50
are relevant. the Air Force Office of Scientific AFOSR 84-0208 to the University the Air Force Office of Scientific AFOSR 88-0040 to the Florida
0 1990, Elsevier Science Publishers
associated
random
variables.
coherent
structure
function.
The ith component is said to be relevant if there exists at least one configuration of the other (n - 1) components such that the system is in the failed state if the ith component is in the failed state and in the functioning state otherwise. Below we assume that X is associated; that is, for every pair of coordinatewise nondecreasing functions ( f, g) defined on [wn + Iw, cov[f(X),
SW)]
ao.
(1)
It is wellknown that if the X, are mutually independent, then X is associated. The main inequality obtained in this note is cov[ xx,
- q(x),
‘k(x)]
> 0.
(2)
Note that the nonstrict version of (2) follows easily from the facts that EX, - \k(X) is coordinatewise nondecreasing in X, and X associated implies (1). The approach in Barlow and Proschan (1981) assumes and uses independence of the components in a crucial manner. As seen from the discussion above, association seems to be the natural condition; we prove ineauality (2) in this more general setting. It should be noted that the covari-
B.V. (North-Holland)
215
Volume
9, Number
STATISTICS
3
& PROBABILITY
ante inequality (2) by itself is not enough for proving the S-shaped property of the reliability function when the components are dependent. However, (2) is an important step towards obtaining such a result. Our approach is based on a result which states that if a bivariate dustribution exhibits positive quadrant dependence, then the uncrrelatedness implies independence. The result is an immediate consequence of a covariance formula due to Hoeffding (see Lehmann, 1966, Lemma 2). In general, the present approach relies directly on the notions of association, independence, and coherence. It is hoped that it will provide a better insight.
LEmERS
March
Due to assumptions CX,=n
1990
on \k made in the theorem,
j
‘k(X)=1
*
@x)=0.
and xx,=0 so that P[Cx,-P(X)=01 >P[CX;=O]
>, I!!JP[X,=O] 1
>o,
(6)
where the second inequality in (6) follows the association of X. Similarly,
from
2. Results P[CX,-‘k(X)=n-1]
>P[CX,=n]
Lemma. Suppose the pair (U, V) satisfies the positive quadrant dependence (PQD) condition, that is P[U>u,
V~u]>,P[u>u]P[V/u],
(3)
for every pair of reals (u, v). Then cov[U,
V] =o
*
U, V are independent.
0
Remark. Suppose the pair (U, V) is associated. Then from (1) it is clear that (U, V) satisfies PQD condition (3). Theorem. Let \k correspond to a system with n (>, 2) associated components X,, X,, . . . , X,, such that for every i, 0 < P[ X, = l] < 1. Further, suppose that q is coordinatewise nondecreasing and 0 < P[q(X) = 0] -C 1. Then cov[xX, - q(X), 9(X)] = 0 implies that only one out about of the n components is relevant for \k. Proof. In view of the lemma and remark above, CX, - \k(X) and 9(X) are independent. Thus P[‘k(X)=O,
From (6) and (7), it follows that the left sides of (4) and (5) are positive. Hence, P[CX,=n-1,
*(X)=0]
>O
(8)
and P[CX,=l,
\k(X)=lj
>O.
(9)
Inequality (9) implies that there is a component j such that the system works even if component j is the only working component. On the other hand, (8) implies the existance of a component whose failure causes system failure even if all other components are funtioning. Due to the nondecreasing character of 9, the latter component has to be j. Thus j is the only relevant component, as claimed. 0
References (4
xx,-*(x)=0]
=P[9(X)=l]P[CX,-*(x)=0]. 216
(7)
CT-*(X)=n-l]
=P[\k(X)=O]P[CX,--*(X)=n--I],
P[*(x)=l,
>O.
(5)
Barlow, R.E. and F. Proschan (1981), Statistical Theory of Reliability and Life Testing (To Begin With, Silver Spring, MD). Lehmann, E.L. (1966), Some concepts of dependence, Ann. Math. Statist. 37, 1137-1153.