Stochastic ordering properties for systems with dependent identically distributed components

Authors


Correspondence to: Jorge Navarro, Facultad de Matemáticas, Universidad de Murcia, 30100 Murcia, Spain.

E-mail: jorgenav@um.es

Abstract

In this paper, we obtain ordering properties for coherent systems with possibly dependent identically distributed components. These results are based on a representation of the system reliability function as a distorted function of the common component reliability function. So, the results included in this paper can also be applied to general distorted distributions. The main advantage of these results is that they are distribution-free with respect to the common component distribution. Moreover, they can be applied to systems with component lifetimes having a non-exchangeable joint distribution. Copyright © 2012 John Wiley & Sons, Ltd.

1 Introduction

The study on ordering properties of coherent systems is a relevant subject in reliability and survival theories. The classical signature-based mixture representation obtained by Samaniego [1] (see also [2]) for systems with independent and identically distributed (IID) components was used by Kochar et al. [3] to obtain some stochastic ordering properties. These ordering properties are distribution-free; that is, they do not depend on the common component distribution. Samaniego's representation was extended by Navarro et al. [4] to the case of systems with exchangeable components, that is, systems with component lifetimes having a joint distribution invariant under permutations. Moreover, they obtain distribution-free ordering properties used to compare systems having different numbers of exchangeable components. Other ordering properties and bounds for systems were obtained in [5-14]. Unfortunately, Samaniego's representation does not necessarily hold when the components are not exchangeable (see Example 5.1 in [4]). In this case, some approximations are obtained in [15], and some particular ordering results have been obtained for specific coherent systems and component distributions (see, e.g., [16-19]).

In this paper, we study ordering properties of coherent systems with identically distributed (ID) components. Note that this case is more general than the exchangeable case. To this purpose, we obtain a representation of the system reliability inline image as a distorted function of the common component reliability inline image; that is, inline image, where h is an increasing function which depends on the system structure and the survival copula of the joint distribution of the component lifetimes. Then, we obtain ordering properties for distorted distributions and use these results to obtain ordering properties for coherent systems. As we use a different approach to that used in [4], these results also have interest for systems with exchangeable (or IID) components. Moreover, we study some stochastic orders not studied in [4]. Our results can also be applied to study general distorted distributions in other fields. Distorted functions have been the subject of a wider interest because of their use in the rank-dependent expected utility model (see [20, 21]). Denneberg [22] and Wang [23, 24] introduced distorted distributions in insurance pricing and financial risk managements. Wang and Young [25] connected these types of distributions with stochastic orders. Recently, Sordo and Suárez-Llorens [26] have obtained some new results about stochastic orderings of distorted distributions. Also, some ordering results were obtained in Section 3.2 of Khaledi and Shaked's work [27].

The rest of the paper is organized as follows. In Section 2, we give the main results of the paper with the general ordering results for distorted distributions and coherent systems with ID components. Specifically, we study properties of the usual stochastic order, hazard rate order, the reversed hazard rate order, the likelihood ratio order, the increasing convex order, the total time on test transform order, and the excess wealth order. In Section 3, we give some examples to show how to apply our general results to specific systems. These examples include systems with IID components and systems with dependent ID components. In Section 4, we give some conclusions and open questions for future research.

Throughout the paper, we use the terms increasing and decreasing in a wide sense, that is, a function g is increasing (decreasing) if g(x) ≤ g(y)() for all x ≤ y. Whenever we consider an expectation (or a conditional random variable), we assume that it exists.

2 The main results

Let us consider a coherent system with lifetime T = ϕ(X1, … ,Xn) based on possibly dependent components with lifetimes X1, … ,Xn, where ϕ is the structure function (see [29], Chapter 1). Let us assume that X1, … ,Xn are identically distributed (ID) with a common reliability function inline image for i = 1, … ,n. The component lifetimes X1, … ,Xn can be dependent, and this dependence will be represented by the joint reliability (or survival) function of (X1, … ,Xn)

display math

This function can be written using the well-known Sklar's copula representation as

display math(2.1)

where K is the survival or reliability copula (K is a multivariate distribution function with uniform marginal distributions in (0,1)). This representation is very convenient in this context because the different kinds of components are represented by the reliability function inline image and the dependence among the components is represented by the copula K (see, e.g., [28]).

It is well known (see, e.g., [29]) that the lifetime of a coherent system can be written as

display math(2.2)

where inline image is the lifetime of the series system with components in Pj, and P1, … ,Pr are the minimal path sets of the system. A path set is a set of indices P, such that if all the components in P work, then the system works. A minimal path set is a path set which does not contain other path sets. For example, the minimal path sets of T = min(X1,max(X2,X3)) are P1 = {1,2}and P2 = {1,3}. The minimal path sets only depend on the system structure function ϕ and vice versa. Then, the system reliability can be obtained by using the minimal path set representation (2.2) and the inclusion–exclusion formula as follows:

display math

This representation can be traced back to [30] (see also [31]. Notice that this representation is a generalized mixture representation; that is, inline image is a linear combination of reliability functions with positive and negative coefficients (see, e.g., [32]). Using this representation, we can obtain the following theorem which is the key to obtain the results included in this paper.

Theorem 1. Let T = ϕ(X1, … ,Xn) be the lifetime of a coherent system based on possibly dependent components with lifetimes X1, … ,Xn, having a common reliability function inline image. Then, the system reliability function can be written as

display math(2.3)

where h only depends on ϕ and on the survival copula of (X1, … ,Xn).

Proof. Recall that the system reliability can be written as

display math

Moreover, from Equation (2.1), the reliability function of the series system with components in P can be written as

display math

where xi = t if i ∈ P, and xi = 0 if i ∉ P. Therefore, the system reliability function can be written as Equation (2.3), where h is a function which only depends on P1, … ,Pr and on K.□

A similar representation holds for the respective distribution functions

display math

where g(z) = 1 − h(1 − z). Note that h and g depend on both ϕ and K, but they do not depend on inline image. Moreover, h (or g) is an increasing function in (0,1) from h(0) = 0 to h(1) = 1. If T is a series system, then h(u) = K(u, … ,u); that is, it is the diagonal section of the copula K. If K is exchangeable (i.e., permutation invariant) and the components are ID, then inline image is exchangeable, and from [31], we have

display math

where a = (a1, … ,an) is called the minimal signature of the system. In particular, in the IID case, K is the product copula, and the function h is a polynomial, called domination or reliability polynomial, given by inline image. In the general case, the function h in Equation (2.3) is not necessarily a polynomial, and then it can be called structure and dependence function (because it contains the information about both the system structure and the dependence among the components) or domination function.

For example, for the system with lifetime T = min(X1,max(X2,X3)), we have

display math(2.4)

where

display math

If K is exchangeable, K(u,u,1) = K(u,1,u) and then

display math

that is, its minimal signature is a = (0,2, − 1). In particular, in the IID case, K is the product copula (i.e., K(x,y,z) = xyz), and the domination polynomial is given by

display math

Our purpose is to compare the lifetimes of two coherent systems T1 = ϕ1(X1, … ,Xn) and T2 = ϕ2(Y1, … ,Ym) with structure functions ϕ1 and ϕ2 and with identically distributed component lifetimes X1, … ,Xn and Y1, … , Ym, having common continuous reliability functions inline image and inline image, respectively. If h1 and h2 are the respective domination functions, our results are of the form

display math

where A is certain class of functions and (1) and (2) are certain stochastic orders.

Now, we recall the definitions of the stochastic orders considered in this paper.

Definition 1. Let X and Y be two nonnegative random variables with respective distribution functions F and G and respective reliability functions inline image and inline image. The quantile function of F is defined by F − 1(p) = sup{x : F(x) ≤ p}. The quantile function of G is defined in a similar way.

  1. X is said to be smaller than Y in the usual stochastic order (denoted by X ≤STY ) if inline image for all t.

  2. X is said to be smaller than Y in the hazard rate order (denoted by X ≤HRY ) if inline image is increasing in t.

  3. X is said to be smaller than Y in the reversed hazard rate order (denoted by X ≤RHRY ) if G(t) ∕ F(t) is increasing in t.

  4. If X and Y have absolutely continuous distributions with respective probability density functions (PDF) f and g, then X is said to be smaller than Y in the likelihood ratio order (denoted by X ≤LRY ) if g(t) ∕ f(t) is increasing in t in the union of their supports.

  5. X is said to be smaller than Y in the increasing convex order (denoted by X ≤ICXY ) if

    display math
  6. X is said to be smaller than Y in the total time on the test transform order (denoted by X ≤TTTY ) if

    display math
  7. X is said to be smaller than Y in the dispersive order (denoted by X ≤DISPY ) if

    display math
  8. X is said to be smaller than Y in the excess wealth order (denoted by X ≤EWY ) if

    display math

The basic properties of these orders can be seen in [33]. If X and Y have absolutely continuous distributions, then X ≤HRY is equivalent to the ordering of their respective hazard (or failure) rate functions inline image and inline image. Moreover, this ordering can be interpreted in the reliability context by using the following equivalence

display math

that is, their respective residual lifetimes are stochastic (ST)-ordered. We also have (see [33] ,p. 17) the following equivalence

display math

Analogously, X ≤RHRY is equivalent to the ordering of their respective reversed hazard rate functions f ∕ F and g ∕ G. This ordering can be interpreted by using the following equivalence

display math

that is, their respective inactivity times (t − X | X < t) and (t − Y | Y < t) are ST-ordered. We also have (see [33] ,p. 37) the following equivalence

display math

Finally, the likelihood ratio order is equivalent to the ordering of their respective Glaser's functions, that is,

display math

and it can be interpreted in this context by using the following equivalence

display math

We also have (see [33] ,p. 43) the following equivalences

display math

and

display math

These orders are related as follows:

display math(2.5)

Note that the LR, HR, and RHR orders imply the ST order. When the ST order between X and Y does not hold, we can use weaker orderings for comparing X and Y. One option is to use the increasing convex order. It is well known (see [33] ,Section 4) that

display math

The ICX order appears in reliability as a useful tool for characterizing aging notions. For example, it is well known (see [33] ,Theorem 4.A.51) that a nonnegative random variable X is DMRL (IMRL) (i.e., X has a decreasing (increasing) mean residual life function inline image, see [29]) if, and only if,

display math

Another option in comparing two random variables when the stochastic order fails is to use the total time on test (TTT) transform order. This order satisfies

display math

When X and Y have the same mean, we have

display math

An important difference between the TTT and excess wealth (EW) orders is that the EW order is location independent, whereas the TTT transform order is location dependent. The excess wealth order is also connected with the stochastic order as follows. If uX and uY denote the right endpoints of the supports of X and Y, respectively, we have

display math

(see [34]). The dispersive order is the most commonly used order in comparing the variability of random variables. It is well known that

display math

For the properties of the dispersive order and the excess wealth order, see Sections 3.B and 3.C in [33]. The dispersive order and the excess wealth order are also relevant to reliability. For example, we have the following property. If X is a random variable with support of the form (a, ∞ ), where a > − ∞ , then X is IFR (DFR) (i.e., X has an increasing (decreasing) hazard rate function inline image, see [29]) if, and only if,

display math

and X is DMRL (IMRL) if, and only if,

display math

Now, we can obtain the main results of the paper. First, we compare a system with DID components to the system with the same structure and IID components having the same reliability function. The comparisons are distribution-free with respect to the common component reliability.

Theorem 2. Let T = ϕ(X1, … ,Xn) be the lifetime of a coherent system on the basis of DID components with lifetimes X1, … ,Xn, having a common reliability function inline image. Let TI = ϕ(Y1, … ,Yn) be the lifetime of a coherent system on the basis of IID components with lifetimes Y1, … ,Yn, having a common reliability function inline image. Let h and hI be their respective domination functions. Then, we have the following properties.

  1. T ≤STTI (ST) for all inline image if, and only if, h(u) ∕ hI(u) 1 () in (0,1).

  2. T ≤HRTI (HR) for all inline image if, and only if, h(u) ∕ hI(u) increases (decreases) in (0,1).

  3. T ≤RHRTI (RHR) for all inline image if, and only if, (1 − h(u)) ∕ (1 − hI(u)) increases (decreases) in (0,1).

  4. T ≤LRTI (LR) for all inline image if, and only if, inline image is convex (concave) in (0,1).

The proof is straightforward from Equation (2.3) and the equivalences for the orders given previously.

Next, we give a general result to compare two different coherent systems having DID components with the same component reliability function.

Theorem 3. Let T1 = ϕ1(X1, … ,Xn) be the lifetime of a coherent system on the basis of DID components with lifetimes X1, … ,Xn, having a common reliability function inline image. Let T2 = ϕ2(Y1, … ,Ym) be the lifetime of a coherent system on the basis of DID components with lifetimes Y1, … ,Ym, having a common reliability function inline image. Let h1 and h2 be their respective domination functions. Then, we have the following properties.

  1. T1 STT2 (ST) for all inline image if, and only if, h1(u) ∕ h2(u) 1 () in (0,1).

  2. T1 HRT2 (HR) for all inline image if, and only if, h1(u) ∕ h2(u) increases (decreases) in (0,1).

  3. T1 RHRT2 (RHR) for all inline image if, and only if, (1 − h1(u)) ∕ (1 − h2(u)) increases (decreases) in (0,1).

  4. T1 LRT2 (LR) for all inline image if, and only if, inline image is concave (convex) in (0,1).

The proof is straightforward from Equation (2.3) and from the equivalences for the orders given previously. Notice that it can be used to compare systems with different numbers of components. Moreover, it can be used to compare a system with its components (taking h1(u) = u). The proofs of (ii) and (iii) can also be obtained from Equation (2.3) and Propositions 3.6 and 3.8 in [27], respectively. We must say, here, that when (iv) is used, it might be quite difficult to obtain an explicit expression inline image. So, one might use a numerical approach to compute inline image. An alternative condition is given in the following result.

Theorem 4. Under the assumptions of Theorem 3, when h1 and h2 are differentiable, then T1 LRT2 (LR) for all absolutely continuous reliability function inline image if, and only if, inline image increases (decreases) in (0,1).

Proof. From Equation (2.3), the probability density function of T1 can be written as

display math

where inline image is the component density function. Analogously, for T2, we have

display math

Then, from the definition, T1 LRT2 holds if, and only if, inline image increases in (0, ∞ ), which is equivalent to inline image increases in (0,1).□

The proof can also be obtained from Equation (2.3) and Proposition 3.10 in [27]. Now, let us compare systems with the same structure and with DID components having the same copula but different common distributions. Similar ordering properties for systems with independent not necessarily identically distributed components were given in [35].

Theorem 5. Let inline image and inline image be the lifetimes of two coherent systems with the same structure and with DID component lifetimes having the same copula and common absolutely continuous reliability functions inline image and inline image, respectively. Let h be the common domination function, and let us assume that it is twice differentiable. Then, we have the following properties.

  1. If X1 STY1, then T1 STT2.

  2. If X1 HRY1 and uh ′ (u) ∕ h(u) decreases in (0,1), then T1 HRT2.

  3. If X1 RHRY1 and (1 − u)h ′ (u) ∕ (1 − h(u)) increases in (0,1), then T1 RHRT2.

  4. If X1 LRY1 and uh ′ ′ (u) ∕ h ′ (u) is nonnegative and decreasing in (0,1), then T1 LRT2.

  5. If X1 ICXY1 and h(u) is concave in (0,1), then T1 ICXT2.

Proof. The proof of (i) is immediate.

To prove (ii), note that from Equation (2.3), the hazard rate function of T1 can be written as

display math

where α(u) = uh ′ (u) ∕ h(u). Analogously, the hazard rate of T2 can be written as

display math

Then, if X1 HRY1, we have inline image. Moreover, from Equation (2.5), we have X1 STY1, that is, inline image. Hence, if α is decreasing in (0,1), we have

display math

Therefore, using the fact that α is nonnegative in (0,1), we have

display math

To prove (iii), note that from Equation (2.3), the reversed hazard rate function of T1 can be written as

display math

where β(u) = (1 − u)h ′ (u) ∕ (1 − h(u)). Analogously, the reversed hazard rate of T2 can be written as

display math

Then, if X1 RHRY1, we have inline image. Moreover, from Equation (2.5), we have X1 STY1, that is, inline image. Hence, if β is increasing in (0,1), we have

display math

Therefore, using the fact that β is nonnegative in (0,1), we have

display math

To prove (iv), note that from Equation (2.3), the Glaser's function inline image of T1 can be written as

display math

where γ(u) = uh ′ ′ (u) ∕ h ′ (u). Analogously, the Glaser's function η2 of T2 can be written as

display math

Then, if X1 LRY1, we have − f ′ (t) ∕ f(t)  − g ′ (t) ∕ g(t). Moreover, from Equation (2.5), we have X1 HRY1 and X1 STY1, that is, inline image and inline image. Hence, if γ is nonnegative and decreasing in (0,1), we have

display math

Therefore, we have η1(t) ≥ η2(t).

In order to prove (v), let us assume that X1 ICXY1 holds. From Theorem 2.1 of [36], it follows that

display math(2.6)

for all convex distortion function g. Equivalently, taking into account that inline image for all p ∈ (0,1) and that g(p) is a convex distortion function if and only if h(p) = 1 − g (1 − p) is a concave distortion function, we see that Equation (2.6) can be written as

display math(2.7)

for all concave distortion function h. Note that, given a concave distortion h and u ∈ (0,1] , the function

display math

is also a concave distortion. Thus, it follows from Equation (2.7) that

display math

for all concave distortion function hu and for all u ∈ (0,1], or, equivalently,

display math

for all concave distortion function h and for all u ∈ [0,1]. This is equivalent, from Lemma 4.2 of [37], to

display math

for all concave distortion function h and for all x ≥0. In particular, this property holds for the common domination function of systems T1 and T2, and hence, T1 ICXT2.□

It is easy to see that the conditions given in (ii)–(v) are not necessarily true for all the coherent systems. Hence, the HR, RHR, LR, and ICX orders are not necessarily preserved under the formation of coherent systems (or distorted distributions). For example, in the IID case, for the system with lifetime T = min(X1,max(X2,X3)), we have hI(u) = 2u2 − u3, and inline image is decreasing in (0,1). Hence, the HR order is preserved in the IID case for this particular system. However, the following example shows that the HR order is not necessarily preserved under the formation of coherent systems with IID components.

Example 1. Let us consider the coherent systems with lifetimes T1 = max(X1,min(X2,X3)) and T2 = max(Y1,min(Y2,Y3)), where X1,X2,X3 are IIDs with common reliability function

display math

for t ≥0, and Y1,Y2,Y3 are IIDs with common reliability function

display math

for t ≥0. It is easy to see that X1 HRY1. In both cases, the domination polynomial is hI(u) = u + u2 − u3. and hence. inline image is increasing-decreasing in (0,1). A straightforward calculation of the hazard rate functions h1 and h2 of the systems T1 and T2 gives that

display math

and

display math

Hence, T1 and T2 are not HR-ordered, and the HR order is not preserved in the IID case for this particular system structure.

Because the usual stochastic order implies that the increasing convex order and the reciprocal is not true, (v) of the previous theorem provides one way of ordering T1 and T2 when X1 and Y1 are not stochastically ordered. For the same reason (because the usual stochastic order implies the TTT order), it also becomes of interest to obtain conditions for ordering T1 and T2 under the condition X1 TTTX2. This is performed in the following theorem.

Theorem 6. Let inline image and inline image be the lifetimes of two coherent systems with the same structure and with DID component lifetimes having the same copula and common continuous reliability functions inline image and inline image respectively. Let us assume that they have 0 as the common left endpoint of their supports. Let us assume that the common domination function h is strictly increasing in (0,1). Then, we have the following properties.

  1. If X1 TTTY1 and h is convex, then T1 TTTT2.

  2. If T1 TTTT2 and h is concave, then X1 TTTY1.

The proof is obtained from Equation (2.3) and Theorem 2.9 and Example 2.10 in [38].

The following result compares the variability of the lifetimes of two coherent systems with the same structure.

Theorem 7. Let inline image and inline image be the lifetimes of two coherent systems with the same structure and with DID component lifetimes having the same copula and common continuous and strictly increasing reliability functions, inline image and inline image, respectively. Let h be the common domination function, and let us assume that it is continuous and strictly increasing in (0,1). Then, we have the following properties.

  1. X1 DISPY1 if, and only if, T1 DISPT2.

  2. If X1 EWY1 and h is concave, then T1 EWT2.

Proof. To prove (i), we first assume that X1 DISPY1 holds. Hence, using inline image and inline image for all p ∈ (0,1), it follows

display math

for all 0 < p < q < 1. Therefore,

display math

for all 0 < p < q < 1. Hence, using Equation (2.3), we get

display math

for all 0 < p < q < 1, and we have T1 DISPT2.

The reverse property in (i) is obtained in a similar way by using that from Equation (2.3); we have inline image and inline image.

To prove (ii), let us assume that X1 EWY1 holds. Equivalently, − X1 LIR − Y1 holds, where LIR denotes the location independent riskier (LIR) order (see [33] ,p. 165). From Theorem 2.1, (ii), of [37], it follows

display math

for all concave distortion function h,where inline image and inline image are the distribution functions of − X1 and − Y1,respectively. Equivalently, we have

display math

for all concave distortion function h. The change of variable t = − u yields

display math

for all concave distortion function h. By applying this property to the common domination function h of these systems, we get T1 EWT2.□

Finally, by combining the results given above, we obtain the following theorem which can be used to compare (in some orders) systems having different structures and DID components with different common distributions.

Theorem 8. Let inline image and inline image be the lifetimes of two coherent systems with DID component lifetimes having common absolutely continuous reliability functions inline image and inline imagerespectively. Let h1 and h2 be the respective domination functions, and let us assume that they are differentiable. Then, we have the following properties.

  1. If X1 STY1 and h1(u) ≤ h2(u) in (0,1), then T1 STT2.

  2. If X1 HRY1, h1(u) ∕ h2(u) increases in (0,1) and either inline image or inline image decreases in (0,1), then T1 HRT2.

  3. If X1 RHRY1, (1 − h1(u)) ∕ (1 − h2(u)) increases in (0,1) and either inline image or inline image increases in (0,1), then T1 RHRT2.

  4. If X1 LRY1 and

    display math(2.8)

    for 1 ≥ u1 ≥ u2 0, 1 ≥ v1 ≥ v2 0, and 1 ≥ u1 ∕ v1 ≥ u2 ∕ v2, then T1 LRT2.

Proof. The proof of (i) is immediate.

To prove (ii), note that if we assume that inline image decreases in (0,1), then from Theorem 5, we have

display math

Moreover, if h1(u) ∕ h2(u) increases in (0,1), from Theorem 3, (ii), we have

display math

Hence, T1 HRT2. The proof of the other case is similar.

The proof of (iii) is analogous to the proof of (ii) by using the results for the RHR order given in Theorems 3 and 5.

To prove (iv), let us assume X1 LRY1. Then, f ∕ g is decreasing. Moreover, we have X1 HR,STY1, that is, inline image is decreasing and inline image. Then, for s ≤ t and inline image, inline image, inline image, and inline image, we have 1 ≥ u1 ≥ u2 0, 1 ≥ v1 ≥ v2 0, and 1 ≥ u1 ∕ v1 ≥ u2 ∕ v2. Hence, from Equation (2.8), we have

display math

that is, the function inline image is nonnegative and decreasing. Therefore,

display math

is decreasing, and hence, T1 LRT2.□

Notice that if Equation (2.8) holds, then inline image increases (i.e., h1 is convex) in (0,1). Moreover, Equation (2.8) holds when inline image increases (h1 is convex) and inline image decreases (h2 is concave) in (0,1). However, note that Equation (2.8) might hold when inline image is not decreasing (see Example 2).

3 Examples

In this section, we show how to apply our theoretical results to specific systems and copulas. We consider both systems with IID components and systems with DID components having different dependence copulas. In the first example, we study a system under both options.

Example 2. Let us consider the coherent system with lifetime T = min(X1, max(X2,X3)). Then, from Equation (2.4), its domination function is

display math

In particular, in the IID case, hI(u) = 2u2 − u3. It is easy to see that hI(u) ∕ u is increasing, and hence, from Theorem 3, (ii), T ≤HRX1 when T has IID components for any common component reliability function. However, hI(u) is neither convex nor concave in (0,1), and hence, T and X1 are not necessarily LR-ordered. These results were obtained in [4] (see Systems 1 and 5 in Figures 2 and 3 of [4]) by using a different approach (through using signatures). Analogously, if we want to compare T with X1:2, whose domination function in the IID case is h1:2(u) = u2, we have that

display math

is a concave function, and hence, from Theorem 3, (iv), X1:2 LRT in the IID case for any component reliability function (see Systems 2 and 5 in Figure 3 of [4]). However, to compare T with other systems (different to a series system), we need to use the alternative condition given in Theorem 4. For example, let us consider the 2-out-of-3 system with lifetime X2:3 and domination function h2:3(u) = 3u2 − 2u3 (in the IID case). It is easy to see that inline image is increasing in (0,1), and hence, we have T ≤LRX2:3 in the IID case for any component reliability function.

As we have already mentioned, from Theorem 5, (ii), we obtain that the HR order is preserved under the formation of this system in the IID case; that is, if X1 HRY1, then

display math

when both systems have IID components. It is easy to see that inline image takes positive and negative values in (0,1); so, we do not know if the LR order is preserved. It is easy to see that Equation  (2.8) holds in the IID case for T1 = min(X1,X2,X3) and T2 = min(Y1,max(Y2,Y3)), and hence, we have T1 LRT2 whenever X1 LRY1.

Let us assume now that (X1,X2,X3) are dependent with a Farlie–Gumbel–Morgenstern (FGM) joint reliability function given by

display math

where α ∈ [ − 1,1], F is any marginal distribution function, and inline image. Then, the FGM survival copula is

display math(3.1)

and the domination function of the system T = min(X1,max(X2,X3)) is

display math

First, let us compare T with X1. It is easy to see that h(u) ∕ u is increasing for all α ∈ [ − 1,1]; hence, from Theorem 3, we obtain T ≤HRX1. However, it can be seen that h is neither convex nor concave; so, T and X1 are not LR-ordered. Analogously, it can be seen that T and X1 are not RHR-ordered because (1 − h(1 − u)) ∕ u is neither increasing nor decreasing in (0,1).

Now, let us compare T with TI, that is, the system with the same structure and IID components (or, equivalently, with the system obtained with the FGM copula when α = 0). It is easy to see that h(u) ∕ hI(u) is neither increasing nor decreasing in (0,1), and hence, T and TI are not HR-ordered. However, for α > 0 ( < ), h(u) ∕ hI(u) 1 () for all u ∈ (0,1), and hence, from Theorem 2, (i), we obtain TI STT (ST) for α > 0 ( < ).

Now let us compare the systems obtained with different values for the dependence parameter α. It is easy to see that h increases with α, and hence, we have Tα STTβ for α ≤ β, where Tα and Tβ are the system lifetimes obtained with the FGM survival copulas Kα and Kβ, respectively. However, hα(u) ∕ hβ(u) is neither increasing nor decreasing in (0,1), and hence, Tα and Tβ are not HR-ordered. Analogously, it can be seen that they are not RHR-ordered. Of course, then, they are not LR-ordered.

Finally, by plotting uh ′ (u) ∕ h(u), we see that it is decreasing for any α ∈ [ − 1,1], and hence, from Theorem 5, (ii), the HR order is preserved under the formation of this system, that is, if X1 HRY1, then

display math

when both systems have DID components having the FGM survival copula. However, (1 − u)h ′ (u) ∕ (1 − h(u)) is increasing-decreasing in (0,1) for any α ∈ [ − 1,1], and hence, the RHR order is not necessarily preserved. Analogously, it can be seen that uh ′ ′ (u) ∕ h ′ (u) takes positive and negative values when u ∈ (0,1), and hence, the LR order is not necessarily preserved.

In the following example, we show that our results can also be applied to systems with DID components having non-exchangeable copulas. Notice that, in this case, Samaniego's representation does not hold, and hence, we cannot use the ordering properties given in [4].

Example 3. Let us consider the system studied in the first example T = min(X1,max(X2,X3)), but with component lifetimes having the copula defined in Example 2 of [39] for n = 3, given by

display math(3.2)

where α ∈ [ − 0.5,0.5]. Hence, the domination function of T is

display math

First, let us compare T with X1. A straightforward calculation proves that

display math

where | 8u2 − 7u + 1 | 2 for u ∈ (0,1). Hence, (h(u) ∕ u) ′ 0, and h(u) ∕ u is increasing in (0,1) for all α ∈ [ − 0.5,0.5]. Then, from Theorem 3, (ii), we obtain T ≤HRX1. However, it can be seen that h is neither convex nor concave in (0,1). So, T and X1 are not LR-ordered. As the copula K is not exchangeable, these ordering properties cannot be obtained from the results given in [4] because the joint distribution of (X1,X2,X3) is not exchangeable.

Now, let us compare T with TI, that is, the system with IID components. It is easy to see that h(u) and hI(u) are not ordered in (0,1), and hence, T and TI are not ST-ordered. The same happens for the systems obtained with the copula given in (3.2) for two different values for the dependence parameter α in [ − 0.5,0.5].

Finally, by plotting uh ′ (u) ∕ h(u), we see that it is decreasing for any α ∈ [ − 0.25,0.5], and hence, from Theorem 5, (ii), we have that the HR order is preserved; that is, if X1 HRY1, then

display math

when both systems have DID components with the copula given in (3.2). However, we do not know if it is preserved for α ∈ [ − 0.5, − 0.25). Analogously, it can be seen that uh ′ ′ (u) ∕ h ′ (u) takes positive and negative values in (0,1) for α ∈ [ − 0.5,0.5]. Hence, the LR order is not necessarily preserved.

The domination polynomial of a series system with IID components is trivially convex. In the two following examples, we analyze this property for different dependence copulas.

Example 4. Let us consider the series system T1 = min(X1,X2,X3) with ID component lifetimes having an FGM survival copula given by Equation (3.1), where α ∈ [ − 1,1]. The domination function of the series system is equal to the diagonal section of the survival copula, that is,

display math

A straightforward computation shows that

display math

It is easy to see that | 5u3 − 10u2 + 6u − 1 | 1 for u ∈ (0,1). Hence, (α(5u3 − 10u2 + 6u − 1) + 1) 0 for α ∈ [ − 1,1], and therefore, h is convex in (0,1). Then, using Theorem 3, (iv), we have X1 LRT1. Analogously, if T2 = min(Y1,Y2,Y3) is another series system with ID component lifetimes having the FGM survival copula (3.1), such that X1 TTTY1, then from Theorem 6, (i), we have T1 TTTT2.

Example 5. Let us consider the series system T1 = min(X1,…,Xn) with ID component lifetimes having a Clayton–Oakes (CO) survival copula given by

display math(3.3)

where θ > 1. The parameter θ governs the strength of the positive dependence, and it is worth mentioning that θ = 1 represents the independence copula. The family given by Equation (3.3) has been widely studied in reliability and biostatistics literature. Then, the domination function of the series system is given by

display math

A straightforward computation shows that

display math

which is an increasing function in (0,1) for θ > 1 and n = 2,3, … . Therefore, h is a convex function in (0,1). Hence, using Theorem 3, (iv), we have X1 LRT1. Analogously, if T2 = min(Y1,…,Yn) is another series system with ID component lifetimes having the CO survival copula given in Equation (3.3), such that X1 TTTY1, then from Theorem 6, (i), we have T1 TTTT2.

In the IID case, it is easy to see that the domination polynomials of the series and parallel systems are convex and concave, respectively. The question is if there are other systems with IID components having convex or concave domination polynomials. The answer is given in the following example.

Example 6. Let us consider a system with IID components and lifetime

display math

Then, its domination function is given by h(u) = 2u3 − u4. A straightforward computation shows that h ′ ′ (u) = 12u − 12u2 > 0 for u ∈ (0,1), and hence, h is a convex function in (0,1). Then, from Theorem 6, (i), the TTT order is preserved; that is, if

display math

is the lifetime of a system having four IID components and X1 TTTY1, then T1 TTTT2.

Let us consider now the system T3 = max(X1,X2,min(X3,X4)) having four IID components. Then, the domination function is given by h(u) = 2u − 2u3 + u4. A straightforward computation shows that h ′ ′ (u) = − 12u + 12u2 < 0 for u ∈ (0,1), and hence, h is a concave function in (0,1). Then, if

display math

is the lifetime of a system having four IID components and X1 ICXY1, from Theorem 5, (v), we get T3 ICXT4. Analogously, using Theorem 7, (ii), if X1 EWY1, then T3 EWT4. Also, note that from Theorem 6, (ii), we have that if T3 TTTT4, then X1 TTTY1.

The domination functions for all the coherent systems with 1–4 IID components are given in Table 1. It can be seen that the only convex domination functions are that of the series systems and the one given in the preceding example. Analogously, the only concave domination functions are that of parallel systems and that given in the preceding example.

Table 1. Domination functions for all the coherent systems with 1–4 IID components.
NT = ϕ(X1, … ,Xn)h(u)
1X(1:1) = X1u
2X1:2 = min(X1,X2)u2
3X2:2 = max(X1,X2)2u − u2
4X1:3 = min(X1,X2,X3)u3
5min(X1,max(X2,X3))2u2 − u3
6X2:3 (2-out-of-3)3u2 − 2u3
7max(X1,min(X2,X3))u + u2 − u3
8X3:3 = max(X1,X2,X3)3u − 3u2 + u3
9X1:4 = min(X1,X2,X3,X4)u4
10max(min(X1,X2,X3),min(X2,X3,X4))2u3 − u4
11min(X2:3,X4)3u3 − 2u4
12min(X1,max(X2,X3),max(X2,X4))u2 + u3 − u4
13min(X1,max(X2,X3,X4))3u2 − 3u3 + u4
14X2:4 (2-out-of-4)4u3 − 3u4
15max(min(X1,X2),min(X1,X3,X4),min(X2,X3,X4))u2 + 2u3 − 2u4
16max(min(X1,X2),min(X3,X4))2u2 − u4
17max(min(X1,X2),min(X1,X3),min(X2,X3,X4))2u2 − u4
18max(min(X1,X2),min(X2,X3),min(X3,X4))3u2 − 2u3
19max(min(X1,max(X2,X3,X4)),min(X2,X3,X4))3u2 − 2u3
20min(max(X1,X2),max(X1,X3),max(X2,X3,X4))4u2 − 4u3 + u4
21min(max(X1,X2),max(X3,X4))4u2 − 4u3 + u4
22min(max(X1,X2),max(X1,X3,X4),max(X2,X3,X4))5u2 − 6u3 + 2u4
23X3:4 (3-out-of-4)6u2 − 8u3 + 3u4
24max(X1,min(X2,X3,X4))u + u3 − u4
25max(X1,min(X2,X3),min(X2,X4))u + 2u2 − 3u3 + u4
26max(X2:3,X4)u + 3u2 − 5u3 + 2u4
27max(X1,X2,min(X3,X4))2u − 2u3 + u4
28X4:4 = max(X1,X2,X3,X4)4u − 6u2 + 4u3 − u4

4 Conclusions

We have shown that the reliability function of a coherent system with dependent identically distributed components can be written as a distorted function of the common component reliability function. This representation is used to obtain some ordering properties for this kind of system. We have studied the main orders, but our approach can also be applied to other orders. Even more, our results can also be applied to general distorted distributions. We have applied this procedure to several examples which show some relevant properties. For example, we have seen that the HR order is not preserved under the formation of coherent systems with IID components.

The results obtained here prove several interesting properties. They also show some open problems for future research. For example, we do not know if other orders such as the LR are preserved under the formation of coherent systems. Analogously, we know that the domination polynomials of series systems with IID components are convex, and hence, the TTT order is preserved. However, we do not know if this property holds for series systems with DID components.

Acknowledgements

JN is partially supported by Ministerio de Ciencia y Tecnología de España under grant MTM2009-08311 and by Fundación Séneca (C.A.R.M.) under grant 08627/PI/08. YA is partially supported by Ministerio de Ciencia e Innovación under grant MTM2010-20774-C03-03. MAS and ASL are partially supported by Ministerio de Ciencia e Innovación under grant MTM2009-08326 and by Consejería de Economía, Innovación y Ciencia under grant P09-SEJ-4739.

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