Comparisons of series and parallel systems with components sharing the same copula

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1 APPLIED STOCHASTIC MODELS IN BUSINESS AND INDUSTRY Appl. Stochastic Models Bus. Ind. (2009) Published online in Wiley InterScience ( Comparisons of series and parallel systems with components sharing the same copula Jorge Navarro 1,, and Fabio Spizzichino 2 1 Facultad de Matemáticas, Universidad de Murcia, Murcia, Spain 2 Department of Mathematics, University La Sapienza, Rome, Italy SUMMARY The paper is devoted to study stochastic comparisons of series and parallel systems with vectors of component lifetimes sharing the same copula. We show that, under some conditions on the common copula, the series system with heterogeneous components is worse than the series system with homogeneous components having a common reliability function, which is equal to the average of the reliability functions of the heterogeneous components. However, we show that this property is not necessarily true for arbitrary copulas. We obtain similar properties for parallel systems and for general coherent systems. For these purposes, we introduce in our analysis the notion of the mean function of a copula. Copyright q 2009 John Wiley & Sons, Ltd. Received 11 April 2009; Revised 6 October 2009; Accepted 6 October 2009 KEY WORDS: series systems; stochastic comparisons; coherent systems; Schur-concave copulas; mean functions 1. INTRODUCTION The stochastic comparisons of the system lifetimes is a very relevant topic in engineering and survival studies. These comparisons can be used to choose the best system structure under different criteria or to study where to place the different components in a system structure. Several results can be seen in [1 16] and in the references therein. The reliability of a system depends on the structure of the system as well as on the multivariate distribution of the component lifetimes. Several papers deal with comparisons among systems with Correspondence to: Jorge Navarro, Facultad de Matemáticas, Universidad de Murcia, Murcia, Spain. jorgenav@um.es Contract/grant sponsor: Ministerio de Ciencia y Tecnología; contract/grant number: MTM Contract/grant sponsor: Fundación Séneca; contract/grant number: 08627/PI/08 Contract/grant sponsor: Progetto di Ricerca Università Sapienza 2008 Interazione e Dipendenza nei Modelli Stocastici Copyright q 2009 John Wiley & Sons, Ltd.

2 J. NAVARRO AND F. SPIZZICHINO the same distribution of lifetimes and with different structures. We are essentially interested, for a given structure, in the comparison among different distributions and, in particular, among different vectors of marginal distributions. It is useful to start such a study by concentrating our attention on the most basic structure, that is, by considering the case of series systems. A series system is a system without redundancy, i.e. a system which works whenever all of its components work. The reliability function of a series system lifetime with independent component lifetimes is the product of the reliability functions of its component lifetimes. Hence, the reliability of a series system with independent non-necessarily identically distributed (INID) component lifetimes is smaller than that of the series system with independent and identically distributed (IID) component lifetimes having a common reliability function equal to the average of the reliability functions of the heterogeneous components (an intuitive property). This property was obtained by Di Crescenzo [17] as a version of Parrondo paradox because a series system with heterogeneous components is worse than a series system with components randomly chosen from many copies of these heterogeneous components, i.e. in this case the heterogeneity is bad for the system. This property is reversed for parallel systems with independent components (also an intuitive property). However, in many practical situations, the components in a system are dependent because they share the same environment. In this situation one can expect that these properties would still hold when we compare systems with dependent components and when the random vectors of components lifetimes have the same copula (that is when they share the same dependence structure). In this paper we show that these properties are true under some conditions for the copula. However, we also show that they are not necessarily true for all copulas (a counterintuitive property). Moreover (both for the cases of series and of parallel systems) we provide some results to compare the systems containing heterogeneous-dependent components with the systems containing homogeneous-dependent components. The remainder of the paper is organized as follows. In Section 2 we introduce the notation and some preliminary results on copulas. The main results for series systems, parallel systems and general coherent systems are given in Sections 3 5, respectively. In Section 6, we give some examples to illustrate the use of our theoretical results. Some conclusions are given in Section 7. Throughout the paper, the terms increasing and decreasing are used in the non-strict sense, that is, a function g :R n R is increasing (decreasing) when g(x) ( ) g(y) for all x y, where x=(x 1, x 2,...,x n ), y=(y 1, y 2,...,y n ) and x y means that x i y i for i =1,2,...,n. 2. NOTATION AND PRELIMINARY RESULTS Let (X 1, X 2,...,X n ) be a random vector. We assume that X 1, X 2,...,X n are the lifetimes of the n components of a system and hence we assume X i 0fori =1,2,...,n. However,wewould like to note that some of the results given in this paper hold without this assumption. The joint reliability (or survival) function of (X 1, X 2,...,X n ) will be denoted by and the joint distribution function by F(x 1, x 2,...,x n )= P(X 1 >x 1, X 2 >x 2,...,X n >x n ) F(x 1, x 2,...,x n )= P(X 1 x 1, X 2 x 2,...,X n x n ).

3 COMPARISONS OF SERIES AND PARALLEL SYSTEMS We assume that F is continuous. The (marginal) reliability function of X i is given by F i (t)= P(X i >t)= F(t i ), where t i =(x 1, x 2,...,x n ), x i =t and x j =0for j =i. The (marginal) distribution function of X i is given by F i (t)= P(X i t)=1 F i (t). From Sklar s theorem (see, e.g. [18, p. 18]), the joint reliability function can be written as and the joint distribution function as F(x 1, x 2,...,x n )= K (F 1 (x 1 ), F 2 (x 2 ),...,F n (x n )) F(x 1, x 2,...,x n )=C(F 1 (x 1 ), F 2 (x 2 ),...,F n (x n )), where K is the survival copula and C is the connecting (or usual) copula. A copula is a multivariate distribution function of a random vector with uniform marginal distributions in (0,1). Note that the survival copula K is a copula, that is, it is a distribution function and it is not a reliability function. In fact, the survival copula of the random vector (X 1, X 2,...,X n ) is the distribution function of the random vector (F 1 (X 1 ), F 2 (X 2 ),...,F n (X n )), wheref i is the marginal reliability function of X i. The component lifetimes (X 1, X 2,...,X n ) are independent if and only if K =C =C I,where C I (u 1,u 2,...,u n )= n u i i=1 is the independent copula. We shall use the following known properties on copulas (see [18]). Lemma 2.1 If C is a copula, then C L C C U,where ( C L (u 1,u 2,...,u n )=max 1 n + n and for 0 u i 1andi =1,2,...,n. i=1 ) u i,0 C U (u 1,u 2,...,u n )=min(u 1,u 2,...,u n ) The functions C L and C U are called Fréchet-Hoeffding bounds or minimal and maximal copulas, respectively. Note that C U is always a copula but C L is only a copula when n =2. A vector (u 1,u 2,...,u n ) is said to be majorized by another vector (v 1,v 2,...,v n ) (written as (u 1,u 2,...,u n ) m (v 1,v 2,...,v n )) if n j=1 u j = n j=1 v j and i j=1 u j:n i j=1 v j:n for i =1,2,...,n 1, where x 1:n, x 2:n,...,x n:n are the components of (x 1, x 2,...,x n ) rearranged in an increasing order. This partial ordering is used below to define the concepts of Schurconcavity/convexity. Applied to different types of multivariate functions, the concepts of Schur-concavity/convexity give properties that are very relevant in the field of reliability (see [19 21] and references therein). In this paper, we make a specifically different use of these concepts of Schur-concavity/convexity as a possible property of the survival or connecting copulas of the lifetimes of the components in a system.

4 J. NAVARRO AND F. SPIZZICHINO Definition 2.2 Let g :R n R be a real-valued function, then g is Schur-concave (Schur-convex) if g(u 1,u 2,...,u n ) g(v 1,v 2,...,v n ) ( ) whenever (u 1,u 2,...,u n ) m (v 1,v 2,...,v n ). It is well known that the copulas C I,C L and C U are Schur-concave and that the unique Schurconvex copula is C L (see [18, p. 104]). If a copula is Schur-concave, then it is exchangeable, that is C(u 1,u 2,...,u n )=C(u σ(1),u σ(2),...,u σ(n) ) for all permutations σ. Therefore, non-exchangeable copulas are not Schur-concave. However, many exchangeable copulas are Schur-concave. For example, Durante and Sempi [22] proved that Archimedean copulas are Schur-concave. The Archimedean copulas are the copulas of the form ( n ) C(u 1,u 2,...,u n )= 1 i ), i=1 (u where :[0,1] [0,+ ) is a continuous strictly decreasing convex function with (1)=0 (and such that C is actually a copula). Durante and Papini [23] proposed a bivariate concept of weakly Schur-concave (weakly Schurconvex) copula. This concept can be extended to the multivariate case as follows. We write it for general real-valued functions. Definition 2.3 Let g :R n R be a real-valued function, then g is weakly Schur-concave (weakly Schur-convex) if g(u 1,u 2,...,u n ) g(u,u,...,u) ( ) for all (u 1,u 2,...,u n ),whereu = n i=1 u i /n. Of course, all the Schur-concave (Schur-convex) copulas are weakly Schur-concave (weakly Schur-convex). These concepts can be characterized using the concept of mean function. Thisis an old concept previously considered by different authors (see [24, 25] and the references therein). Definition 2.4 Let g :R n R be a real-valued function, then the mean function associated with g is any function m g :R n R such that g(u 1,u 2,...,u n )= g(z, z,...,z) for all (u 1,u 2,...,u n ),wherez =m g (u 1,u 2,...,u n ). Note that a function g might have several mean functions (e.g. when n =2 the copula C L has several mean functions since it is constant for u 1 +u 2 1). If g is increasing in every component, then m g is an increasing function. The weakly Schur-concave/convex functions (copulas) can be characterized as follows. The proof is immediate.

5 COMPARISONS OF SERIES AND PARALLEL SYSTEMS Lemma 2.5 If m g is the mean function associated with an increasing function g,theng is weakly Schur-concave (weakly Schur-convex) if and only if m g (u 1,u 2,...,u n ) 1 n n u i i=1 ( ). The mean function associated with the independent copula C I is ( n ) 1/n m CI (u 1,u 2,...,u n )= u i i=1 for all 0 u i 1andi =1,2,...,n, that is, it is the geometric mean. As the geometric mean is less or equal to the arithmetic mean, C I is weakly Schur-concave (actually it is Schur-concave). Analogously, the mean function associated with the function C L is m CL (u 1,u 2,...,u n )= 1 n n u i i=1 for all 0 u i 1andi =1,2,...,n, that is, it is the arithmetic mean. Thus C L is weakly Schurconstant, that is, it is both weakly Schur-convex and weakly Schur-concave (actually it is Schurconstant). The mean function associated with the maximal copula C U is m CU (u 1,u 2,...,u n )=min(u 1,u 2,...,u n ) for all 0 u i 1andi =1,2,...,n. Thus C U is weakly Schur-concave (actually it is Schur-concave). If C is Archimedean, then ( ) 1 n m C (u 1,u 2,...,u n )= 1 (u i ). n As mentioned above these are always the cases where C is Schur-concave and then, in particular, weakly Schur-concave. Remark 2.6 If C is a continuous copula with strictly increasing diagonal section δ(u)=c(u,u,...,u), then i=1 m C (u 1,u 2,...,u n )=δ 1 (C(u 1,u 2,...,u n )) (1) is a strictly increasing continuous mean function such that m C (u,u,...,u)=u for 0 u 1, where δ 1 is the inverse function of the diagonal section δ(u) of the copula. Natural applications of the diagonal sections of copulas in the fields of reliability and order statistics can be seen in [18, 26 28] and in the references therein. We shall use the following stochastic order. Its properties can be seen in [29]. Definition 2.7 A random variable X is said to be smaller than another random variable Y in the stochastic order (written as X ST Y ) if their distribution functions satisfy F X F Y.

6 J. NAVARRO AND F. SPIZZICHINO 3. MAIN RESULTS FOR SERIES SYSTEMS Let (X 1, X 2,...,X n ) be the random vector of the lifetimes of n components in a coherent system. For this vector we use the notation introduced in the preceding section. We denote by K the survival copula of (X 1, X 2,...,X n ) and by F i the marginal reliability function of X i for i =1,2,...,n. The series system lifetime is X 1:n =min(x 1, X 2,...,X n ) and its reliability function is given by where F 1:n (t) = P(X 1:n >t) = P(X 1 >t, X 2 >t,...,x n >t) = F(t,t,...,t) = K (F 1 (t), F 2 (t),...,f n (t)) = K (φ(t),φ(t),...,φ(t)), (2) φ(t)=m K (F 1 (t), F 2 (t),...,f n (t)) (3) and m K is the mean function of K. Notice that if m K is an increasing right-continuous function such that m K (0,0,...,0)=0 andm K (1,1,...,1)=1, then for any choice of F 1, F 2,...,F n,the function φ in (3) turns out to be a reliability function on [0,+ ). In this case, the function φ can be called the mean reliability function associated with K and F 1, F 2,...,F n.ifthesurvival copula K has a strictly increasing continuous diagonal, then φ can be obtained from (1). Thus we can obtain the following result for series systems with components sharing the same copula. Theorem 3.1 If (X 1, X 2,...,X n ) and (Y 1,Y 2,...,Y n ) have the same survival copula K with mean function m K, then X 1:n ST Y 1:n holds if and only if m K (F 1 (t), F 2 (t),...,f n (t)) m K (G 1 (t), G 2 (t),...,g n (t)) for all t, wheref i and G i are the reliability functions of X i and Y i, respectively. Proof From (2), the reliability function of X 1:n can be written as F 1:n (t) = K (F 1 (t), F 2 (t),...,f n (t)) = K (φ 1 (t),φ 1 (t),...,φ 1 (t)), where φ 1 (t)=m K (F 1 (t), F 2 (t),...,f n (t)). Analogously, for the reliability function of Y 1:n we have G 1:n (t) = K (G 1 (t), G 2 (t),...,g n (t)) = K (φ 2 (t),φ 2 (t),...,φ 2 (t)), where φ 2 (t)=m K (G 1 (t), G 2 (t),...,g n (t)). Hence, as K is an increasing function, F 1:n G 1:n if and only if φ 1 (t) φ 2 (t) for all t.

7 COMPARISONS OF SERIES AND PARALLEL SYSTEMS When the survival copula is equal to the independent copula K =C I (i.e. when the components are independent), we have that X 1:n ST Y 1:n if and only if (F 1 (t)f 2 (t)...f n (t)) 1/n (G 1 (t)g 2 (t)...g n (t)) 1/n (a well-known result). However, for n =2 and the minimal copula K =C L,wehaveX 1:2 ST Y 1:2 if and only if F 1 (t)+ F 2 (t) 2 G 1(t)+G 2 (t). 2 In the following theorem we give a general result for series systems whose vectors of component lifetimes have different copulas. Theorem 3.2 If (X 1, X 2,...,X n ) and (Y 1,Y 2,...,Y n ) have the survival copulas K X and K Y such that K X K Y and m K X (F 1 (t), F 2 (t),...,f n (t)) m KY (G 1 (t), G 2 (t),...,g n (t)) (4) for all t, wheref i is the reliability function of X i and G i is the reliability function of Y i,then X 1:n ST Y 1:n. Proof From (2), (4) and K X K Y,wehave F 1:n (t) = K X (F 1 (t), F 2 (t),...,f n (t)) = K X (φ 1 (t),φ 1 (t),...,φ 1 (t)) K X (φ 2 (t),φ 2 (t),...,φ 2 (t)) K Y (φ 2 (t),φ 2 (t),...,φ 2 (t)) = K Y (G 1 (t), G 2 (t),...,g n (t)) = G 1:n (t), where φ 1 (t)=m K X (F 1 (t), F 2 (t),...,f n (t)) and φ 2 (t)=m KY (G 1 (t), G 2 (t),..., G n (t)). In particular, from Lemma 2.1, we obtain that series systems improve with dependency, that is, X L 1:n ST X 1:n ST X U 1:n, where X1:n L and XU 1:n represent the lifetimes of the series systems with component lifetimes having the same marginal reliability functions as X 1, X 2,...,X n and having survival copulas C L and C U, respectively. Although X1:n L is not a proper series system lifetime when n>2 (asc L is not a copula), its reliability function C L (F 1 (t), F 2 (t),...,f n (t)) can be used as a lower bound for the reliability function of X 1:n. As an immediate consequence of Theorem 3.1, we can easily obtain the following result about comparisons between heterogeneous and homogeneous series systems.

8 J. NAVARRO AND F. SPIZZICHINO Theorem 3.3 If (X 1, X 2,...,X n ) and (Y 1,Y 2,...,Y n ) have the same survival copula K with mean function m K, then X 1:n ST Y 1:n holds if and only if m K (F 1 (t), F 2 (t),...,f n (t)) G(t) for all t, wheref i is the reliability function of X i and G is the reliability function of Y i for i =1,2,...,n. Note that if φ 1 (t)=m K (F 1 (t), F 2 (t),...,f n (t)) is a reliability function, then X 1:n = ST Z 1:n, where (Z 1, Z 2,...,Z n ) has survival copula K and marginal reliability functions equal to φ 1 (t). The following result proves that the series systems improve when their component lifetimes are identically distributed whenever the survival copula is weakly Schur-concave. Theorem 3.4 If (X 1, X 2,...,X n ) and (Y 1,Y 2,...,Y n ) have the same survival copula K, K is weakly Schurconcave (weakly Schur-convex), F i is the reliability function of X i and G = n i=1 F i /n is the reliability function of Y i for i =1,2,...,n, thenx 1:n ST Y 1:n ( ST ). Proof From (2), we have F 1:n (t) = K (F 1 (t), F 2 (t),...,f n (t)) = K (φ 1 (t),φ 1 (t),...,φ 1 (t)) K (G(t), G(t),...,G(t)) ( ) = G 1:n (t), where φ 1 (t)=m K (F 1 (t), F 2 (t),...,f n (t)) and m K is the mean function associated with K and where the inequality is obtained using that if K is weakly Schur-concave (weakly Schur-convex) then, from Lemma 2.5, φ 1 (t) G(t) ( ) holds. Many copulas are weakly Schur-concave (see [22]). Thus, in many situations, a series system with heterogeneous components improves if we replace its components by ID components with common reliability functions equal to the average of the reliability functions of the heterogeneous components, that is, Parrondo paradox holds for series systems with components having a weakly Schur-concave survival copula. However, Example 6.1 proves that this property is not necessarily true for any copulas. In fact, as already indicated by Theorem 3.4, it can be reversed if the survival copula is a weakly Schur-convex copula different from the minimal copula C L (see Example 6.2). The conditions in the preceding results can be simplified in the case of Archimedean copulas. For example, condition (4) reduces to ( 1 1 X n n i=1 ) ( 1 X (F i ) 1 Y n n i=1 ) Y (G i ), where X and Y are the generators of the Archimedean copulas K X and K Y, respectively.

9 COMPARISONS OF SERIES AND PARALLEL SYSTEMS 4. MAIN RESULTS FOR PARALLEL SYSTEMS In the following theorems we give the results for the parallel systems similar to that obtained in the preceding section for the series systems. We omit the proofs. They can be obtained by using the fact that for the parallel system lifetime X n:n =max(x 1, X 2,...,X n ), the distribution function is given by F n:n (t) = P(T t) = P(X 1 t, X 2 t,...,x n t) = F(t,t,...,t) = C(F 1 (t), F 2 (t),...,f n (t)) (5) = C( (t), (t),..., (t)), (6) where C is the connecting copula, F i is the distribution function of X i, (t)=m C (F 1 (t), F 2 (t),...,f n (t)) and m C is the mean function of the copula C. Theorem 4.1 If (X 1, X 2,...,X n ) and (Y 1,Y 2,...,Y n ) have the same connecting copula C with mean function m C, then X n:n ST Y n:n holds if and only if m C (F 1 (t), F 2 (t),...,f n (t)) m C (G 1 (t), G 2 (t),...,g n (t)) for all t,wheref i is the distribution function of X i and G i is the distribution function of Y i. The proof is given in the Appendix. Theorem 4.2 If (X 1, X 2,...,X n ) and (Y 1,Y 2,...,Y n ) have the connecting copulas C X and C Y such that C X C Y,and m CX (F 1 (t), F 2 (t),...,f n (t)) m CY (G 1 (t), G 2 (t),...,g n (t)) for all t, wheref i is the distribution function of X i and G i is the distribution function of Y i,then X n:n ST Y n:n. Theorem 4.3 If (X 1, X 2,...,X n ) and (Y 1,Y 2,...,Y n ) have the same connecting copula C with mean function m C,thenX n:n ST Y n:n holds if and only if m C (F 1 (t), F 2 (t),...,f n (t)) G(t) for all t, wheref i is the distribution function of X i and G is the distribution function of Y i for i =1,2,...,n. Theorem 4.4 If (X 1, X 2,...,X n ) and (Y 1,Y 2,...,Y n ) have the same connecting copula C, C is weakly Schurconvex (weakly Schur-concave), F i is the distribution function of X i and G =(1/n) n i=1 F i is the reliability function of Y i for i =1,2,...,n, thenx n:n ST Y n:n ( ST ).

10 J. NAVARRO AND F. SPIZZICHINO Hence a parallel system with heterogeneous components having a weakly Schur-concave copula gets worse if we replace its components by ID components with common distribution functions equal to the average of the distribution functions of the heterogeneous components. Example 6.1 proves that this property is not necessarily true for any copulas and Theorem 4.4 proves that it can be reversed if the connecting copula is a weakly Schur-convex copula different from the minimal copula C L (see Example 6.2). 5. MAIN RESULTS FOR GENERAL COHERENT SYSTEMS In this section we show how the preceding results can be extended to coherent systems different from series or parallel systems. From [30], we know that the reliability function of a coherent system can be written as a linear combination of the reliability functions of the series systems obtained from its component lifetimes. Therefore, if T =Ψ(X 1, X 2,...,X n ) is the lifetime of a coherent system with a structure described by the function Ψ and with component lifetimes (X 1, X 2,...,X n ) having the survival copula K, then its reliability function can be written as F T (t)= p(f 1 (t), F 2 (t),...,f n (t)), where F i is the reliability function of X i and p is a function that only depends on K and Ψ. For example, for the coherent system lifetime T =min(x 1,max(X 2, X 3 )) we obtain F T (t) = P(X 1 >t, X 2 >t)+ P(X 1 >t, X 3 >t) P(X 1 >t, X 2 >t, X 3 >t) = F(t,t,0)+ F(t,0,t) F(t,t,t) = K (F 1 (t), F 2 (t),1)+ K (F 1 (t),1, F 3 (t)) K (F 1 (t), F 2 (t), F 3 (t)) = p(f 1 (t), F 2 (t), F 3 (t)), (7) where p(x, y, z)= K (x, y,1)+ K (x,1, z) K (x, y, z). The function p can be called the domination function. It will be denoted by p K,Ψ when it is necessary to stress its dependence on the functions K and Ψ. If the components are independent, then p is a polynomial that only depends on the system structure and it is called the structure reliability function in [31]. If the components are IID, then the polynomial p is called the domination polynomial. It is easy to see that, in the general case, the function p is increasing in every component, p(0,0,...,0)=0 and p(1,1,...,1)=1. Of course, it is a copula if the system is a series system (actually it is the survival copula). However, in the general case, it is not necessarily a copula and hence it might have different properties. Despite this, it is easy to see that the preceding results for series systems can also be applied for a general coherent system by replacing the survival copula K by the domination function p = p K,Ψ. They can be stated as follows. The proofs are analogous. Theorem 5.1 If T 1 =Ψ(X 1, X 2,...,X n ) and T 2 =Ψ(Y 1,Y 2,...,Y n ) are the lifetimes of two coherent systems and (X 1, X 2,...,X n ) and (Y 1,Y 2,...,Y n ) have the same survival copula K, then T 1 ST T 2 holds if and only if m p (F 1 (t), F 2 (t),...,f n (t)) m p (G 1 (t), G 2 (t),...,g n (t))

11 COMPARISONS OF SERIES AND PARALLEL SYSTEMS for all t, wheref i and G i are the reliability functions of X i and Y i, respectively, and m p is the mean function associated with the domination function p obtained from K and Ψ. The proof is given in the Appendix. Theorem 5.2 If T 1 =Ψ 1 (X 1, X 2,...,X n ) and T 2 =Ψ 2 (Y 1,Y 2,...,Y n ) are the lifetimes of two coherent systems with domination functions p 1 and p 2 such that p 1 p 2 and m p1 (F 1 (t), F 2 (t),...,f n (t)) m p2 (G 1 (t), G 2 (t),...,g n (t)) for all t, wheref i is the reliability function of X i and G i is the reliability function of Y i,then T 1 ST T 2. Theorem 5.3 If T 1 =Ψ(X 1, X 2,...,X n ) and T 2 =Ψ(Y 1,Y 2,...,Y n ) are the lifetimes of two coherent systems and (X 1, X 2,...,X n ) and (Y 1,Y 2,...,Y n ) have the same survival copula K,thenT 1 ST T 2 holds if and only if m p (F 1 (t), F 2 (t),...,f n (t)) G(t) for all t,wheref i is the reliability function of X i, G is the reliability function of Y i for i =1,2,...,n and m p is the mean function associated with the domination function p obtained from K and Ψ. Theorem 5.4 If T 1 =Ψ(X 1, X 2,...,X n ) and T 2 =Ψ(Y 1,Y 2,...,Y n ) are the lifetimes of two coherent systems, (X 1, X 2,...,X n ) and (Y 1,Y 2,...,Y n ) have the same survival copula K, the domination function p K,Ψ obtained from K and Ψ is weakly Schur-concave (weakly Schur-convex), F i is the reliability function of X i and G = n i=1 F i /n is the reliability function of Y i for i =1,2,...,n, then T 1 ST T 2 ( ST ). 6. EXAMPLES In the first example we use the copula given in Example 2 of Durante and Sempi [22] to show that the conclusion of Theorem 3.4 does not necessarily hold when the copula is neither weakly Schur-concave nor weakly Schur-convex. Example 6.1 Let X 1:2 and Y 1:2 be two series system lifetimes with component lifetimes having the following joint survival copula uv/2 if 0 u 1 2,0 v 1 2 u(3v 1)/2 if 0 u 1 2, 1 2 <v 1 K DS (u,v)=. v(3u 1)/2 if 1 2 <u 1,0 v 1 2 (uv +u +v 1)/2 if 1 2 <u 1, 1 2 <v 1

12 J. NAVARRO AND F. SPIZZICHINO Durante and Sempi [22] proved that K DS is exchangeable but it is not Schur-concave. In fact, it is not weakly Schur-concave as and it is not weakly Schur-convex as K DS ( 2 9, 2 3 )= 9 81 > 8 81 = K DS( 4 9, 4 9 ) K DS ( 1 4, 1 2 )= < = K DS( 3 8, 3 8 ). Therefore, if X 1 and X 2 have different reliability functions F 1 and F 2 with support (0, ), and Y 1 and Y 2 are ID with common reliability function (F 1 + F 2 )/2, then X 1:2 and Y 1:2 are not necessarily ST-ordered. For example, if X 1 and X 2 have exponential distributions with means 1and 1 2, respectively, then G 1:2(0.2)=0.522>0.519= F 1:2 (0.2) and G 1:2 (0.4)=0.217<0.227= F 1:2 (0.4), that is, X 1:2 and Y 1:2 are not ST-ordered (see Figure 1). Similar results can be obtained if we consider parallel systems with connecting copula equal to K DS. In the second example we use the copula given in Durante and Papini [23, p. 1382] to show that the conclusion of Theorem 3.4 can be reversed when the copula is weakly Schur-convex. In this case, the series system with heterogeneous components is ST-better than the associated series system with homogeneous components (a counterintuitive property). Example 6.2 Let X 1:2 and Y 1:2 be two series system lifetimes with component lifetimes having the following joint survival copula: min(u,v 1 2 ) if 0 u 1 2, 1 2 <v 1 K DP (u,v)= min(u 1 2,v) if 1 2 <u 1,0 v 1. 2 max(u +v 1,0) otherwise Figure 1. Reliability functions F 1:2 (continuous line) and G 1:2 (dashed line) for the series systems obtained in Example 6.1. Note that they are not ST-ordered.

13 COMPARISONS OF SERIES AND PARALLEL SYSTEMS Durante and Papini [23] proved that K DP is weakly Schur-convex and that it is neither Schurconvex nor weakly Schur-concave. Hence, from Theorem 3.4, X 1:2 ST Y 1:2 whenever Y 1 and Y 2 are ID with common reliability H such that H G =(F 1 + F 2 )/2, where F i is the reliability function of X i for i =1,2. A similar result can be obtained from Theorem 4.4 if we consider parallel systems with connecting copula equal to K DP. The following example, inspired by some arguments in [24], points out a special application of the notion of mean function in our context. Example 6.3 We consider here the case of a coherent system formed with n components, all of them of the same type. The components at our disposal, however, are not new: they are used and they are of different ages, namely x 1, x 2,...,x n.iff denotes the reliability function of the component lifetimes when they are new and if the survival copula K does not depend on x 1, x 2,...,x n, then we have that the reliability function of the lifetime T of the system made with the used components is ( ) F(x 1 +t) F T (t)= p, F(x 2 +t),..., F(x n +t), F(x 1 ) F(x 2 ) F(x n ) where p depends on the structure of the system and on the copula of the components, but it does not depend on x 1, x 2,...,x n.ifm p is the mean function of p, then F T (t)= p (φ(t),φ(t),...,φ(t)), where ( ) F(x 1 +t) φ(t)=m p, F(x 2 +t),..., F(x n +t). F(x 1 ) F(x 2 ) F(x n ) Let us denote by T the lifetime of the analogous system with new components. Hence, from Theorem 5.3, T ST T for all x 1, x 2,...,x n 0 if and only if F(t) φ(t) for all t 0 and all x 1, x 2,...,x n 0. (8) We notice that, for a fixed system and for a fixed copula such that m p (x, x,...,x)= x for all x, the condition (8) is equivalent to the new better than used (NBU) property of the components in the system (i.e. F(t) F(x +t)/f(x) for all x,t 0). In fact, if (8) holds, then taking x 1 = x 2 = =x n = x, we obtain F(t) φ(t) ( ) F(x +t) F(x +t) F(x +t) = m p,,..., F(x) F(x) F(x) F(x +t) =. F(x)

14 J. NAVARRO AND F. SPIZZICHINO Conversely, if F is NBU, then F(t) F(x i +t)/f(x i ) holds for all t 0 andallx 1, x 2,...,x n 0. Then using m p as an increasing function, we have ( ) F(x 1 +t) φ(t) = m p, F(x 2 +t),..., F(x n +t) F(x 1 ) F(x 2 ) F(x n ) m p (F(t), F(t),...,F(t)) = F(t). Finally, the following example is related to the application of our results to systems with structure functions, different from series and parallel system structures. Example 6.4 Let T 1 =min(x 1,max(X 2, X 3 )) and T 2 =min(y 1,max(Y 2,Y 3 )) be two coherent system lifetimes with component lifetimes having survival copula equal to the maximal copula, K =C U. Then, from (7), the reliability of T 1 is given by F T1 (t)= p(f 1 (t), F 2 (t), F 3 (t)), where F i is the reliability function of X i, i =1,2,3, and p(x, y, z)=min(x, y)+min(x, z) min(x, y, z) for 0 x, y, z 1. As p(z, z, z)= z, the mean function associated with p is m p (x, y, z)= p(x, y, z). The fact that K =C U means that the lifetimes are increasing deterministic functions one of the other. This implies that, if all Y i have the same reliability G for i =1,2,3, then T 2 =Y 1 =Y 2 =Y 3 Figure 2. Reliability functions F T1 (continuous line) and F T2 (dashed line) for the series systems obtained in Example 6.4. Note that they are not ST-ordered.

15 COMPARISONS OF SERIES AND PARALLEL SYSTEMS (with probability one). Therefore, if Y i has reliability function G for i =1,2,3, then T 1 ST T 2 ( ST ) if and only if G(t) min(f 1 (t), F 2 (t))+min(f 1 (t), F 3 (t)) min(f 1 (t), F 2 (t), F 3 (t)) ( ). It is easy to see that p is neither weakly Schur-concave nor weakly Schur-convex and hence T 1 and T 2 are not necessarily ST-ordered when G =(F 1 + F 2 + F 3 )/3. For example, if the component lifetimes have reliability functions F 1 (t)=e t (exponential), F 2 (t)=(1+t) 2 and F 3 (t)=(1+ t/2) 3 (Pareto type II) for t 0, and G =(F 1 + F 2 + F 3 )/3, then F T1 (1)=0.296<0.305= F T2 (1) and F T1 (2)=0.125>0.124= F T2 (2), that is, T 1 and T 2 are not ST-ordered (see Figure 2). However, note that if the component lifetimes in T 1 are ST-ordered, then the reliability of the system is equal to the reliability of the worst component and hence and T 1 ST T 2 hold. F T1 (t)=min(f 1 (t), F 2 (t), F 3 (t)) (F 1 (t)+ F 2 (t)+ F 3 (t))/3= F T2 (t) 7. DISCUSSIONS AND CONCLUDING REMARKS We have obtained some procedures to compare stochastically series, parallel and general coherent systems with independent or dependent components. Our results are based on copula representations and some related concepts such as weakly Schur-concavity/convexity and mean function. The copulas are used to model the dependence of the components in the system. Of course, the case of systems with dependent components is very relevant in practice. The results for general coherent systems are based on the associated domination functions which are not copulas but have some similar properties. For these purposes we introduce in our analysis the notions of mean functions of a copula and of domination functions. The results in this paper are theoretical but they can be applied to any real industrial systems with series, parallel or other coherent structures. We must say that if the structure of a coherent system is complex, then it would be difficult to analyze the corresponding domination function. Moreover, in order to apply our results to data obtained from real systems, it is necessary to develop the corresponding inference procedures. This work is left for future research projects. APPENDIX Proof of Theorem 4.1 If (X 1, X 2,...,X n ) and (Y 1,Y 2,...,Y n ) have the same connecting copula C with mean function m C,then and F n:n (t) = C(F 1 (t),...,f n (t)) = C(m C (F 1 (t),...,f n (t)),...,m C (F 1 (t),...,f n (t))) G n:n (t) = C(G 1 (t),...,g n (t)) = C(m C (G 1 (t),...,g n (t)),...,m C (G 1 (t),...,g n (t))).

16 J. NAVARRO AND F. SPIZZICHINO Hence X n:n ST Y n:n holds if and only if m C (F 1 (t), F 2 (t),...,f n (t)) m C (G 1 (t), G 2 (t),...,g n (t)) for all t. Proof of Theorem 5.1 If T 1 =Ψ(X 1, X 2,...,X n ) and T 2 =Ψ(Y 1,Y 2,...,Y n ) are two coherent systems and their components lifetimes have the same survival copula K,then and F T1 (t) = p(f 1 (t),...,f n (t)) = p(m p (F 1 (t),...,f n (t)),...,m p (F 1 (t),...,f n (t))) G n:n (t) = p(g 1 (t),...,g n (t)) = p(m p (G 1 (t),...,g n (t)),...,m p (G 1 (t),...,g n (t))), where m p is the mean function associated with the domination function p obtained from K and Ψ. Hence, as p is increasing, T 1 ST T 2 holds if and only if m p (F 1 (t), F 2 (t),...,f n (t)) m p (G 1 (t), G 2 (t),...,g n (t)) for all t. ACKNOWLEDGEMENTS The authors wish to thank the anonymous reviewers for several helpful comments. We are especially grateful to the reviewer who suggested the proof of the NBU property in Example 6.3. JN is partially supported by Ministerio de Ciencia y Tecnología under grant MTM and Fundación Séneca under grant 08627/PI/08 and FS is partially supported by Progetto di Ricerca Università Sapienza 2008 Interazione e Dipendenza nei Modelli Stocastici. REFERENCES 1. Navarro J, Balakrishnan N. Study of some measures of dependence between order statistics and systems. Journal of Multivariate Analysis 2010; 101(1): Barlow RE, Proschan F. Statistical Theory of Reliability and Life Testing. Reliability Models. International Series in Decision Processes. Series in Quantitative Methods for Decision Making. Holt, Rinehart and Winston, Inc.: New York, Montreal, Que., London, Kochar S, Mukerjee H, Samaniego FJ. The signature of a coherent system and its application to comparison among systems. Naval Research Logistics 1999; 46: Navarro J, Shaked M. Hazard rate ordering of order statistics and systems. Journal of Applied Probability 2006; 43: Dugas MR, Samaniego FJ. On optimal system designs in reliability-economics frameworks. Naval Research Logistics 2007; 54: Bairamov I, Arnold BC. On the residual lifelengths of the remaining components in an n k +1 out of n system. Statistics and Probability Letters 2008; 78: Bhattacharya D, Samaniego FJ. On the optimal allocation of components within coherent systems. Statistics and Probability Letters 2008; 78:

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