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Statistics & Probability Letters 72 (2005) 179–185

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A note on comparisons among coherent systems withdependent components using signatures$

Jorge Navarroa,�, Jose M. Ruiza, Carlos J. Sandovalb

aFac. de Matematicas, Universidad de Murcia, 30100 Murcia, SpainbUniversidad Catolica San Antonio de Murcia, 30107 Murcia, Spain

Received 23 January 2004; accepted 7 December 2004

Abstract

We extend some ordering properties for coherent systems with independent and identically distributedcomponents given by Kochar et al. (Naval Res. Logist. 46 (1999) 507) to coherent systems with (possibly)dependent components by using the concept of Samaniego’s signatures.r 2005 Elsevier B.V. All rights reserved.

Keywords: Ordering; Coherent systems; k-out-of-n systems; Order statistics; Signatures

1. Introduction

Recently, some new results have been obtained on systems with dependent components (see Huand Hu, 1998; Papadatos, 2001; Rychlik, 1994, 1995, 2001). In this area, it is interesting tocompare systems from properties of their components. For example, Mi and Shaked (2002)showed that if the components are stochastically ordered, then the corresponding k-out-of-nsystems are stochastic ordered. If the random vectors ðX 1; . . . ;X nÞ and ðY 1; . . . ;Y nÞ represent thecomponents life lengths, and X ði;nÞ and Y ði;nÞ represent the ðn � i þ 1Þ-out-of-n systems life lengths(order statistics), Mi and Shaked’s result can be written as follows.

see front matter r 2005 Elsevier B.V. All rights reserved.

spl.2004.12.017

upported by Ministerio de Ciencia y Tecnologia under Grant BFM2003-02947.

ding author. Fax: +34968364182.

ress: jorgenav@um.es (J. Navarro).

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Theorem 1. If ðX 1; . . . ;X nÞpstðY 1; . . . ;Y nÞ; then X ði;nÞpstY ði;nÞ for i ¼ 1; . . . ; n:

We refer the reader to Shaked and Shanthikumar (1994) for the definitions of univariate andmultivariate orders used in this paper.A coherent system whose components life lengths are given by ðX 1; . . . ;X nÞ can be represented

as T ¼ FðX 1; . . . ;X nÞ; where F is a coherent life function (see Esary and Marshall (1970) orBarlow and Proschan (1975, p. 12) for the definitions and properties of coherent life functions).When the components are independent and identically distributed (i.i.d.), Samaniego (1985)

(see also Kochar et al., 1999) defined the signature of a coherent system T ¼ FðX 1; . . . ;X nÞ as thevector p ¼ ðp1; . . . ; pnÞ; where

pi ¼ PrðT ¼ X ði;nÞÞ, (1)

showing that

pi ¼# of orderings for which the ith failure causes system failure

n!. (2)

Samaniego (1985) also showed that a coherent system with i.i.d. components and signature p isa mixture of k-out-of-n systems with weights pi; and hence the system reliability can be written as

RT ðtÞ ¼ PrðTXtÞ ¼Xn

i¼1

piRði;nÞðtÞ, (3)

where Rði;nÞðtÞ ¼ PrðX ði;nÞXtÞ can be computed from

Rði;nÞðtÞ ¼Xi�1j¼0

n

j

� �Rn�jðtÞð1� RðtÞÞ j, (4)

where RðtÞ is the reliability function for the components.Other properties on signatures are given in Boland (2001) and Boland and Samaniego (2003).

Kochar et al. (1999) used these results to obtain some comparisons among coherent systems withi.i.d. components whose signatures are ordered. In this paper, we extend these results to coherentsystems with (possibly) dependent components.

2. Main results

In this paper, we assume that T ¼ FðX 1; . . . ;X nÞ is a coherent system with possibly dependentcomponents and absolutely continuous joint reliability function

Rðx1; . . . ;xnÞ ¼ PrðX 1Xx1; . . . ;X nXxnÞ.

First, we note that the definition of signature given in (1) extends to general coherent systems.Moreover, when the components have an absolutely continuous exchangeable joint distribution,(3) also holds and, hence, any coherent system is a mixture of k-out-of-n systems. Thus, we canobtain properties for coherent systems from properties of k-out-of-n systems and mixtures. Forexample, if all the k-out-of-n systems are DFR, then all the coherent systems with thesecomponents are DFR. We can use Theorem 4.B.10.d in Shaked and Shanthikumar (1994) toextend Theorem 1 as follows.

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Theorem 2. If ðX 1; . . . ;X nÞpstðY 1; . . . ;Y nÞ; then FðX 1; . . . ;X nÞpstFðY 1; . . . ;Y nÞ for any coherent

function F:

Unfortunately, in this case the signature cannot necessarily be computed from (2) and itdepends on Rðx1; . . . ;xnÞ: However, the signature does not depend on Rðx1; . . . ;xnÞ and (2) holdsfor the very important case when the components of the system are exchangeable (i.e.Rðx1; . . . ;xnÞ is symmetric in x1; . . . ; xn), since then all these orderings for the components have thesame probability, and hence

PrðX 1o � � �oX nÞ ¼ PrðXpð1Þo � � �oXpðnÞÞ

holds for any permutation p: Thus, all the signatures of coherent systems given in Kochar et al.(1999) and Shaked and Suarez-Llorens (2003) continue to hold.

Example 3. Samaniego (1985) showed in the i.i.d. components case the signature of T ¼

minðX 1;maxðX 2;X 3ÞÞ is ð13; 23; 0Þ: The same holds for any exchangeable joint distribution.

However, in general, it is obtained by noting that

p1 ¼ PrðX 1oX 2oX 3Þ þ PrðX 1oX 3oX 2Þ,

p2 ¼ 1� p1 and p3 ¼ 0; since

ðX 1oX 2oX 3Þ ) T ¼ X ð1Þ,

ðX 1oX 3oX 2Þ ) T ¼ X ð1Þ,

ðX 2oX 1oX 3Þ ) T ¼ X ð2Þ,

ðX 2oX 3oX 1Þ ) T ¼ X ð2Þ,

ðX 3oX 1oX 2Þ ) T ¼ X ð2Þ,

ðX 3oX 2oX 1Þ ) T ¼ X ð2Þ.

If the components are exchangeable, then we have an expression similar to (4) to compute Rði;nÞ:

Rði;nÞðtÞ ¼Xi�1j¼0

n

j

� �PrðX 1ot; . . . ;X jot;X jþ1Xt; . . . ;X nXtÞ. (5)

We note that the signatures associated to coherent systems can be seen as discrete distributions(pi ¼ PrðZ ¼ iÞ for i ¼ 1; 2; . . . ; n) and, hence, they can also be st (hr,lr)-ordered by using thegeneral definitions given in Shaked and Shanthikumar (1994). The following theorem extendsTheorem 3 in Kochar et al. (1999).

Theorem 4. Let p1 and p2 be the signatures of two coherent systems with exchangeable componentsT1 ¼ F1ðX 1; . . . ;X nÞ and T2 ¼ F2ðY 1; . . . ;Y nÞ: If ðX 1; . . . ;X nÞpstðY 1; . . . ;Y nÞ and p1pstp2; thenT1pstT2:

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Proof. If pk ¼ ðpk1; . . . ; pknÞ; k ¼ 1; 2; then p1pstp2 implies

Xn

i¼1

gðiÞp1ipXn

i¼1

gðiÞp2i

for any increasing function gðiÞ: As X ði;nÞpstX ðiþ1;nÞ; gðiÞ ¼ RXði;nÞðtÞ ¼ PrðX ði;nÞXtÞ is an increasing

function in i for all t and hence, from (3) and Theorem 1, we have

PrðT1XtÞ ¼Xn

i¼1

p1iRXði;nÞðtÞp

Xn

i¼1

p2iRXði;nÞðtÞp

Xn

i¼1

p2iRYði;nÞðtÞ ¼ PrðT2XtÞ,

which implies F1ðX 1; . . . ;X nÞpstF2ðY 1; . . . ;Y nÞ: &

The proof can also be obtained from Theorem 1.A.6 in Shaked and Shanthikumar (1994). Notethat the signatures do not depend on the joint distributions.The following theorem extends Theorem 4 in Kochar et al. (1999).

Theorem 5. Let p1 and p2 be the signatures of two coherent systems T1 ¼ F1ðX 1; . . . ;X nÞ andT2 ¼ F2ðX 1; . . . ;X nÞ with common exchangeable components. If X ði;nÞphrX ðiþ1;nÞ for i ¼ 1; . . . ; n �

1 and p1phrp2; then T1phrT2:

Proof. If t1ot2; we define aðiÞ ¼ Rði;nÞðt2Þ and bðiÞ ¼ Rði;nÞðt1Þ and we note that bðiÞand aðiÞ=bðiÞ are nondecreasing functions of i since X ði;nÞphrX ðiþ1;nÞ: Hence, if we applyLemma 1 in Kochar et al. (1999) to the distributions defined by the signatures by using p1phrp2;we obtainPn

i¼1 p2iRði;nÞðt2ÞPni¼1 p1iRði;nÞðt2Þ

X

Pni¼1 p2iRði;nÞðt1ÞPni¼1 p1iRði;nÞðt1Þ

(6)

for t1ot2; where pk ¼ ðpk1; . . . ; pknÞ; k ¼ 1; 2: Finally, (6) implies that RT2ðtÞ=RT1

ðtÞ isnondecreasing in t and hence T1phrT2: &

The following result extends Theorem 5 in Kochar et al. (1999).

Theorem 6. Let p1 and p2 be the signatures of two coherent systems T1 ¼ F1ðX 1; . . . ;X nÞ and

T2 ¼ F2ðX 1; . . . ;X nÞ with common exchangeable components. If X ði;nÞplrX ðiþ1;nÞ for i ¼ 1; . . . ; n �

1 and p1plrp2; then T1plrT2:

Proof. For j ¼ 1; 2; the density of Tj is

f jðtÞ ¼Xn

i¼1

pjif ði;nÞðtÞ.

If for c40; we consider the function

gðtÞ ¼ f 2ðtÞ � cf 1ðtÞ ¼Xn

i¼1

ðp2i � cp1iÞf ði;nÞðtÞ,

it is sufficient to prove that g has at most one change of sign from negative to positive as tincreases. Since p1plrp2; p2i=p1i is nondecreasing in i and hence the sequence fp2i � cp1ig has atmost one change of sign from negative to positive as i goes from 1 to n. Since, X ði;nÞplrX ðiþ1;nÞ;

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f ðiþ1;nÞðtÞ=f ði;nÞðtÞ is nondecreasing in t for any i ¼ 1; . . . ; n � 1: That is, f ðiÞðtÞ is TP2 in ði; tÞ andfrom Karlin (1968) we have that g has at most one change of sign from negative to positive as tincreases. This completes the proof. &

The proof can be also obtained from (3) and Theorem 1.C.11 in Shaked and Shanthikumar(1994).

Remark 7. We note that X ði;nÞpstX ðiþ1;nÞ holds for any multivariate reliability function butwe do not know if X ði;nÞplrX ðiþ1;nÞ or X ði;nÞphrX ðiþ1;nÞ holds in general. We also note thatTheorems 5 and 6 hold if we change T2 ¼ F2ðX 1; . . . ;X nÞ by T2 ¼ F2ðY 1; . . . ;Y nÞ

with ðX 1; . . . ;X nÞ¼stðY 1; . . . ;Y nÞ or for distributions which are discrete mixtures of k-out-of-nsystems.

The following result is similar to Theorem 6 in Kochar et al. (1999) but, in this case, we need toadd a stronger condition.

Theorem 8. Let T1 ¼ FðX 1; . . . ;X nÞ and T2 ¼ FðY 1; . . . ;Y nÞ be two coherent systems with thesame structure but different exchangeable components. If X ði;nÞphrY ðj;nÞ for all i; j ¼ 1; . . . ; n; thenT1phrT2:

The proof is obtained directly from (3) and Theorem 1.B.8 in Shaked and Shanthikumar (1994).In practice, it is not easy to check the hypotheses included in our theorems. However,

sometimes we can check them numerically. In the following example, we show that theseconditions hold in a specific Gumbel exponential model.

Example 9. If we consider coherent systems with four exchangeable components with the samejoint reliability function given by

Rðx1;x2; x3;x4Þ ¼ expð�x1 � x2 � x3 � x4 � 0:5x1x2x3x4Þ,

for xiX0; i ¼ 1; 2; 3; 4 (Gumbel distribution), then

Rð1;4ÞðtÞ ¼ e�4t�0:5t4 ,

Rð2;4ÞðtÞ ¼ 4e�3t � 3e�4t�0:5t4 ,

Rð3;4ÞðtÞ ¼ 6e�2t � 8e�3t þ 3e�4t�0:5t4

and

Rð4;4ÞðtÞ ¼ 4e�t � 6e�2t þ 4e�3t � e�4t�0:5t4

hold. Hence, X ði;nÞphrX ðiþ1;nÞ and X ði;nÞplrX ðiþ1;nÞ hold for i ¼ 1; 2; 3 (see Figs. 1 and 2,respectively). Thus, if the signatures are hr-ordered (lr), then the corresponding coherent systemsare also hr-ordered (lr).

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Fig. 2. Plots for f ði;4ÞðtÞ=f ðiþ1;4ÞðtÞ for i ¼ 1; 2; 3 and a Gumbel joint distribution. As they are decreasing, X ði;4ÞplrX ðiþ1;4Þ

holds for i ¼ 1; 2; 3:

Fig. 1. Hazard rates for k-out-of-4 systems ðk ¼ 1; 2; 3; 4Þ and Gumbel joint distribution.

J. Navarro et al. / Statistics & Probability Letters 72 (2005) 179–185184

Acknowledgements

The authors gratefully acknowledge the many helpful suggestions made by the referee and theassociate editor Prof. Boland during the preparation of the paper.

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