Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

Preservation of ILR and IFR aging classes in sums of dependent random variables

Preservation of ILR and IFR aging classes in sums of dependent random variables INTRODUCTIONIn different applied probability fields, like reliability theory or actuarial sciences, various concepts of aging have been proposed and studied to characterize and classify (in terms of “aging classes”) the distributions of random lifetimes of components, devices or individuals. Apart for the usefulness in the understanding of the intrinsic meaning of aging and its different effects on lifetimes, these classes serve for a number of purposes. They can be used, for example, to provide bounds for survival probabilities or expected residual lifetimes, or to define optimal replacement policies under different constrains. They can also serve to evaluate risks in insurance contracts. We refer the reader to the monograph Barlow and Proschan1 for a detailed description of the main aging classes and their applications in reliability modeling. Some among the most well‐known of these classes are recalled in the next section.Because of their usefulness in the applicative fields mentioned above, one can find in the literature a list of papers which study these classes and their properties. Several of them deal with their preservations under different stochastic operations, such as the closures with respect to construction of coherent systems, mixtures or sums. Concerning sums of random lifetimes, it is well known that the increasing failure rate (IFR) aging class is preserved under convolutions, that is, if X and Y are two independent random variables with distributions in the IFR class, then also X+Y has an IFR distribution (see, e.g., Barlow and Proschan1(p. 100)). On the contrary, the decreasing failure rate (DFR) class is not preserved under convolutions. These properties can be different for other transformations. For example, the DFR class is preserved under the formation of mixtures but the IFR class is not, and both of them are not preserved under the formation of coherent systems with i.i.d. components. However, the New Better than Used (NBU) class is preserved for these systems (see Barlow and Proschan1(p. 187)), but it is not preserved when the independence assumption is not satisfied (see Navarro et al.2). Analogously, the increasing likelihood ratio (ILR) class (equivalent to logconcavity of the densities) is preserved (with some conditions) by the formation of parallel and series systems (see Navarro and Shaked3), while the preservation with respect to convolutions was proved by Karlin and Proschan4 (see also Belzunce et al.5(p. 7)). Conditions for such preservations in distorted distributions and systems were recently obtained in Navarro et al.,2 and additional results for convolutions can be found in Alimohammadi et al.6 Properties of conditional distributions of sums are given in Pellerey and Navarro.7It must be pointed out that for almost all the results available in the literature dealing with preservation of aging classes under sums, the independence among summands is required. However, since in general the sums reduce the uncertainty, it is expectable to observe sums that are “more” IFR (or more ILR) than their summands, also when the assumption of independence is unsatisfied (in particular, this intuition seems especially true for negatively dependent random variables).The purpose of the present paper is to investigate this insight in more detail, in order to provide new conditions for preservation of some of the most well known aging classes under sums of dependent random variables. The new statements can be added to similar ones recently obtained in Navarro and Sarabia8 for the IFR/DFR classes. The results described here are mainly based on preservation properties for distorted distributions and C‐convolutions, which are copula representations for the distributions of sums of dependent random variables (see Section 2). In particular, such representations, as well as other tools, are used here to provide new conditions for the closure under sums of IFR and ILR aging classes. Some properties for the limiting behavior of the failure rate of the sum are provided as well, together with new conditions for the closure under maximum of dependent IFR and ILR lifetimes.The rest of the paper is structured as follows. Basic definitions and some preliminary results are introduced in Section 2. The main results for ILR and IFR classes are placed in Sections 3 and 4, respectively. Illustrative examples and counterexamples dealing with a specific kind of sums modeled with relevation transforms are given in Section 5. Similar results, but dealing with preservation of ILR and IFR aging classes with respect to maximum of non‐independent variables, are presented in Section 6.Throughout the paper the notions increasing and decreasing are used in a wide sense, that is, they mean non‐decreasing and non‐increasing, respectively.PRELIMINARIESAging classesFirst we recall the following definition.1DefinitionWe say that f:ℝn→ℝ+ is logconcave (logconvex) if logf(λx+(1−λ)y)≥(≤)λlogf(x)+(1−λ)logf(y)for all λ∈(0,1) and all x,y∈ℝn.Given a nonnegative random variable (lifetime) X with distribution function F(t)=Pr(X≤t) and survival (reliability) function F‾(t)=1−F(t)=Pr(X>t), let us denote with f(t)=F′(t) for t≥0 (zero elsewhere) its probability density function (pdf), and with r(t)=f(t)/F‾(t), for t≥0, its failure (or hazard) rate function, assuming they exist (from now on, whenever we use f or r, we are tacitly assuming that X has an absolutely continuous distribution). Then we say that:X (or F) is ILR (DLR) if f(t) is logconcave (logconvex) for t≥0;X (or F) is IFR (DFR) if F‾(t) is logconcave (logconvex) for t≥0;X (or F) is NBU (NWU) if F‾(t)F‾(s)≥F‾(t+s) for all s,t≥0.The IFR (DFR) property is equivalent to have an increasing (decreasing) failure rate function. It is also equivalent to have Xs≥STXt (≤ST) for all t≥s≥0, where ≥ST represents the usual stochastic order (see Shaked and Shanthikumar9) and Xt=(X−t|X>t) represents the residual lifetime of X at time t. Analogously, the NBU (NWU) property is equivalent to have X≥STXt (≤ST) for all t≥0. The ILR (DLR) class is equivalent to have Xs≥LRXt (≤LR) for all t≥s≥0, where ≥LR represents the likelihood ratio order (see Shaked and Shanthikumar9). The ILR and DLR classes are also equivalent to the respective monotonicity properties of the function η(t)=−f′(t)/f(t) for t≥0. Clearly, since the likelihood ratio order implies the usual stochastic order, ILR⇒IFR⇒NBU and, when the support of X is (l,∞), DLR⇒DFR⇒NWU.The IFR, NBU, and ILR classes represent the usual (positive or natural) aging properties where the older units have worse performance than the younger ones. The reverse scenario holds for the dual classes DFR, NWU, and DLR.It must be pointed out that, even if some notions like IFR and NBU usually refer to non‐negative variables, on the contrary the ILR notion also applies to variables that are not necessarily positive, and some of the statements described below hold true even in this case.Some additional (related) properties, defined here, will be considered through the following sections. Given a nonnegative random variable X, and denoted with R(t)=−logF‾(t)=∫0tr(s)ds its cumulative failure rate (or hazard rate) function, then we say that:X (or F) is IFRA (increasing in failure rate average) if R(t)/t is increasing for t≥0;X (or F) is LFR (logconcave in the failure rate) if r(t) is logconcave for t≥0;X (or F) is LCFR (logconcave in the cumulative failure rate) if R(t) is logconcave for t≥0.While the IFRA class is well‐known, being the largest class closed with respect to construction of coherent systems (see Barlow and Proschan1), the LCFR and LFR classes have been considered few times in the literature. Their usefulness in describing aging properties has been pointed out in Pellerey et al.,10 where they have been used to describe sufficient conditions for a nonhomogeneous Poisson processes to satisfy specific monotonicity properties. Note that, in Pellerey et al.,10 the LFR and LCFR classes were denoted as (A.3) and (A.2), respectively, and their relationships with the ILR and IFR classes have been fully described. These relationships are summarized in Table 1. The dual classes (describing negative aging rather positive aging) can be considered as well.1TABLERelationships among positive aging classesLFR⇒ILR⇒IFRA⇓⇓⇓LCFR⇒IFR⇒NBUApart for applications in reliability theory, aging classes can be of interest also in other contexts. The following simple result shows the relevance of the ILR property for the estimation of the parameter in a scale family ℱG of a nonnegative random variable. Recall that, given a distribution function G, we say that F belongs to the scale family of distributions ℱG determined by G if F(t)=G(θt) for all t, where θ>0 is the parameter of the model. Some examples are the exponential, the Weibull (with a fixed shape parameter) and the Pareto (with a fixed shape parameter) distributions.1PropositionIf X is a nonnegative random variable with a distribution function F∈ℱG and G is absolutely continuous and ILR, then the maximum likelihood estimator (MLE) for θ is unique.ProofIf X1,…,Xn are i.i.d. with distribution function F, then the likelihood function for θ is ℓ(θ)=f(x1)…f(xn)=θng(θx1)…g(θxn),where g is the pdf of G. To maximize ℓ in (0,∞) is equivalent to maximize ℓ˜(θ)=log(ℓ(θ))=nlog(θ)+∑i=1nlog(g(θxi))for θ>0. Then ℓ˜′(θ)=nθ+∑i=1nxig′(θxi)g(θxi).As G is ILR, then g′/g is decreasing and so ℓ′(θ) is strictly decreasing in (0,∞) for all x1,…,xn>0. Therefore, ℓ(θ) is strictly concave. Thus it has a unique maximum value in (0,∞).The class of ILR distributions contains many of the most commonly used parametric distributions, thus it is a rich and flexible nonparametric class of distributions. Further, the MLE for the density exists and can be computed with readily available algorithms (see, e.g., the review in Walther11).The following statement, which is theorem 1 in Block et al.,12 will be also used in the following sections. It essentially affirms that, in the limit, the hazard rate of a convolution converges to the smallest of the hazards of the summands.2PropositionBlock et al.12Let X and Y be two independent random variables with hazard rates rX and rY, respectively. Assume that limt→∞rX(t)=a, limt→∞rY(t)=b with 0<a<b<∞ and that rX is bounded. Then the failure rate rS of the convolution S=X+Y satisfies limt→∞rS(t)=a.C‐convolutionsFirst, we recall here the notion of C‐convolutions, introduced in Cherubini et al.,13 that will be used in the following sections to prove some of the main results. Let (X,Y) be two possibly dependent random variables with joint distribution F(x,y)=Pr(X≤x,Y≤y).From copula theory (see, e.g., Durante and Sempi14 or Nelsen15), it is well known that this joint distribution can be written as F(x,y)=C(F(x),G(y)),where F(x)=Pr(X≤x) and G(y)=Pr(Y≤y) are the marginal distribution functions of X and Y, respectively, and where C is a copula function. This representation (by means of a copula C) is unique when F and G are continuous (by the Sklar's theorem, see e.g., Durante and Sempi14). Also note that given a copula C, the right‐hand side of this expression provides a proper bivariate distribution function for any marginal distribution functions F and G.Then, from Cherubini et al.13 (see also Navarro and Sarabia8), the distribution function H of the sum S=X+Y can be obtained as H(t)=∫−∞∞f(x)Pr(Y≤t−x|X=x)dx=∫−∞∞f(x)∂1C(F(x),G(t−x))dx,where ∂1C represents the partial derivative of C with respect to its first argument. If X and Y are nonnegative, then the limits in the integral reduces to 0 and t, respectively. If F−1 is the inverse function of F, then this expression can be rewritten as H(t)=∫01∂1C(u,G(t−F−1(u)))du.This distribution is called the C‐convolution and it is represented as H=F∗CG in Cherubini et al.13 If C is the product copula, we get the usual expression for the convolution (i.e., the distribution of the sum of two independent random variables).In some models it is better to use survival functions instead of distribution functions. For them, we have the expression1H‾(t)=Pr(S>t)=∫−∞∞f(x)∂1C^(F‾(x),G‾(t−x))dx,where H‾=1−H, F‾=1−F, and G‾=1−G are the survival functions of S, X, and Y, respectively, and where C^ is a copula, called survival copula, that allows to represent the joint survival function of (X,Y) as F‾(x,y)=Pr(X>x,Y>y)=C^(F‾(x),G‾(y)).In particular, for nonnegative random variables Equation (1) reduces to2H‾(t)=F‾(t)+∫0tf(x)∂1C^(F‾(x),G‾(t−x))dx.The exponential distribution plays a central role in reliability theory and survival analysis, representing units without aging. If both marginal distributions coincide and are exponentials, that is, if F‾(t)=G‾(t)=e−λt for t≥0 and λ>0, then Equation (2) reduces to3H‾(t)=e−λt+∫e−λt1∂1C^(v,e−λt/v)dv=q‾(e−λt)for t≥0, where q‾(u)=u+∫u1∂1C^v,uvdvfor u∈[0,1]. This kind of representations are called distorted distributions, and they are studied in the following subsection. A similar representation can be obtained when just G is an exponential distribution (see theorem 2.2 in Navarro and Sarabia8). In the bivariate case such a representation can be stated as follows.3PropositionIf (X,Y) has survival copula C^, F‾ is strictly decreasing in its support (0,∞) and G‾(t)=exp(−λt)=ψ(F‾(t)) for t≥0 and λ>0, where ψ(u)=exp(−λF‾−1(u)) for 0<u<1, then the survival function of S=X+Y can be written as H‾(t)=q‾(F‾(t)) with4q‾(u)=u+∫u1∂1C^v,ψ(u)ψ(v)dvfor u∈[0,1].ProofFirst we note that F‾ can be written as F‾(t)=ψ−1(exp(−λt)) for all t≥0, where ψ−1(u)=F‾(−log(u)/λ) is strictly increasing for u∈(0,1). Hence, exp(−λt)=ψ(F‾(t)) for t≥0, where ψ is a strictly increasing distortion function. Then, from (2), we getH‾(t)=F‾(t)+∫0tf(x)∂1C^(F‾(x),exp(λx−λt))dx=F‾(t)+∫0tf(x)∂1C^F‾(x),ψ(F‾(t))ψ(F‾(x))dxfor all t≥0. Finally, by doing the change v=F‾(x), we conclude the proof.1RemarkThe representation based on function ψ considered above is important in order to get formula (4) for the distortion function q‾ (see next subsection). For specific copulas and functions ψ, we can obtain explicit expressions for q‾ useful to study the preservation of IFR and ILR aging classes (see Example 4), while numerical approximations of the solutions of (4) can be used in other cases. Note that the distortion representation obtained in the preceding proposition can also be expressed in terms of G‾ (exponential model) if we are able to get an explicit expression of ψ−1 (the inverse function of ψ). Thus, F‾(t)=ψ−1(exp(−λt)) and H‾(t)=q‾(F‾(t))=q‾(ψ−1(exp(−λt)))=q‾2(G‾(t))for t≥0, where q‾2(u)=q‾(ψ−1(u)) for u∈[0,1]. This approach is used in Example 4.Distorted distributionsThe distorted distributions were introduced in the theory of choice under risk (see Yaari16). They will be used in the next sections to prove the main results on closure with respect to sums of dependent random lifetimes. This is their formal definition.2DefinitionWe say that Fq is a distorted distribution function from a distribution function F if Fq(t)=q(F(t)) for all t, where q:[0,1]→[0,1] is a distortion function, that is, q is continuous, increasing, and satisfies q(0)=0 and q(1)=1.We will use the notation Fq=q∘F for the composition. Note that the respective survival functions satisfy a similar relationship, that is, one can write F‾q=q‾∘F‾, where q‾(u)=1−q(1−u) for u∈[0,1] is another distortion function called dual distortion function.Clearly, the representations in (3) and in Proposition 3 are distortion representations for H‾. There are others well known models that can be represented as distorted distribution, like the proportional hazard rate (PHR) Cox model with q‾(u)=uθ for θ>0, or the proportional reversed hazard rate (PRHR) model with q(u)=uθ for θ>0. It can also be proved that the distributions of coherent systems with identically distributed components, are also distortions of the common components' distribution function (see, e.g., Navarro et al.2).We say that an aging class 𝒞 is preserved by a distortion function q if Fq=q∘F belongs to 𝒞 for all F in 𝒞. For example, it is easy to see that the PHR model preserves the IFR and DFR classes. The preservation of aging classes under distortions, such as for the ILR/DLR, IFR/DFR, NBU/NWU, and IFRA/DFRA classes, were studied in Navarro et al.2 In particular, in that paper (see remark 2.3) it is proved that if the IFR class is preserved, then the NBU and IFRA classes are preserved as well. An extension of the result obtained there for the IFR/DFR classes is stated in the following proposition.4PropositionIf Fq=q∘F for a differentiable distortion function q and an absolutely continuous distribution function F, and α(u)=uq‾′(u)/q‾(u) for u∈[0,1], then the following conditions are equivalent:i) The IFR (DFR) class is preserved by q; ii) The IFR (DFR) class is preserved by q for the standard exponential distribution; iii) The function α is decreasing (increasing) in (0,1).ProofWe are going to prove the proposition for the IFR class (the proof for the DFR class is analogous). First, we note that the hazard rate function rq of Fq can be obtained as5rq(t)=q‾′(F‾(t))q‾(F‾(t))f(t)=α(F‾(t))r(t),where r=f/F‾ is the hazard rate of F. Clearly, (i) implies (ii).To prove that (ii) implies (iii), we note that for the standard exponential distribution r(t)=1 for t≥0. Then we know that rq(t)=α(F‾(t))=α(e−t)is increasing for t≥0. Therefore, α is decreasing in [0,1].Finally, to prove that (iii) implies (i), we note that if α is decreasing, then α(F‾(t)) is increasing in t. This function is also nonnegative. Therefore, if r is increasing, then rq is increasing as well, since from (5) it is the product of two nonnegative increasing functions.The preceding proposition shows that the IFR and DFR classes are both preserved with respect to a distortion q if and only if the corresponding α is constant, which leads to the PHR model. In the other cases, if the IFR is preserved, then the DFR class is not (and vice versa). It may also happen that neither of them are preserved (when α is not monotone).For the ILR class we have the following new result, to be added to the one already described in Navarro et al..2 To provide its statement, we need the following definition.3DefinitionWe say that a function ϕ:ℝ→ℝ+ preserves logconcave functions if ϕ∘f is logconcave for any logconcave function f .For example, the functions ϕc(t)=tc preserve logconcave functions for any c>0.5PropositionIf Fq=q∘F for a differentiable distortion function q and q‾′ preserves logconcave survival functions, then the ILR class is preserved by q.ProofFirst, we note that logfq can be written as log(fq(t))=log(q‾′(F‾(t)))+log(f(t))for all tin the support of F.Note that if F is ILR, then F is also IFR, that is, F‾ is logconcave. Hence, we have that q‾′∘F‾ is logconcave. As, by assumption, f is logconcave, we have that fq is logconcave (since logfq is the sum of two concave functions).It should be noted here that q‾′ is not a distortion function. Also note that we do not have necessary and sufficient conditions for the preservation of the ILR class (as we have for the IFR/DFR classes). However, Proposition 5 holds for not necessarily positive variables too. Other sufficient conditions for the preservation of ILR class can be found in Navarro et al..2Similar conditions can be obtained for the classes LFR and LCFR. They can be stated as follows. The proofs are analogous, and therefore omitted.6PropositionIf Fq=q∘F for a differentiable distortion function q and an absolutely continuous distribution function F, and if the function α in Proposition 4 preserves logconcave survival functions, then the LFR class is preserved by q.7PropositionIf Fq=q∘F for a distortion function q and an IFR distribution function F and the function β=−logq‾ preserves logconcave survival functions, then Fq is LCFR. In particular, under this condition for β, the LCFR class is preserved by q.We must say that it is not easy to determine if a function ϕ preserves logconcave functions. A result similar to Proposition 4 can be obtained for increasing functions ϕ.The distorted distributions can also be used to get stochastic comparisons, see for example, Navarro et al..17 They can be stated as follows. The definitions and main properties of the stochastic (ST), hazard rate (HR), reversed hazard rate (RHR), and likelihood ratio (LR) orders can be found in Belzunce et al.5 and Shaked and Shanthikumar.98PropositionIf Ti has the survival function q‾i(F‾(t)) and the distribution function qi(F(t)) for i=1,2, then the following properties hold:(i) T1≤STT2 for all F if and only if q‾2≥q‾1 (or q2≤q1) in (0,1); (ii) T1≤HRT2 for all F if and only if q‾2/q‾1 decreases in (0,1); (iii) T1≤RHRT2 for all F if and only if q2/q1 increases in (0,1); (iv) T1≤LRT2 for all absolutely continuous distribution functions F if and only if q‾2′/q‾1′ decreases (or q2′/q1′ increases) in (0,1).In particular, from (5), the condition in (ii) can be replaced by α2(u)≤α1(u) for all u∈(0,1], where αi(u)=uq‾i′(u)/q‾i(u) for i=1,2.PRESERVATION RESULTS FOR THE ILR CLASSIn general, if X1,…,Xn are ILR, then one cannot affirm that also S=X1+⋯+Xn is ILR (see, e.g., Sections 4 and 5.1 below). Conditions for this preservation property are discussed here. In the first result we study the preservation of the ILR class when (X1,…,Xn) is a random vector having a logconcave probability density function. To this purpose we need Prékopa's theorem (see Prékopa18) which is stated in the following lemma.1LemmaPrékopa's theoremSuppose that g:ℝm×ℝk→[0,∞) is a logconcave function and thath(x)=∫ℝkg(x,y)dyis finite for each x∈ℝm. Then h is logconcave on ℝm.Now we are ready to state the following preservation result.1TheoremLet (X1,…,Xn) be a random vector having a logconcave pdf f. Then S=X1+⋯+Xn is ILR.ProofLet us consider the random vector (Y1,…,Yn) with Y1=S and Yj=Xj for j=2,…,n. Then the pdf g of (Y1,…,Yn) is g(y1,…,yn)=f(y1−y2−⋯−yn,y2,…,yn)for all y1,…,yn. Therefore, g is also logconcave, since logg(λy+(1−λ)z)=logg(λy1+(1−λ)z1,…,λyn+(1−λ)zn)=logf(λ(y1−y2−⋯−yn)+(1−λ)(z1−z2−⋯−zn),λy2+(1−λ)z2,…,λyn+(1−λ)zn)≥λlogf(y1−y2−⋯−yn,y2,…,yn)+(1−λ)logf(z1−z2−⋯−zn,z2,…,zn)=λlogg(y)+(1−λ)logg(z)for all λ∈(0,1) and all y,z∈ℝn. Finally, we note that the pdf h of S can be obtained as h(x)=∫ℝn−1g(x,y2,…,yn)dy2…dyn,thus, from Lemma 1, we have that h is logconcave.It must be pointed out that Theorem 1, as well as the following two statements, applies to any vector of random variables, not necessarily non‐negative. Multivariate normal, Wishart, Dirichlet, uniform distributions over compact and convex sets and elliptically contoured distributions are examples of logconcave densities (see Hu and Li19 and Walther11).As an immediate consequence of the preceding theorem we obtain the following result, proved in Karlin and Proschan.41CorollaryIf X1,…,Xn are independent and ILR, then X1+⋯+Xn is ILR.The preceding theorem can be extended to linear combinations as follows.2TheoremLet (X1,…,Xn) be a random vector having a logconcave pdf f. Then S=a1X1+⋯+anXn is ILR for all a1,…,an∈ℝ.ProofThe result is trivial if a1=⋯=an=0. Let us assume that a1≠0 (the proof is similar in the other cases) and let us consider the random vector (Y1,…,Yn) with Y1=S and Yj=Xj for j=2,…,n. Then the pdf g of (Y1,…,Yn) is g(y1,…,yn)=1|a1|fy1−a2y2−⋯−anyna1,y2,…,yn.Therefore, g is also logconcave (see the similar proof in the preceding theorem). Finally, we note that the pdf h of S can be obtained as h(x)=∫ℝn−1g(x,y2,…,yn)dy2…dynand so, from Lemma 1, we have that h is logconcave.In particular, this result proves that any sum of the Xi is ILR when (X1,…,Xn) has a logconcave joint pdf. Note that this property can be applied to L‐statistics, that is, to linear combinations of order statistics, when its joint pdf is logconcave. When the order statistics are obtained from i.i.d. random variables with a common univariate pdf f (the usual case), then it is well known that its joint pdf is logconcave whenever f is logconcave, see proposition 5 in An.20 This property can be extended as follows. If (X1,…,Xn) have a joint logconcave pdf fn and they are exchangeable (i.e., fn is permutation invariant), then the joint pdf of the order statistics gn is gn(x1,…,xn)=n!fn(x1,…,xn)for x1<⋯<xn (zero elsewhere), and so it is logconcave as well. This is an important result since the distribution of any L‐statistic obtained from them is ILR. Hence its pdf can be estimated by using MLE, see Walther.11In particular, we can apply this property to a single order statistic and therefore to the extremes min(X1,…,Xn) and max(X1,…,Xn). Note that this result is related to proposition 3 in Navarro and Shaked,3 which proves that if (X1,…,Xn) is exchangeable (i.e., it is permutation symmetric in law) and it has a logconcave pdf, then all the minima (series systems) and maxima (parallel systems) formed with the Xi are also ILR. A similar result is obtained in proposition 1 of that article for the IFR/DFR classes (without the assumption of exchangeability).In a similar way, this property can also be applied to linear combinations of generalized order statistics and the related concepts obtained from them (e.g., generalized spacings). Here we first need to prove the logconcave property for their joint pdf which is given by gn(x1,…,xn)=kγf(xn)F‾k−1(xn)∏i=1n−1f(xi)F‾mi(xi),where γ,k>0 (see, e.g., Cramer and Kamps21). Hence, clearly, it is logconcave when f is logconcave and mi≥0 for i=1,…,n−1. More conditions to get this property were obtained in Chen et al..22 The same property holds for linear combinations of progressively censored order statistics, see (1) in Cramer.23PRESERVATION RESULTS FOR THE IFR CLASSThe following example shows that IFR and ILR classes are not always preserved with respect to sums of dependent random variables.1ExampleLet us consider the random vector (X,Y) with uniform distribution over the set S=[0,1/2]2∪[1/2,1]2, that is, f(x,y)=2 for (x,y)∈S (zero elsewhere). Note that X and Y have a positive dependence and that the marginal distributions are uniform over (0,1) (i.e., they define a copula). Hence X and Y are both IFR and ILR. A straightforward calculation shows that the survival function H‾ of S=X+Y is H‾(s)=1−s2,for0≤s≤1/2;3/4−(s−1/2)2,for1/2<s≤1;1/2−(s−1)2,for1<s≤3/2;(2−s)2,for3/2<s≤2;and its pdf h is h(s)=2s,for0≤s≤1/2;2−2s,for1/2<s≤1;2s−2,for1<s≤3/2;4−2s,for3/2<s≤2;(zero elsewhere). Note that h is not logconcave (see Figure 1, left), that is, S is not ILR.The hazard rate of S is rS(s)=2s1−s2,for0≤s≤1/2;2−2s3/4−(s−1/2)2,for1/2<s≤1;2s−21/2−(s−1)2,,for1<s≤3/2;4−2s(2−s)2,for3/2<s≤2.It is plotted in Figure 1, right, and there we can see that it is not increasing, that is, S is not IFR.1FIGUREPlots of the pdf h (left) and the hazard rate rS for the sum of two dependent random variables with uniform distributions over the interval (0,1) (see Example 1)However, the IFR class is preserved in many models, even if, unfortunately, it does not seem possible to find a very general result such as Theorem 1 for the ILR class. Nevertheless, there are cases in which something can be proved, as in the following example, which deals with a model that can describe both positive or negative dependence. As the IFR class is preserved for this model, then the DFR class is not preserved (from Proposition 4).2ExampleLet us consider a random vector (X,Y) with a Farlie‐Gumbel‐Morgenstern (FGM) survival copula defined by6C^(u,v)=uv[1+θ(1−u)(1−v)]for u,v∈[0,1] and −1≤θ≤1. The independent case is represented by θ=0 while θ>0 (resp. <) represents a positive (negative) dependence (see, e.g., Nelsen15).If we also assume that X and Y have a common exponential distribution with hazard rate λ>0, then a straightforward calculation from (3) gives H‾(t)=q‾θ(F‾(t)) with F‾(t)=e−λt for t≥0 and q‾θ(u)=u−ulogu+θu−3+3u−logu−2ulogufor u∈[0,1] (see also Navarro and Sarabia8). As expected, in the independent case (θ=0), we obtain q‾0(u)=u−ulogu for u∈[0,1].Hence γθ(u)=q‾θ′(u)=−logu−θ4−4u+logu+4uloguand αθ(u)=uq‾θ′(u)q‾θ(u)=−logu−θ4−4u+logu+4ulogu1−logu+θ−3+3u−logu−2ulogufor u∈[0,1]. A straightforward calculation shows that both functions are decreasing for any θ∈[−1,1]. They are plotted in Figure 2 for θ=−1,−0.5,0,0.5,1. As γθ is decreasing, we have X≤LRX+Y from Proposition 8, (iv). Also, as αθ is strictly decreasing, from Proposition 4 we have that X+Y is IFR and not DFR for all λ>0 and all θ∈[−1,1].In Figure 3 we plot logh (left) and the hazard rate rS of the sum (right) for θ=−1,−0.5,0,1,0.5,1. In this figure we can see that IFR and ILR classes are preserved. As a consequence, NBU and IFRA classes are preserved as well (see Navarro et al.2). However, as mentioned above, the DFR class is not preserved. Moreover, from (5), we have that limu→0+αθ(u)=1andlimt→∞rS(t)=λfor all θ∈[−1,1]. Therefore, Proposition 2 for convolutions (θ=0) can be extended for the specific C‐convolution defined by the FGM survival copula for all −1≤θ≤1 when they have a common exponential distribution.It can be seen that this sum does not preserve LFR for θ=1 (see Figure 4, left). However, the plot of logRS (see Figure 4, right), where RS(t)=−logH‾(t)=∫0trS(x)dx,for t≥0, shows that LCFR is preserved in this dependence model.2FIGUREPlots of γθ(u)=q‾θ′(u) (left) and αθ(u)=uq‾θ′(u)/q‾θ(u) for the sum S=X+Y of two exponential distributions with a FGM survival copula with θ=−1,−0.5 (black), 0 (red, convolution), and 0.5,1 (blue)3FIGUREPlots of logh (left) and the hazard rate rS (right) for the sum S=X+Y of two dependent standard exponential distributions with a FGM survival copula with θ=−1,−0.5 (black), 0 (red, convolution), and 0.5,1 (blue). The dashed line represents the common hazard rate of X and Y and the limiting behavior of rS4FIGUREPlots of logrS (left) and logRS (right) for the sum S=X+Y of two dependent standard exponential distributions with a FGM survival copula with θ=−1,−0.5 (black), 0 (red, convolution), 0.5 (blue) and 1 (green). The dashed line represents the log of the common hazard rate of X and Y and the limiting behavior of logrSAlternative conditions for a sum of two dependent lifetimes to be IFR are described in the next statement.3TheoremLet (X,Y) be a random vector with survival copula C^ and marginal survival functions F‾ and G‾. Let S=X+Y. i) If X is ILR andΔ1(x,s)=∂1C^(F‾(x),G‾(s−x))is logconcave, then S is IFR. ii) If Y is ILR andΔ2(y,s)=∂2C^(F‾(s−y),G‾(y))is logconcave, then S is IFR.ProofLet us prove (i) (the proof of (ii) is symmetric). From (2), the survival function of S can be written as7H‾(s)=∫−∞∞f(x)∂1C^(F‾(x),G‾(s−x))dx=∫−∞∞f(x)Δ1(x,s)dxIt is easy to see that if Δ1(x,s) is logconcave and X is ILR, then f(x)Δ1(x,s) is also logconcave. Then the proof is completed by applying Prékopa's theorem (i.e., Lemma 1) to (7).The following numerical example shows how to use Theorem 3 to study the preservation of the IFR class.3ExampleLet us consider the sum of two dependent random variables X and Y having exponential and Rayleigh (Weibull) distributions with F‾(t)=exp(−t) and G‾(t)=exp(−t2) for t≥0 and with the FGM survival copula C^ defined in (6). Its partial derivative is8∂1C^(u,v)=v1+θ(1−2u)(1−v).Hence L(x,s)=logΔ(x,s)=log(G‾(s−x))+log1+θ(1−2F‾(x))(1−G‾(s−x)),that is, L(x,s)=−(s−x)2+log1+θ(1−2exp(−x))(1−exp(−(s−x)2)),for 0≤x≤s. Its 3D plot for θ=1 is shown in Figure 5, left. There, we can see that, in this case, L(x,s) is concave. Moreover, the exponential distribution is ILR. Hence, by applying Theorem 3, (i), we obtain that S=X+Y is IFR for θ=1. On the contrary, for θ=−1 the function L(x,s) is not concave, being, for example, L(0.5,0)=−0.06845,L(0.5,0.15)=−0.05031 and L(0.5,0.3)=−0.02315. The 3D plot for θ=−1 is shown in Figure 5, right. Thus, Theorem 3 cannot be applied to this case.5FIGUREPlots of the function L(x,s) in Example 3 for θ=1 (left) and θ=−1 (right)In the following example we show how to use Propositions 3 and 4 to check the preservation of the IFR class in sums of dependent random variables when just one of them has an exponential distribution.4ExampleLet us consider a random vector (X,Y) with the FGM survival copula defined in (6). Let us assume that Y has a standard exponential distribution and that the survival function of X is F‾(t)=1−1−exp(−t)for t≥0. Note that the survival function of Y can be represented as G‾(t)=exp(−t)=ψ(F‾(t))for ψ(u)=2u−u2. Hence, the survival function of S=X+Y can be written as H‾(t)=q‾1(F‾(t)), where q‾1 is obtained from (4) as q‾1(u)=u+∫u1∂1C^v,ψ(u)ψ(v)dvfor u∈[0,1]. In particular, for the FGM copula we get the expression in Equation (8) and q‾1(u)=u+ψ(u)∫u11ψ(v)dv+θψ(u)∫u11−2vψ(v)dv+θψ2(u)∫u11−2vψ2(v)dvfor u∈[0,1]. A straightforward calculation leads to q‾1(u)=u+12ψ(u)log2u−1−12θψ(u)log(u(2−u)3)−θψ(u)1+u2−ψ(u)+14ψ(u)logu2−ufor u∈[0,1]. By plotting α1(u)=uq‾1′(u)/q‾1(u) we see that it is decreasing for all θ∈[0,1]. However, we cannot apply Proposition 4 to this representation since F‾ is not IFR (actually it is strictly DFR). Instead, we are going to use a representation based on G‾ which is IFR. Note that F‾(t)=β(G‾(t)) for β(u)=1−1−u for u∈[0,1]. Then the survival function of S can be obtained as H‾(t)=q‾1(F‾(t))=q‾1(β(G‾(t)))=q‾2(G‾(t)),where q‾2(u)=q‾1(β(u)) for u∈[0,1]. Hence α2(u)=uq‾2′(u)q‾2(u)=uq‾1′(β(u))β′(u)q‾1(β(u))for u∈(0,1]. This function is plotted in Figure 6, left, for a sample of possible values of θ, that is, θ=−1,0.5,0,0.5,1. There we can see that α2 is decreasing for such values of θ, except for the case θ=−1. Then, from Proposition 4, S is IFR for θ=0.5,0,0.5,1. The hazard rate functions of X and Y (dashed lines) and S (continuous lines) are plotted in Figure 6, right. There we can see that S is IFR for such a sample of values for θ, except for θ=−1 (green line).6FIGUREPlots of the function α2 (left) and the hazard rate function rS (right) for the sum S=X+Y in Example 4 and θ=−1 (green), −0.5 (black), 0 (red, convolution), 0.5,1 (blue). The dashed lines represent the hazard rate functions of X (decreasing line) and Y (constant line)RELEVATION TRANSFORM MODELSA model of interest in reliability theory which deals with sums of not independent lifetimes is the relevation transformation. This model is considered here as an example where the results described in previous sections can be applied, but it is of interest also because it provides meaningful counterexamples. Under a relevation transform (see, e.g., Krakowski24 or Belzunce et al.25), the unit with lifetime X is replaced (or repaired) when it fails at a time x≥0 by a unit Y having survival function G‾ but with the same age as X, that is, with survival function G‾x(y)=Pr(Y>y|X=x)=Pr(Y−x>y|Y>x)=G‾(x+y)G‾(x)for y≥0. Then X and Y are dependent and the survival function H‾(t)=Pr(S>t) of S=X+Y can be obtained as9H‾(t)=F‾(t)+∫0tG‾x(t−x)f(x)dx=F‾(t)+∫0tG‾(t)G‾(x)f(x)dx.This is formula (1.1) in Krakowski,24 in which the notation H‾=F‾#G‾ is introduced. In a relevation transform with perfect repair, the unit with lifetime X is replaced when it fails at a time x by an independent new unit with lifetime Y having survival function G‾x(y)=G‾(y). Then10H‾(t)=F‾∗F‾(t)=F‾(t)+∫0tG‾(t−x)f(x)dx,which is the survival function of the sum of two independent random variables (convolution). More realistic cases are considered in the following subsections.Relevation transform with minimal repairIn a relevation transform with minimal repair, the unit X is replaced when it fails at a time x by a unit Yx=(Y−x|Y>x) having survival function F‾ but with the same age as X, that is, G‾x(y)=F‾(x+y)/F‾(x) for y≥0. This is equivalent to minimally repair the broken unit to be like it was before its failure. In this case, from (9), H‾(t)=F‾#F‾(t)=F‾(t)+∫0tF‾(t)F‾(x)f(x)dx=F‾(t)−F‾(t)logF‾(t).This is formula (3.1) in Krakowski24 with the notation F‾#F‾ for the relevation transform under minimal repair. Note that for the exponential distribution, this expression coincides with the one obtained for the usual convolutions from (10).Note that it is a distorted distribution with dual distortion q‾(u)=u−ulogu for u∈[0,1]. The pdf of S is h(t)=−f(t)logF‾(t) and its hazard rate is rS(t)=h(t)H‾(t)=−logF‾(t)1−logF‾(t)r(t)for t≥0, where r=f/F‾ is the hazard rate of F‾. Hence q‾′(u)=−logu and α(u)=−logu/(1−logu) for u∈(0,1). As α is decreasing in (0,1), the IFR class is preserved by the relevation transform with minimal repair, which is a well known result (see the references in Pellerey et al.10). As the IFR class is preserved, so are the NBU and IFRA classes. As α≤1, then S=X+Y≥HRX for all F‾. Moreover, from (5), we get limt→∞rS(t)r(t)=limu→0+α(u)=1.Therefore, Proposition 2, which deals with the limit of the hazard rate of the sum of independent variables (thus, with convolutions), can be extended for the specific C‐convolution defined by the relevation transform.However, q‾′(u)=−logu does not always preserve logconcave survival functions. Thus the ILR class is not always preserved. Sufficient conditions for that preservation in this model were obtained in Pellerey et al.,10 where it is proved that the ILR class is preserved when F is also LCFR.For example, we observe that the ILR class is preserved for a positively truncated standard normal (Gaussian) distribution as can be seen in Figure 7, left. Note that the normal distribution satisfies condition LFR, which implies condition LCFR. However, this is not the case for a uniform distribution over (0,1) (see Figure 7, right) since the uniform distribution is not LCFR. Therefore, Theorem 1 cannot be extended to arbitrary vectors, that is, the ILR class is not always preserved.7FIGUREPlots of logh for the relevation transform under minimal repair for a truncated standard Gaussian (normal) distribution (left) and for a uniform distribution over (0,1) (right)Relevation transform with imperfect repairIn a relevation transform with imperfect repair, the unit X is replaced when it fails at a time x by a unit Yx=(Y−x|Y>x) having survival function G‾=F‾θ for θ>0 and θ≠1 but with the same age as X (see Brown and Proschan26 and Bhattacharjee27 for details). In this case, from (9), H‾θ(t)=F‾(t)+∫0tF‾θ(t)F‾θ(x)f(x)dx=F‾(t)+F‾θ(t)F‾1−θ(t)θ−1−1θ−1for t≥0. Note that it is also a distorted distribution with dual distortion q‾θ(u)=θθ−1u−1θ−1uθfor u∈[0,1]. Also note that it is a negative mixture of F‾ and F‾θ and that limθ→∞q‾θ(u)=u, limθ→0+q‾θ(u)=1, limθ→1q‾θ(u)=u−ulogu for u∈(0,1) (i.e., a relevation transform with minimal repair, as expected). Their plots for different values of θ can be seen in Figure 8, left.8FIGUREPlots of q‾θ (left) and αθ (right) for a relevation transform with imperfect repair and θ=0.25,0.5,0.75 (black), θ=1 (red, minimal repair) and θ=1.5,2,3,5 (blue)The pdf of S in this model is hθ(t)=θθ−11−F‾θ−1(t)f(t)and its hazard rate is rθ(t)=αθ(F‾(t))r(t), where r=f/F‾ is the hazard rate of F‾ and αθ(u)=θ1−uθ−1θ−uθ−1for u∈[0,1]. The plots of αθ can be seen in Figure 8, right. As they are ordered, the respective sums are ordered in hazard rate sense and, as they are strictly decreasing, from Proposition 4 it follows that the IFR class is preserved (and the DFR class is not preserved). This is a well‐known property, proved in Brown and Proschan.26 As a consequence, NBU and IFRA classes are preserved as well. Even more, since αθ≤1, then S≥HRX for all F‾ and all θ>0. Moreover, from (5), limt→∞rθ(t)r(t)=limu→0+αθ(u)=1forθ>1;θforθ<1.Therefore, Proposition 2 for convolutions can be extended for the specific C‐convolution defined by the relevation transform with imperfect repair and θ>1. However, this property does not hold when 0<θ<1. The respective hazard rates for a standard exponential distribution can be seen in Figure 9, right. Note that they have different limiting behavior.9FIGUREPlots of loghθ (left) and hazard rate rθ (right) for a relevation transform with imperfect repair with a baseline standard exponential distributions and θ=0.25,0.5,0.75 (black), θ=1 (red, minimal repair) and θ=1.5,2,3,5 (blue). The dashed lines represent the limiting behavior for t→∞However, q‾θ′(u)=θθ−11−uθ−1does not always preserve logconcave functions. So we do not know if the ILR class is preserved. For example, in the case of a standard exponential distribution, the ILR class is preserved (see Figure 9, left). As q‾θ′(u) is decreasing in u, in this dependence model, from Proposition 8, (iv), we also get S≥LRX for all F‾ and all θ>0.2RemarkThe sequential order statistics fit to a relevation transform model with different distributions, see Cramer and Kamps.21 However, they do not fit to a minimal (or imperfect) repair model. In the case of a PHR model for the baseline distributions, they can be written as distortions. This case was studied in Burkschat and Navarro.28 So we can also apply to them the new results for distortions obtained here.CLOSURES WITH RESPECT TO MAXIMUMThe closure properties of maximum also have interest in several applicative areas. Some general results for IFR/DFR and ILR/DLR classes were obtained in Hu and Li19 and Navarro and Shaked.3 The preservation for other classes, such as NBU, DMRL and NBUE, was studied in Abouammoh and El‐Neweihi29 and Navarro.30 By using Proposition 4 we can obtain the following result, dealing with the preservation of the IFR class in maxima of identically distributed (i.d.) components.4TheoremIf X1,…,Xn are i.d. with copula C, then the following conditions are equivalent:i) The IFR (DFR) class is preserved by M=max(X1,…,Xn); ii) The IFR (DFR) class is preserved by M=max(X1,…,Xn) for the standard exponential distribution; iii) The function α(u)=uδ′(1−u)/(1−δ(1−u)) is decreasing (increasing) in (0,1), where δ(u)=C(u,…,u) is the diagonal section of C.ProofThe distribution function of M is FM(t)=Pr(X1≤t,…,Xn≤t)=C(F(t),…,F(t))=δ(F(t)),where F is the common distribution function of X1,…,Xn. Hence, its survival function is F‾M(t)=1−δ(F(t))=1−δ(1−F‾(t))=q‾(F‾(t))for q‾(u)=1−δ(1−u). Then the proof is completed by applying Proposition 4.It is well known that the IFR property is preserved in maxima of i.i.d. random variables (see Esary and Proschan31), that is, the property in (iii) above holds for the product copula. This property can be extended for k‐out‐of‐n systems (order statistics) with i.i.d. components (see Esary and Proschan31). It is also true for other copulas (see next example). However, it can be easily verified that this preservation property is not always true.5ExampleLet M=max(X1,X2) with (X1,X2) having the same marginal distribution function F and the following Clayton‐Oakes copula11C(u,v)=uvu+v−uvfor u,v∈[0,1]. Its diagonal section is δ(u)=C(u,u)=u2−ufor u∈[0,1]. Hence the dual distortion function of M is q‾(u)=1−δ(1−u)=1−1−u1+u=2u1+uand the associated α function is α(u)=11+ufor u∈[0,1]. As α is strictly decreasing, the IFR class is preserved by M but the DFR class is not. Then the NBU and IFRA classes are preserved as well. Moreover, the limiting behavior of the hazard rate of M can be determined as limt→∞rM(t)r(t)=limu→0+α(u)=limu→0+11+u=1.Also note that Xi≤HRM holds for any F (since α(u)≤1 for all u∈[0,1]). Even more, as q‾′ is decreasing, then Xi≤LRM holds for any F from Proposition 8, (iv).It is easy to verify that results similar to those described in Example 5 can be provided considering the FGM copula as the connecting copula of (X1,X2) instead of the Clayton‐Oakes copula.For non i.d. components we have the following result. The proof is immediate from proposition 2.5 in Navarro et al..2 The similar results for other aging classes such as DMRL/IMRL, NBU/NWU, and IFRA/DFRA can be obtained from the results for distortions given in Navarro30 and Navarro et al..2 Results for NBUE/NWUE aging classes are still an open question.5TheoremIf X1,…,Xn have distribution functions F1,…,Fn and copula C, and if the following conditions hold:i) X1,…,Xn are IFR (DFR); ii) The functionsαi(u1,…,un)=ui∂iC(1−u1,…,1−un)1−C(1−u1,…,1−un)are decreasing (increasing) in (0,1)n for i=1,…,n;then M=max(X1,…,Xn) is IFR (DFR).For the ILR class we have the following result, that follows from Proposition 5.6TheoremLet (X1,…,Xn) be a random vector with copula C and ILR margins Xi having the same distribution F. Let δ(u)=C(u,…,u) be the diagonal section of C. If δ′ preserves logconcave distribution functions, then M=max(X1,…,Xn) is ILR.ProofObserve that the distribution function of M can be written as FM(t)=δ(F(t)), for all t. Thus its density fM can be written as fM(t)=δ′(F(t))f(t), so that logfM(t)=log(δ′(F(t)))+logf(t)for all t. The term logf is concave by assumption (being F ILR). Moreover, since logconcavity of f implies logconcavity of F (see, e.g., Barlow and Proschan1(p. 77)), if δ′ preserves logconcave distribution functions, then the first term is also concave, and the assertion follows.Note that non‐negativity of X1,…,Xn is not required in Theorem 6. Unfortunately, as stated already above, it is not easy to check if a function preserves logconcavity. However, Theorem 6 provides an immediate tool to check when the maximum of dependent and ILR variables is still ILR, knowing the copula C and the distribution F. For example, if C is the Clayton copula with parameter θ∈(0,∞), so that δ(u)=(2u−θ−1)−1/θ for u∈[0,1], and the Xi are exponentially distributed with the same rate λ, then one can easily verify that the ratio δ′(F(t+s))δ′(F(t))is decreasing in t≥0 for all s≥0, that is, that δ′(F(t)) is logconcave. Here δ′(u)=2u−1−θ(2u−θ−1)−(1+θ−1) for u∈[0,1]. Thus, the maximum is ILR.Similarly, if the connecting copula C is an AMH copula (see (4.2.3) in Nelsen15(p. 116)) with parameter θ∈[−1,1), so that δ(u)=u2/(1−θ(1−u)2), and the Xi are exponentially distributed with the same rate, then one can easily verify that the ratio above is decreasing in t≥0 for all s≥0, that is, that δ′(F(t)) is logconcave. Here δ′(u)=(2u(1−θ(1−u)2)−2θ(1−u)t2)/(1−θ(1−u)2)4. Thus, again the maximum is ILR.The preceding theorem can also be applied to copulas having diagonal sections δ(u)=uc for 1<c≤2 (since, as mentioned above, the functions q‾′(u)=δ′(u)=uc−1 preserve logconcave functions for c>1). For example, this property holds for the Cuadras‐Augé family of copulas, see Nelsen.15(p. 15)Load‐sharing models with time‐homogeneous failure ratesHere we consider a particular model dealing with the maximum of dependent variables. In the case of absolute continuity, the joint probability law of n non‐negative random variables X1,…,Xn can be described by means of the family of its Multivariate Conditional Hazard Rates (MCHR) functions. Such a description is alternative but mathematically equivalent to the one expressed in terms of the joint density function of (X1,…,Xn) (see, e.g., the review paper Shaked and Shanthikumar32 and the references cited therein).In a sense, the MCHR functions arise as direct extensions of the univariate concept of hazard rate function for a single non‐negative random variable X. In the bivariate case the family of the MCHR functions is ℒ={λ10(t),λ20(t),λ11(t,x),λ21(t,x)}, where λ10(t)=limΔt→0+Pr[X1>t+Δt|X1>t,X2>t]Δt,λ20(t)=limΔt→0+Pr[X2>t+Δt|X1>t,X2>t]Δt,and, for 0<x<t, λ11(t,x)=limΔt→0+Pr[X1>t+Δt|X1>t,X2=x]Δt,λ21(t,x)=limΔt→0+Pr[X2>t+Δt|X2>t,X1=x]Δt.The functions λ10(t),λ20(t),λ11(t,x), and λ21(t,x) belonging to ℒ can be obtained in terms of the joint density function fX1,X2 of (X1,X2). One can also check that, vice versa, fX1,X2 can be recovered from the knowledge of the family ℒ.In particular, special models arise from the condition that λ11(t,x) and λ21(t,x) do not depend on x. The corresponding models were called Load‐Sharing models, see, for example, Spizzichino33 and references cited therein for more details and further remarks. They were also called models in which the failure rates depend on the working set by some authors, such as in Ross.34 The term load‐sharing model is inspired by the model in which L(t) has the meaning of a level of load to be shared by the surviving components at time t>0, and by letting L(t)=λ10(t)+λ20(t) if both components are alive at time t, or L(t)=λi1(t) if only component i is alive at time t.Note that if X1 and X2 are exchangeable, so that λ10(t)=λ20(t)=λ0(t) and λ11(t)=λ21(t)=λ1(t), then, in agreement to what is stated above, it must be λ1(t)=2λ0(t). However, we will consider here the more general case where λ1(t)=θλ0(t) for some θ>0.Let us thus consider a vector (X1,X2) with distribution defined as in the model above. One can be interested in the distribution of the sum S=X1+X2, having survival function H‾θ depending on the parameter θ. In this case, with straightforward calculations, one can verify that H‾θ(t)=F‾(t)+∫0tF‾θ(t)F‾θ(x)f(x)dxfor t≥0, where F‾(t) is the survival function corresponding to the hazard rate λ0(t), while F‾θ(t) is the survival function corresponding to the hazard rate λ1(t). Thus, the study of the sum of two lifetimes in a load‐sharing model reduces to the case of minimal repair models (see Section 5.1).However, dealing with these models, one is mainly interested in the total time the components can deal with the load to be shared, that is, in M=max(X1,X2). Letting H‾θ be the survival function of M, one has H‾θ(t)=Pr(X1>t,X2>t)+∫0tPr(X1=x,X2>t)dx+∫0tPr(X1>t,X2=x)dxthat is, H‾θ(t)=F‾2(t)(1−2logF‾(t))ifθ=2,and H‾θ(t)=12−θ2F‾θ(t)−θF‾2(t)ifθ≠2for t≥0. In both cases we get distorted distributions.Thus, for θ=2, one has q‾2(u)=u2(1−2logu) and q‾2′(u)=−4ulogu. It follows that α2(u)=−4logu1−2loguwhich is decreasing in (0,1). Therefore, by Proposition 4, if F‾ is IFR (i.e., if λ0(t) is increasing), then M is IFR as well.For θ≠2 one has q‾θ(u)=12−θ(2uθ−θu2) and q‾θ′(u)=12−θ(2θuθ−1−2θu). It follows that αθ(u)=2θuθ−2θu22uθ−θu2,which is also decreasing in (0,1) for every strictly positive θ, with θ≠2. Therefore, again by Proposition 4, if F‾ is IFR (i.e., if λ0(t) is increasing), then M is IFR as well.It is interesting to observe that this property is satisfied also for θ∈(0,1) even if in this case the marginal distributions of X1 and X2 can be not IFR (for example, if λ0(t)=λ∈ℝ+, then at the failure of one component the hazard of the remaining one reduces to θλ, which is smaller than λ).For some specific choices of λ0(t), and therefore of F‾, one can verify that M is also ILR. For example, if θ=2 and λ0(t)=bktk−1 for t≥0 (which is the case of a Weibull distribution), then logq‾2′(F‾(t))=log(4btkexp(−btk))=log(4b)+klogt−btkfor t≥0. Since this function is concave for any positive b and k then, by Proposition 5, one has that M is also ILR.ACKNOWLEDGMENTSThe authors would like to thank the editors and the anonymous reviewers for several helpful suggestions and tasks. Jorge Navarro is supported in part by Ministerio de Ciencia e Innovación of Spain under Grant PID2019‐103971GB‐I00/AEI/10.13039/501100011033 and Franco Pellerey by the GNAMPA research group of INdAM (Istituto Nazionale Di Alta Matematica), Italy, and by the Grant Progetto di Eccellenza, CUP: E11G18000350001, Italy.DATA AVAILABILITY STATEMENTData sharing not applicable to this article as no datasets were generated or analyzed during the current study.REFERENCESBarlow RE, Proschan F. Statistical Theory of Reliability and Life Testing: Probability Models. Holt, Rinehart and Winston, INC; 1975.Navarro J, del Águila Y, Sordo MA, Suárez‐Llorens A. Preservation of reliability classes under the formation of coherent systems. Appl Stoch Model Bus Ind. 2014;30:444‐454.Navarro J, Shaked M. Some properties of the minimum and the maximum of random variables with joint logconcave distributions. Metrika. 2010;71:313‐317.Karlin S, Proschan F. Pólya type distributions of convolutions. Ann Math Stat. 1960;31:721‐736.Belzunce F, Martínez‐Riquelme C, Mulero J. An Introduction to Stochastic Orders. Academic Press; 2015.Alimohammadi M, Alamatsaz MH, Cramer E. Convolutions and generalization of logconcavity: implications and applications. Naval Res Log. 2016;63:109‐123.Pellerey F, Navarro J. Stochastic monotonicity of dependent variables given their sum. Test. 2021;1‐19. doi:10.1007/s11749-021-00789-5Navarro J, Sarabia JM. Copula representations for the sums of dependent risks: models and comparisons. Probab Eng Inf Sci. 2020;1‐21. doi:10.1017/S0269964820000649Shaked M, Shanthikumar JG. Stochastic Orders. Springer Series in Statistics. Springer; 2007.Pellerey F, Shaked M, Zinn J. Nonhomogeneous Poisson processes and logconcavity. Probab Eng Infor Sci. 2000;14:353‐373.Walther G. Inference and modeling with log‐concave distributions. Stat Sci. 2009;24:319‐327.Block H, Langberg N, Savits T. The limiting failure rate for a convolution of life distributions. J Appl Probab. 2015;52:894‐898.Cherubini U, Mulinacci S, Romagnoli S. A copula‐based model of speculative price dynamics in discrete time. J Mult Anal. 2011;102:1047‐1063.Durante F, Sempi C. Principles of Copula Theory. CRC/Chapman & Hall; 2016.Nelsen RB. An Introduction to Copulas. 2nd ed. Springer; 2006.Yaari ME. The dual theory of choice under risk. Econometrica. 1987;55:95‐115.Navarro J, del Águila Y, Sordo MA, Suárez‐Llorens A. Stochastic ordering properties for systems with dependent identically distributed components. Appl Stoch Model Bus Ind. 2013;29:264‐278. doi:10.1002/asmb.1917Prékopa A. A note on logarithmic concave measures. Acta Sci Math. 1973;34:335‐343.Hu T, Li Y. Increasing failure rate and decreasing reversed hazard rate properties of the minimum and maximum of multivariate distributions with log‐concave densities. Metrika. 2007;65:325‐330.An MY. Logconcavity versus logconvexity: a complete characterization. J Eco Theory. 1998;80:350‐369.Cramer E, Kamps U. Marginal distributions of sequential and generalized order statistics. Metrika. 2003;58:293‐310.Chen H, Xie H, Hu T. Log‐concavity of generalized order statistics. Stat Probab Lett. 2009;79:396‐399.Cramer E. Logconcavity and unimodality of progressively censored order statistics. Stat Probab Lett. 2004;68:83‐90.Krakowski M. The relevation transform and a generalization of the Gamma distribution function. Revue Française D'Automatique, Informatique et Recherche Opérationnelle Mai. 1973;V‐2:107‐120.Belzunce F, Martínez‐Riquelme C, Mercader JA, Ruiz JM. Comparisons of policies based on relevation and replacement by a new one unit in reliability. Test. 2021;30:211‐227.Brown M, Proschan F. Imperfect repair. J Appl Probab. 1983;20:851‐859.Bhattacharjee MC. New results for the Brown‐Proschan model of imperfect repair. J Stat Plann Infer. 1987;16:305‐316.Burkschat M, Navarro J. Stochastic comparisons of systems based on sequential order statistics via properties of distorted distributions. Probab Eng Infor Sci. 2018;32:246‐274.Abouammoh A, El‐Neweihi E. Closure of NBUE and DMRL under the formation of parallel systems. Stat Probab Lett. 1986;4:223‐225.Navarro J. Preservation of DMRL and IMRL aging classes under the formation of order statistics and coherent systems. Stat Probab Lett. 2018;137:264‐268.Esary J, Proschan F. Relationship between system failure rate and component failure rates. Technometrics. 1963;5:183‐189.Shaked M, Shanthikumar JG. Multivariate conditional hazard rate functions ‐ an overview. Appl Stoch Model Bus Ind. 2015;31:285‐296.Spizzichino F. Reliability, signature, and relative quality functions of systems under time‐homogeneous load‐sharing models. Appl Stoch Model Bus Ind. 2019;35:158‐176.Ross SM. A model in which component failure rates depend on the working set. Naval Res Log. 1984;31:297‐300. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Stochastic Models in Business and Industry Wiley

Preservation of ILR and IFR aging classes in sums of dependent random variables

Loading next page...
 
/lp/wiley/preservation-of-ilr-and-ifr-aging-classes-in-sums-of-dependent-random-oYamnNVMI8

References (35)

Publisher
Wiley
Copyright
© 2022 John Wiley & Sons, Ltd.
ISSN
1524-1904
eISSN
1526-4025
DOI
10.1002/asmb.2657
Publisher site
See Article on Publisher Site

Abstract

INTRODUCTIONIn different applied probability fields, like reliability theory or actuarial sciences, various concepts of aging have been proposed and studied to characterize and classify (in terms of “aging classes”) the distributions of random lifetimes of components, devices or individuals. Apart for the usefulness in the understanding of the intrinsic meaning of aging and its different effects on lifetimes, these classes serve for a number of purposes. They can be used, for example, to provide bounds for survival probabilities or expected residual lifetimes, or to define optimal replacement policies under different constrains. They can also serve to evaluate risks in insurance contracts. We refer the reader to the monograph Barlow and Proschan1 for a detailed description of the main aging classes and their applications in reliability modeling. Some among the most well‐known of these classes are recalled in the next section.Because of their usefulness in the applicative fields mentioned above, one can find in the literature a list of papers which study these classes and their properties. Several of them deal with their preservations under different stochastic operations, such as the closures with respect to construction of coherent systems, mixtures or sums. Concerning sums of random lifetimes, it is well known that the increasing failure rate (IFR) aging class is preserved under convolutions, that is, if X and Y are two independent random variables with distributions in the IFR class, then also X+Y has an IFR distribution (see, e.g., Barlow and Proschan1(p. 100)). On the contrary, the decreasing failure rate (DFR) class is not preserved under convolutions. These properties can be different for other transformations. For example, the DFR class is preserved under the formation of mixtures but the IFR class is not, and both of them are not preserved under the formation of coherent systems with i.i.d. components. However, the New Better than Used (NBU) class is preserved for these systems (see Barlow and Proschan1(p. 187)), but it is not preserved when the independence assumption is not satisfied (see Navarro et al.2). Analogously, the increasing likelihood ratio (ILR) class (equivalent to logconcavity of the densities) is preserved (with some conditions) by the formation of parallel and series systems (see Navarro and Shaked3), while the preservation with respect to convolutions was proved by Karlin and Proschan4 (see also Belzunce et al.5(p. 7)). Conditions for such preservations in distorted distributions and systems were recently obtained in Navarro et al.,2 and additional results for convolutions can be found in Alimohammadi et al.6 Properties of conditional distributions of sums are given in Pellerey and Navarro.7It must be pointed out that for almost all the results available in the literature dealing with preservation of aging classes under sums, the independence among summands is required. However, since in general the sums reduce the uncertainty, it is expectable to observe sums that are “more” IFR (or more ILR) than their summands, also when the assumption of independence is unsatisfied (in particular, this intuition seems especially true for negatively dependent random variables).The purpose of the present paper is to investigate this insight in more detail, in order to provide new conditions for preservation of some of the most well known aging classes under sums of dependent random variables. The new statements can be added to similar ones recently obtained in Navarro and Sarabia8 for the IFR/DFR classes. The results described here are mainly based on preservation properties for distorted distributions and C‐convolutions, which are copula representations for the distributions of sums of dependent random variables (see Section 2). In particular, such representations, as well as other tools, are used here to provide new conditions for the closure under sums of IFR and ILR aging classes. Some properties for the limiting behavior of the failure rate of the sum are provided as well, together with new conditions for the closure under maximum of dependent IFR and ILR lifetimes.The rest of the paper is structured as follows. Basic definitions and some preliminary results are introduced in Section 2. The main results for ILR and IFR classes are placed in Sections 3 and 4, respectively. Illustrative examples and counterexamples dealing with a specific kind of sums modeled with relevation transforms are given in Section 5. Similar results, but dealing with preservation of ILR and IFR aging classes with respect to maximum of non‐independent variables, are presented in Section 6.Throughout the paper the notions increasing and decreasing are used in a wide sense, that is, they mean non‐decreasing and non‐increasing, respectively.PRELIMINARIESAging classesFirst we recall the following definition.1DefinitionWe say that f:ℝn→ℝ+ is logconcave (logconvex) if logf(λx+(1−λ)y)≥(≤)λlogf(x)+(1−λ)logf(y)for all λ∈(0,1) and all x,y∈ℝn.Given a nonnegative random variable (lifetime) X with distribution function F(t)=Pr(X≤t) and survival (reliability) function F‾(t)=1−F(t)=Pr(X>t), let us denote with f(t)=F′(t) for t≥0 (zero elsewhere) its probability density function (pdf), and with r(t)=f(t)/F‾(t), for t≥0, its failure (or hazard) rate function, assuming they exist (from now on, whenever we use f or r, we are tacitly assuming that X has an absolutely continuous distribution). Then we say that:X (or F) is ILR (DLR) if f(t) is logconcave (logconvex) for t≥0;X (or F) is IFR (DFR) if F‾(t) is logconcave (logconvex) for t≥0;X (or F) is NBU (NWU) if F‾(t)F‾(s)≥F‾(t+s) for all s,t≥0.The IFR (DFR) property is equivalent to have an increasing (decreasing) failure rate function. It is also equivalent to have Xs≥STXt (≤ST) for all t≥s≥0, where ≥ST represents the usual stochastic order (see Shaked and Shanthikumar9) and Xt=(X−t|X>t) represents the residual lifetime of X at time t. Analogously, the NBU (NWU) property is equivalent to have X≥STXt (≤ST) for all t≥0. The ILR (DLR) class is equivalent to have Xs≥LRXt (≤LR) for all t≥s≥0, where ≥LR represents the likelihood ratio order (see Shaked and Shanthikumar9). The ILR and DLR classes are also equivalent to the respective monotonicity properties of the function η(t)=−f′(t)/f(t) for t≥0. Clearly, since the likelihood ratio order implies the usual stochastic order, ILR⇒IFR⇒NBU and, when the support of X is (l,∞), DLR⇒DFR⇒NWU.The IFR, NBU, and ILR classes represent the usual (positive or natural) aging properties where the older units have worse performance than the younger ones. The reverse scenario holds for the dual classes DFR, NWU, and DLR.It must be pointed out that, even if some notions like IFR and NBU usually refer to non‐negative variables, on the contrary the ILR notion also applies to variables that are not necessarily positive, and some of the statements described below hold true even in this case.Some additional (related) properties, defined here, will be considered through the following sections. Given a nonnegative random variable X, and denoted with R(t)=−logF‾(t)=∫0tr(s)ds its cumulative failure rate (or hazard rate) function, then we say that:X (or F) is IFRA (increasing in failure rate average) if R(t)/t is increasing for t≥0;X (or F) is LFR (logconcave in the failure rate) if r(t) is logconcave for t≥0;X (or F) is LCFR (logconcave in the cumulative failure rate) if R(t) is logconcave for t≥0.While the IFRA class is well‐known, being the largest class closed with respect to construction of coherent systems (see Barlow and Proschan1), the LCFR and LFR classes have been considered few times in the literature. Their usefulness in describing aging properties has been pointed out in Pellerey et al.,10 where they have been used to describe sufficient conditions for a nonhomogeneous Poisson processes to satisfy specific monotonicity properties. Note that, in Pellerey et al.,10 the LFR and LCFR classes were denoted as (A.3) and (A.2), respectively, and their relationships with the ILR and IFR classes have been fully described. These relationships are summarized in Table 1. The dual classes (describing negative aging rather positive aging) can be considered as well.1TABLERelationships among positive aging classesLFR⇒ILR⇒IFRA⇓⇓⇓LCFR⇒IFR⇒NBUApart for applications in reliability theory, aging classes can be of interest also in other contexts. The following simple result shows the relevance of the ILR property for the estimation of the parameter in a scale family ℱG of a nonnegative random variable. Recall that, given a distribution function G, we say that F belongs to the scale family of distributions ℱG determined by G if F(t)=G(θt) for all t, where θ>0 is the parameter of the model. Some examples are the exponential, the Weibull (with a fixed shape parameter) and the Pareto (with a fixed shape parameter) distributions.1PropositionIf X is a nonnegative random variable with a distribution function F∈ℱG and G is absolutely continuous and ILR, then the maximum likelihood estimator (MLE) for θ is unique.ProofIf X1,…,Xn are i.i.d. with distribution function F, then the likelihood function for θ is ℓ(θ)=f(x1)…f(xn)=θng(θx1)…g(θxn),where g is the pdf of G. To maximize ℓ in (0,∞) is equivalent to maximize ℓ˜(θ)=log(ℓ(θ))=nlog(θ)+∑i=1nlog(g(θxi))for θ>0. Then ℓ˜′(θ)=nθ+∑i=1nxig′(θxi)g(θxi).As G is ILR, then g′/g is decreasing and so ℓ′(θ) is strictly decreasing in (0,∞) for all x1,…,xn>0. Therefore, ℓ(θ) is strictly concave. Thus it has a unique maximum value in (0,∞).The class of ILR distributions contains many of the most commonly used parametric distributions, thus it is a rich and flexible nonparametric class of distributions. Further, the MLE for the density exists and can be computed with readily available algorithms (see, e.g., the review in Walther11).The following statement, which is theorem 1 in Block et al.,12 will be also used in the following sections. It essentially affirms that, in the limit, the hazard rate of a convolution converges to the smallest of the hazards of the summands.2PropositionBlock et al.12Let X and Y be two independent random variables with hazard rates rX and rY, respectively. Assume that limt→∞rX(t)=a, limt→∞rY(t)=b with 0<a<b<∞ and that rX is bounded. Then the failure rate rS of the convolution S=X+Y satisfies limt→∞rS(t)=a.C‐convolutionsFirst, we recall here the notion of C‐convolutions, introduced in Cherubini et al.,13 that will be used in the following sections to prove some of the main results. Let (X,Y) be two possibly dependent random variables with joint distribution F(x,y)=Pr(X≤x,Y≤y).From copula theory (see, e.g., Durante and Sempi14 or Nelsen15), it is well known that this joint distribution can be written as F(x,y)=C(F(x),G(y)),where F(x)=Pr(X≤x) and G(y)=Pr(Y≤y) are the marginal distribution functions of X and Y, respectively, and where C is a copula function. This representation (by means of a copula C) is unique when F and G are continuous (by the Sklar's theorem, see e.g., Durante and Sempi14). Also note that given a copula C, the right‐hand side of this expression provides a proper bivariate distribution function for any marginal distribution functions F and G.Then, from Cherubini et al.13 (see also Navarro and Sarabia8), the distribution function H of the sum S=X+Y can be obtained as H(t)=∫−∞∞f(x)Pr(Y≤t−x|X=x)dx=∫−∞∞f(x)∂1C(F(x),G(t−x))dx,where ∂1C represents the partial derivative of C with respect to its first argument. If X and Y are nonnegative, then the limits in the integral reduces to 0 and t, respectively. If F−1 is the inverse function of F, then this expression can be rewritten as H(t)=∫01∂1C(u,G(t−F−1(u)))du.This distribution is called the C‐convolution and it is represented as H=F∗CG in Cherubini et al.13 If C is the product copula, we get the usual expression for the convolution (i.e., the distribution of the sum of two independent random variables).In some models it is better to use survival functions instead of distribution functions. For them, we have the expression1H‾(t)=Pr(S>t)=∫−∞∞f(x)∂1C^(F‾(x),G‾(t−x))dx,where H‾=1−H, F‾=1−F, and G‾=1−G are the survival functions of S, X, and Y, respectively, and where C^ is a copula, called survival copula, that allows to represent the joint survival function of (X,Y) as F‾(x,y)=Pr(X>x,Y>y)=C^(F‾(x),G‾(y)).In particular, for nonnegative random variables Equation (1) reduces to2H‾(t)=F‾(t)+∫0tf(x)∂1C^(F‾(x),G‾(t−x))dx.The exponential distribution plays a central role in reliability theory and survival analysis, representing units without aging. If both marginal distributions coincide and are exponentials, that is, if F‾(t)=G‾(t)=e−λt for t≥0 and λ>0, then Equation (2) reduces to3H‾(t)=e−λt+∫e−λt1∂1C^(v,e−λt/v)dv=q‾(e−λt)for t≥0, where q‾(u)=u+∫u1∂1C^v,uvdvfor u∈[0,1]. This kind of representations are called distorted distributions, and they are studied in the following subsection. A similar representation can be obtained when just G is an exponential distribution (see theorem 2.2 in Navarro and Sarabia8). In the bivariate case such a representation can be stated as follows.3PropositionIf (X,Y) has survival copula C^, F‾ is strictly decreasing in its support (0,∞) and G‾(t)=exp(−λt)=ψ(F‾(t)) for t≥0 and λ>0, where ψ(u)=exp(−λF‾−1(u)) for 0<u<1, then the survival function of S=X+Y can be written as H‾(t)=q‾(F‾(t)) with4q‾(u)=u+∫u1∂1C^v,ψ(u)ψ(v)dvfor u∈[0,1].ProofFirst we note that F‾ can be written as F‾(t)=ψ−1(exp(−λt)) for all t≥0, where ψ−1(u)=F‾(−log(u)/λ) is strictly increasing for u∈(0,1). Hence, exp(−λt)=ψ(F‾(t)) for t≥0, where ψ is a strictly increasing distortion function. Then, from (2), we getH‾(t)=F‾(t)+∫0tf(x)∂1C^(F‾(x),exp(λx−λt))dx=F‾(t)+∫0tf(x)∂1C^F‾(x),ψ(F‾(t))ψ(F‾(x))dxfor all t≥0. Finally, by doing the change v=F‾(x), we conclude the proof.1RemarkThe representation based on function ψ considered above is important in order to get formula (4) for the distortion function q‾ (see next subsection). For specific copulas and functions ψ, we can obtain explicit expressions for q‾ useful to study the preservation of IFR and ILR aging classes (see Example 4), while numerical approximations of the solutions of (4) can be used in other cases. Note that the distortion representation obtained in the preceding proposition can also be expressed in terms of G‾ (exponential model) if we are able to get an explicit expression of ψ−1 (the inverse function of ψ). Thus, F‾(t)=ψ−1(exp(−λt)) and H‾(t)=q‾(F‾(t))=q‾(ψ−1(exp(−λt)))=q‾2(G‾(t))for t≥0, where q‾2(u)=q‾(ψ−1(u)) for u∈[0,1]. This approach is used in Example 4.Distorted distributionsThe distorted distributions were introduced in the theory of choice under risk (see Yaari16). They will be used in the next sections to prove the main results on closure with respect to sums of dependent random lifetimes. This is their formal definition.2DefinitionWe say that Fq is a distorted distribution function from a distribution function F if Fq(t)=q(F(t)) for all t, where q:[0,1]→[0,1] is a distortion function, that is, q is continuous, increasing, and satisfies q(0)=0 and q(1)=1.We will use the notation Fq=q∘F for the composition. Note that the respective survival functions satisfy a similar relationship, that is, one can write F‾q=q‾∘F‾, where q‾(u)=1−q(1−u) for u∈[0,1] is another distortion function called dual distortion function.Clearly, the representations in (3) and in Proposition 3 are distortion representations for H‾. There are others well known models that can be represented as distorted distribution, like the proportional hazard rate (PHR) Cox model with q‾(u)=uθ for θ>0, or the proportional reversed hazard rate (PRHR) model with q(u)=uθ for θ>0. It can also be proved that the distributions of coherent systems with identically distributed components, are also distortions of the common components' distribution function (see, e.g., Navarro et al.2).We say that an aging class 𝒞 is preserved by a distortion function q if Fq=q∘F belongs to 𝒞 for all F in 𝒞. For example, it is easy to see that the PHR model preserves the IFR and DFR classes. The preservation of aging classes under distortions, such as for the ILR/DLR, IFR/DFR, NBU/NWU, and IFRA/DFRA classes, were studied in Navarro et al.2 In particular, in that paper (see remark 2.3) it is proved that if the IFR class is preserved, then the NBU and IFRA classes are preserved as well. An extension of the result obtained there for the IFR/DFR classes is stated in the following proposition.4PropositionIf Fq=q∘F for a differentiable distortion function q and an absolutely continuous distribution function F, and α(u)=uq‾′(u)/q‾(u) for u∈[0,1], then the following conditions are equivalent:i) The IFR (DFR) class is preserved by q; ii) The IFR (DFR) class is preserved by q for the standard exponential distribution; iii) The function α is decreasing (increasing) in (0,1).ProofWe are going to prove the proposition for the IFR class (the proof for the DFR class is analogous). First, we note that the hazard rate function rq of Fq can be obtained as5rq(t)=q‾′(F‾(t))q‾(F‾(t))f(t)=α(F‾(t))r(t),where r=f/F‾ is the hazard rate of F. Clearly, (i) implies (ii).To prove that (ii) implies (iii), we note that for the standard exponential distribution r(t)=1 for t≥0. Then we know that rq(t)=α(F‾(t))=α(e−t)is increasing for t≥0. Therefore, α is decreasing in [0,1].Finally, to prove that (iii) implies (i), we note that if α is decreasing, then α(F‾(t)) is increasing in t. This function is also nonnegative. Therefore, if r is increasing, then rq is increasing as well, since from (5) it is the product of two nonnegative increasing functions.The preceding proposition shows that the IFR and DFR classes are both preserved with respect to a distortion q if and only if the corresponding α is constant, which leads to the PHR model. In the other cases, if the IFR is preserved, then the DFR class is not (and vice versa). It may also happen that neither of them are preserved (when α is not monotone).For the ILR class we have the following new result, to be added to the one already described in Navarro et al..2 To provide its statement, we need the following definition.3DefinitionWe say that a function ϕ:ℝ→ℝ+ preserves logconcave functions if ϕ∘f is logconcave for any logconcave function f .For example, the functions ϕc(t)=tc preserve logconcave functions for any c>0.5PropositionIf Fq=q∘F for a differentiable distortion function q and q‾′ preserves logconcave survival functions, then the ILR class is preserved by q.ProofFirst, we note that logfq can be written as log(fq(t))=log(q‾′(F‾(t)))+log(f(t))for all tin the support of F.Note that if F is ILR, then F is also IFR, that is, F‾ is logconcave. Hence, we have that q‾′∘F‾ is logconcave. As, by assumption, f is logconcave, we have that fq is logconcave (since logfq is the sum of two concave functions).It should be noted here that q‾′ is not a distortion function. Also note that we do not have necessary and sufficient conditions for the preservation of the ILR class (as we have for the IFR/DFR classes). However, Proposition 5 holds for not necessarily positive variables too. Other sufficient conditions for the preservation of ILR class can be found in Navarro et al..2Similar conditions can be obtained for the classes LFR and LCFR. They can be stated as follows. The proofs are analogous, and therefore omitted.6PropositionIf Fq=q∘F for a differentiable distortion function q and an absolutely continuous distribution function F, and if the function α in Proposition 4 preserves logconcave survival functions, then the LFR class is preserved by q.7PropositionIf Fq=q∘F for a distortion function q and an IFR distribution function F and the function β=−logq‾ preserves logconcave survival functions, then Fq is LCFR. In particular, under this condition for β, the LCFR class is preserved by q.We must say that it is not easy to determine if a function ϕ preserves logconcave functions. A result similar to Proposition 4 can be obtained for increasing functions ϕ.The distorted distributions can also be used to get stochastic comparisons, see for example, Navarro et al..17 They can be stated as follows. The definitions and main properties of the stochastic (ST), hazard rate (HR), reversed hazard rate (RHR), and likelihood ratio (LR) orders can be found in Belzunce et al.5 and Shaked and Shanthikumar.98PropositionIf Ti has the survival function q‾i(F‾(t)) and the distribution function qi(F(t)) for i=1,2, then the following properties hold:(i) T1≤STT2 for all F if and only if q‾2≥q‾1 (or q2≤q1) in (0,1); (ii) T1≤HRT2 for all F if and only if q‾2/q‾1 decreases in (0,1); (iii) T1≤RHRT2 for all F if and only if q2/q1 increases in (0,1); (iv) T1≤LRT2 for all absolutely continuous distribution functions F if and only if q‾2′/q‾1′ decreases (or q2′/q1′ increases) in (0,1).In particular, from (5), the condition in (ii) can be replaced by α2(u)≤α1(u) for all u∈(0,1], where αi(u)=uq‾i′(u)/q‾i(u) for i=1,2.PRESERVATION RESULTS FOR THE ILR CLASSIn general, if X1,…,Xn are ILR, then one cannot affirm that also S=X1+⋯+Xn is ILR (see, e.g., Sections 4 and 5.1 below). Conditions for this preservation property are discussed here. In the first result we study the preservation of the ILR class when (X1,…,Xn) is a random vector having a logconcave probability density function. To this purpose we need Prékopa's theorem (see Prékopa18) which is stated in the following lemma.1LemmaPrékopa's theoremSuppose that g:ℝm×ℝk→[0,∞) is a logconcave function and thath(x)=∫ℝkg(x,y)dyis finite for each x∈ℝm. Then h is logconcave on ℝm.Now we are ready to state the following preservation result.1TheoremLet (X1,…,Xn) be a random vector having a logconcave pdf f. Then S=X1+⋯+Xn is ILR.ProofLet us consider the random vector (Y1,…,Yn) with Y1=S and Yj=Xj for j=2,…,n. Then the pdf g of (Y1,…,Yn) is g(y1,…,yn)=f(y1−y2−⋯−yn,y2,…,yn)for all y1,…,yn. Therefore, g is also logconcave, since logg(λy+(1−λ)z)=logg(λy1+(1−λ)z1,…,λyn+(1−λ)zn)=logf(λ(y1−y2−⋯−yn)+(1−λ)(z1−z2−⋯−zn),λy2+(1−λ)z2,…,λyn+(1−λ)zn)≥λlogf(y1−y2−⋯−yn,y2,…,yn)+(1−λ)logf(z1−z2−⋯−zn,z2,…,zn)=λlogg(y)+(1−λ)logg(z)for all λ∈(0,1) and all y,z∈ℝn. Finally, we note that the pdf h of S can be obtained as h(x)=∫ℝn−1g(x,y2,…,yn)dy2…dyn,thus, from Lemma 1, we have that h is logconcave.It must be pointed out that Theorem 1, as well as the following two statements, applies to any vector of random variables, not necessarily non‐negative. Multivariate normal, Wishart, Dirichlet, uniform distributions over compact and convex sets and elliptically contoured distributions are examples of logconcave densities (see Hu and Li19 and Walther11).As an immediate consequence of the preceding theorem we obtain the following result, proved in Karlin and Proschan.41CorollaryIf X1,…,Xn are independent and ILR, then X1+⋯+Xn is ILR.The preceding theorem can be extended to linear combinations as follows.2TheoremLet (X1,…,Xn) be a random vector having a logconcave pdf f. Then S=a1X1+⋯+anXn is ILR for all a1,…,an∈ℝ.ProofThe result is trivial if a1=⋯=an=0. Let us assume that a1≠0 (the proof is similar in the other cases) and let us consider the random vector (Y1,…,Yn) with Y1=S and Yj=Xj for j=2,…,n. Then the pdf g of (Y1,…,Yn) is g(y1,…,yn)=1|a1|fy1−a2y2−⋯−anyna1,y2,…,yn.Therefore, g is also logconcave (see the similar proof in the preceding theorem). Finally, we note that the pdf h of S can be obtained as h(x)=∫ℝn−1g(x,y2,…,yn)dy2…dynand so, from Lemma 1, we have that h is logconcave.In particular, this result proves that any sum of the Xi is ILR when (X1,…,Xn) has a logconcave joint pdf. Note that this property can be applied to L‐statistics, that is, to linear combinations of order statistics, when its joint pdf is logconcave. When the order statistics are obtained from i.i.d. random variables with a common univariate pdf f (the usual case), then it is well known that its joint pdf is logconcave whenever f is logconcave, see proposition 5 in An.20 This property can be extended as follows. If (X1,…,Xn) have a joint logconcave pdf fn and they are exchangeable (i.e., fn is permutation invariant), then the joint pdf of the order statistics gn is gn(x1,…,xn)=n!fn(x1,…,xn)for x1<⋯<xn (zero elsewhere), and so it is logconcave as well. This is an important result since the distribution of any L‐statistic obtained from them is ILR. Hence its pdf can be estimated by using MLE, see Walther.11In particular, we can apply this property to a single order statistic and therefore to the extremes min(X1,…,Xn) and max(X1,…,Xn). Note that this result is related to proposition 3 in Navarro and Shaked,3 which proves that if (X1,…,Xn) is exchangeable (i.e., it is permutation symmetric in law) and it has a logconcave pdf, then all the minima (series systems) and maxima (parallel systems) formed with the Xi are also ILR. A similar result is obtained in proposition 1 of that article for the IFR/DFR classes (without the assumption of exchangeability).In a similar way, this property can also be applied to linear combinations of generalized order statistics and the related concepts obtained from them (e.g., generalized spacings). Here we first need to prove the logconcave property for their joint pdf which is given by gn(x1,…,xn)=kγf(xn)F‾k−1(xn)∏i=1n−1f(xi)F‾mi(xi),where γ,k>0 (see, e.g., Cramer and Kamps21). Hence, clearly, it is logconcave when f is logconcave and mi≥0 for i=1,…,n−1. More conditions to get this property were obtained in Chen et al..22 The same property holds for linear combinations of progressively censored order statistics, see (1) in Cramer.23PRESERVATION RESULTS FOR THE IFR CLASSThe following example shows that IFR and ILR classes are not always preserved with respect to sums of dependent random variables.1ExampleLet us consider the random vector (X,Y) with uniform distribution over the set S=[0,1/2]2∪[1/2,1]2, that is, f(x,y)=2 for (x,y)∈S (zero elsewhere). Note that X and Y have a positive dependence and that the marginal distributions are uniform over (0,1) (i.e., they define a copula). Hence X and Y are both IFR and ILR. A straightforward calculation shows that the survival function H‾ of S=X+Y is H‾(s)=1−s2,for0≤s≤1/2;3/4−(s−1/2)2,for1/2<s≤1;1/2−(s−1)2,for1<s≤3/2;(2−s)2,for3/2<s≤2;and its pdf h is h(s)=2s,for0≤s≤1/2;2−2s,for1/2<s≤1;2s−2,for1<s≤3/2;4−2s,for3/2<s≤2;(zero elsewhere). Note that h is not logconcave (see Figure 1, left), that is, S is not ILR.The hazard rate of S is rS(s)=2s1−s2,for0≤s≤1/2;2−2s3/4−(s−1/2)2,for1/2<s≤1;2s−21/2−(s−1)2,,for1<s≤3/2;4−2s(2−s)2,for3/2<s≤2.It is plotted in Figure 1, right, and there we can see that it is not increasing, that is, S is not IFR.1FIGUREPlots of the pdf h (left) and the hazard rate rS for the sum of two dependent random variables with uniform distributions over the interval (0,1) (see Example 1)However, the IFR class is preserved in many models, even if, unfortunately, it does not seem possible to find a very general result such as Theorem 1 for the ILR class. Nevertheless, there are cases in which something can be proved, as in the following example, which deals with a model that can describe both positive or negative dependence. As the IFR class is preserved for this model, then the DFR class is not preserved (from Proposition 4).2ExampleLet us consider a random vector (X,Y) with a Farlie‐Gumbel‐Morgenstern (FGM) survival copula defined by6C^(u,v)=uv[1+θ(1−u)(1−v)]for u,v∈[0,1] and −1≤θ≤1. The independent case is represented by θ=0 while θ>0 (resp. <) represents a positive (negative) dependence (see, e.g., Nelsen15).If we also assume that X and Y have a common exponential distribution with hazard rate λ>0, then a straightforward calculation from (3) gives H‾(t)=q‾θ(F‾(t)) with F‾(t)=e−λt for t≥0 and q‾θ(u)=u−ulogu+θu−3+3u−logu−2ulogufor u∈[0,1] (see also Navarro and Sarabia8). As expected, in the independent case (θ=0), we obtain q‾0(u)=u−ulogu for u∈[0,1].Hence γθ(u)=q‾θ′(u)=−logu−θ4−4u+logu+4uloguand αθ(u)=uq‾θ′(u)q‾θ(u)=−logu−θ4−4u+logu+4ulogu1−logu+θ−3+3u−logu−2ulogufor u∈[0,1]. A straightforward calculation shows that both functions are decreasing for any θ∈[−1,1]. They are plotted in Figure 2 for θ=−1,−0.5,0,0.5,1. As γθ is decreasing, we have X≤LRX+Y from Proposition 8, (iv). Also, as αθ is strictly decreasing, from Proposition 4 we have that X+Y is IFR and not DFR for all λ>0 and all θ∈[−1,1].In Figure 3 we plot logh (left) and the hazard rate rS of the sum (right) for θ=−1,−0.5,0,1,0.5,1. In this figure we can see that IFR and ILR classes are preserved. As a consequence, NBU and IFRA classes are preserved as well (see Navarro et al.2). However, as mentioned above, the DFR class is not preserved. Moreover, from (5), we have that limu→0+αθ(u)=1andlimt→∞rS(t)=λfor all θ∈[−1,1]. Therefore, Proposition 2 for convolutions (θ=0) can be extended for the specific C‐convolution defined by the FGM survival copula for all −1≤θ≤1 when they have a common exponential distribution.It can be seen that this sum does not preserve LFR for θ=1 (see Figure 4, left). However, the plot of logRS (see Figure 4, right), where RS(t)=−logH‾(t)=∫0trS(x)dx,for t≥0, shows that LCFR is preserved in this dependence model.2FIGUREPlots of γθ(u)=q‾θ′(u) (left) and αθ(u)=uq‾θ′(u)/q‾θ(u) for the sum S=X+Y of two exponential distributions with a FGM survival copula with θ=−1,−0.5 (black), 0 (red, convolution), and 0.5,1 (blue)3FIGUREPlots of logh (left) and the hazard rate rS (right) for the sum S=X+Y of two dependent standard exponential distributions with a FGM survival copula with θ=−1,−0.5 (black), 0 (red, convolution), and 0.5,1 (blue). The dashed line represents the common hazard rate of X and Y and the limiting behavior of rS4FIGUREPlots of logrS (left) and logRS (right) for the sum S=X+Y of two dependent standard exponential distributions with a FGM survival copula with θ=−1,−0.5 (black), 0 (red, convolution), 0.5 (blue) and 1 (green). The dashed line represents the log of the common hazard rate of X and Y and the limiting behavior of logrSAlternative conditions for a sum of two dependent lifetimes to be IFR are described in the next statement.3TheoremLet (X,Y) be a random vector with survival copula C^ and marginal survival functions F‾ and G‾. Let S=X+Y. i) If X is ILR andΔ1(x,s)=∂1C^(F‾(x),G‾(s−x))is logconcave, then S is IFR. ii) If Y is ILR andΔ2(y,s)=∂2C^(F‾(s−y),G‾(y))is logconcave, then S is IFR.ProofLet us prove (i) (the proof of (ii) is symmetric). From (2), the survival function of S can be written as7H‾(s)=∫−∞∞f(x)∂1C^(F‾(x),G‾(s−x))dx=∫−∞∞f(x)Δ1(x,s)dxIt is easy to see that if Δ1(x,s) is logconcave and X is ILR, then f(x)Δ1(x,s) is also logconcave. Then the proof is completed by applying Prékopa's theorem (i.e., Lemma 1) to (7).The following numerical example shows how to use Theorem 3 to study the preservation of the IFR class.3ExampleLet us consider the sum of two dependent random variables X and Y having exponential and Rayleigh (Weibull) distributions with F‾(t)=exp(−t) and G‾(t)=exp(−t2) for t≥0 and with the FGM survival copula C^ defined in (6). Its partial derivative is8∂1C^(u,v)=v1+θ(1−2u)(1−v).Hence L(x,s)=logΔ(x,s)=log(G‾(s−x))+log1+θ(1−2F‾(x))(1−G‾(s−x)),that is, L(x,s)=−(s−x)2+log1+θ(1−2exp(−x))(1−exp(−(s−x)2)),for 0≤x≤s. Its 3D plot for θ=1 is shown in Figure 5, left. There, we can see that, in this case, L(x,s) is concave. Moreover, the exponential distribution is ILR. Hence, by applying Theorem 3, (i), we obtain that S=X+Y is IFR for θ=1. On the contrary, for θ=−1 the function L(x,s) is not concave, being, for example, L(0.5,0)=−0.06845,L(0.5,0.15)=−0.05031 and L(0.5,0.3)=−0.02315. The 3D plot for θ=−1 is shown in Figure 5, right. Thus, Theorem 3 cannot be applied to this case.5FIGUREPlots of the function L(x,s) in Example 3 for θ=1 (left) and θ=−1 (right)In the following example we show how to use Propositions 3 and 4 to check the preservation of the IFR class in sums of dependent random variables when just one of them has an exponential distribution.4ExampleLet us consider a random vector (X,Y) with the FGM survival copula defined in (6). Let us assume that Y has a standard exponential distribution and that the survival function of X is F‾(t)=1−1−exp(−t)for t≥0. Note that the survival function of Y can be represented as G‾(t)=exp(−t)=ψ(F‾(t))for ψ(u)=2u−u2. Hence, the survival function of S=X+Y can be written as H‾(t)=q‾1(F‾(t)), where q‾1 is obtained from (4) as q‾1(u)=u+∫u1∂1C^v,ψ(u)ψ(v)dvfor u∈[0,1]. In particular, for the FGM copula we get the expression in Equation (8) and q‾1(u)=u+ψ(u)∫u11ψ(v)dv+θψ(u)∫u11−2vψ(v)dv+θψ2(u)∫u11−2vψ2(v)dvfor u∈[0,1]. A straightforward calculation leads to q‾1(u)=u+12ψ(u)log2u−1−12θψ(u)log(u(2−u)3)−θψ(u)1+u2−ψ(u)+14ψ(u)logu2−ufor u∈[0,1]. By plotting α1(u)=uq‾1′(u)/q‾1(u) we see that it is decreasing for all θ∈[0,1]. However, we cannot apply Proposition 4 to this representation since F‾ is not IFR (actually it is strictly DFR). Instead, we are going to use a representation based on G‾ which is IFR. Note that F‾(t)=β(G‾(t)) for β(u)=1−1−u for u∈[0,1]. Then the survival function of S can be obtained as H‾(t)=q‾1(F‾(t))=q‾1(β(G‾(t)))=q‾2(G‾(t)),where q‾2(u)=q‾1(β(u)) for u∈[0,1]. Hence α2(u)=uq‾2′(u)q‾2(u)=uq‾1′(β(u))β′(u)q‾1(β(u))for u∈(0,1]. This function is plotted in Figure 6, left, for a sample of possible values of θ, that is, θ=−1,0.5,0,0.5,1. There we can see that α2 is decreasing for such values of θ, except for the case θ=−1. Then, from Proposition 4, S is IFR for θ=0.5,0,0.5,1. The hazard rate functions of X and Y (dashed lines) and S (continuous lines) are plotted in Figure 6, right. There we can see that S is IFR for such a sample of values for θ, except for θ=−1 (green line).6FIGUREPlots of the function α2 (left) and the hazard rate function rS (right) for the sum S=X+Y in Example 4 and θ=−1 (green), −0.5 (black), 0 (red, convolution), 0.5,1 (blue). The dashed lines represent the hazard rate functions of X (decreasing line) and Y (constant line)RELEVATION TRANSFORM MODELSA model of interest in reliability theory which deals with sums of not independent lifetimes is the relevation transformation. This model is considered here as an example where the results described in previous sections can be applied, but it is of interest also because it provides meaningful counterexamples. Under a relevation transform (see, e.g., Krakowski24 or Belzunce et al.25), the unit with lifetime X is replaced (or repaired) when it fails at a time x≥0 by a unit Y having survival function G‾ but with the same age as X, that is, with survival function G‾x(y)=Pr(Y>y|X=x)=Pr(Y−x>y|Y>x)=G‾(x+y)G‾(x)for y≥0. Then X and Y are dependent and the survival function H‾(t)=Pr(S>t) of S=X+Y can be obtained as9H‾(t)=F‾(t)+∫0tG‾x(t−x)f(x)dx=F‾(t)+∫0tG‾(t)G‾(x)f(x)dx.This is formula (1.1) in Krakowski,24 in which the notation H‾=F‾#G‾ is introduced. In a relevation transform with perfect repair, the unit with lifetime X is replaced when it fails at a time x by an independent new unit with lifetime Y having survival function G‾x(y)=G‾(y). Then10H‾(t)=F‾∗F‾(t)=F‾(t)+∫0tG‾(t−x)f(x)dx,which is the survival function of the sum of two independent random variables (convolution). More realistic cases are considered in the following subsections.Relevation transform with minimal repairIn a relevation transform with minimal repair, the unit X is replaced when it fails at a time x by a unit Yx=(Y−x|Y>x) having survival function F‾ but with the same age as X, that is, G‾x(y)=F‾(x+y)/F‾(x) for y≥0. This is equivalent to minimally repair the broken unit to be like it was before its failure. In this case, from (9), H‾(t)=F‾#F‾(t)=F‾(t)+∫0tF‾(t)F‾(x)f(x)dx=F‾(t)−F‾(t)logF‾(t).This is formula (3.1) in Krakowski24 with the notation F‾#F‾ for the relevation transform under minimal repair. Note that for the exponential distribution, this expression coincides with the one obtained for the usual convolutions from (10).Note that it is a distorted distribution with dual distortion q‾(u)=u−ulogu for u∈[0,1]. The pdf of S is h(t)=−f(t)logF‾(t) and its hazard rate is rS(t)=h(t)H‾(t)=−logF‾(t)1−logF‾(t)r(t)for t≥0, where r=f/F‾ is the hazard rate of F‾. Hence q‾′(u)=−logu and α(u)=−logu/(1−logu) for u∈(0,1). As α is decreasing in (0,1), the IFR class is preserved by the relevation transform with minimal repair, which is a well known result (see the references in Pellerey et al.10). As the IFR class is preserved, so are the NBU and IFRA classes. As α≤1, then S=X+Y≥HRX for all F‾. Moreover, from (5), we get limt→∞rS(t)r(t)=limu→0+α(u)=1.Therefore, Proposition 2, which deals with the limit of the hazard rate of the sum of independent variables (thus, with convolutions), can be extended for the specific C‐convolution defined by the relevation transform.However, q‾′(u)=−logu does not always preserve logconcave survival functions. Thus the ILR class is not always preserved. Sufficient conditions for that preservation in this model were obtained in Pellerey et al.,10 where it is proved that the ILR class is preserved when F is also LCFR.For example, we observe that the ILR class is preserved for a positively truncated standard normal (Gaussian) distribution as can be seen in Figure 7, left. Note that the normal distribution satisfies condition LFR, which implies condition LCFR. However, this is not the case for a uniform distribution over (0,1) (see Figure 7, right) since the uniform distribution is not LCFR. Therefore, Theorem 1 cannot be extended to arbitrary vectors, that is, the ILR class is not always preserved.7FIGUREPlots of logh for the relevation transform under minimal repair for a truncated standard Gaussian (normal) distribution (left) and for a uniform distribution over (0,1) (right)Relevation transform with imperfect repairIn a relevation transform with imperfect repair, the unit X is replaced when it fails at a time x by a unit Yx=(Y−x|Y>x) having survival function G‾=F‾θ for θ>0 and θ≠1 but with the same age as X (see Brown and Proschan26 and Bhattacharjee27 for details). In this case, from (9), H‾θ(t)=F‾(t)+∫0tF‾θ(t)F‾θ(x)f(x)dx=F‾(t)+F‾θ(t)F‾1−θ(t)θ−1−1θ−1for t≥0. Note that it is also a distorted distribution with dual distortion q‾θ(u)=θθ−1u−1θ−1uθfor u∈[0,1]. Also note that it is a negative mixture of F‾ and F‾θ and that limθ→∞q‾θ(u)=u, limθ→0+q‾θ(u)=1, limθ→1q‾θ(u)=u−ulogu for u∈(0,1) (i.e., a relevation transform with minimal repair, as expected). Their plots for different values of θ can be seen in Figure 8, left.8FIGUREPlots of q‾θ (left) and αθ (right) for a relevation transform with imperfect repair and θ=0.25,0.5,0.75 (black), θ=1 (red, minimal repair) and θ=1.5,2,3,5 (blue)The pdf of S in this model is hθ(t)=θθ−11−F‾θ−1(t)f(t)and its hazard rate is rθ(t)=αθ(F‾(t))r(t), where r=f/F‾ is the hazard rate of F‾ and αθ(u)=θ1−uθ−1θ−uθ−1for u∈[0,1]. The plots of αθ can be seen in Figure 8, right. As they are ordered, the respective sums are ordered in hazard rate sense and, as they are strictly decreasing, from Proposition 4 it follows that the IFR class is preserved (and the DFR class is not preserved). This is a well‐known property, proved in Brown and Proschan.26 As a consequence, NBU and IFRA classes are preserved as well. Even more, since αθ≤1, then S≥HRX for all F‾ and all θ>0. Moreover, from (5), limt→∞rθ(t)r(t)=limu→0+αθ(u)=1forθ>1;θforθ<1.Therefore, Proposition 2 for convolutions can be extended for the specific C‐convolution defined by the relevation transform with imperfect repair and θ>1. However, this property does not hold when 0<θ<1. The respective hazard rates for a standard exponential distribution can be seen in Figure 9, right. Note that they have different limiting behavior.9FIGUREPlots of loghθ (left) and hazard rate rθ (right) for a relevation transform with imperfect repair with a baseline standard exponential distributions and θ=0.25,0.5,0.75 (black), θ=1 (red, minimal repair) and θ=1.5,2,3,5 (blue). The dashed lines represent the limiting behavior for t→∞However, q‾θ′(u)=θθ−11−uθ−1does not always preserve logconcave functions. So we do not know if the ILR class is preserved. For example, in the case of a standard exponential distribution, the ILR class is preserved (see Figure 9, left). As q‾θ′(u) is decreasing in u, in this dependence model, from Proposition 8, (iv), we also get S≥LRX for all F‾ and all θ>0.2RemarkThe sequential order statistics fit to a relevation transform model with different distributions, see Cramer and Kamps.21 However, they do not fit to a minimal (or imperfect) repair model. In the case of a PHR model for the baseline distributions, they can be written as distortions. This case was studied in Burkschat and Navarro.28 So we can also apply to them the new results for distortions obtained here.CLOSURES WITH RESPECT TO MAXIMUMThe closure properties of maximum also have interest in several applicative areas. Some general results for IFR/DFR and ILR/DLR classes were obtained in Hu and Li19 and Navarro and Shaked.3 The preservation for other classes, such as NBU, DMRL and NBUE, was studied in Abouammoh and El‐Neweihi29 and Navarro.30 By using Proposition 4 we can obtain the following result, dealing with the preservation of the IFR class in maxima of identically distributed (i.d.) components.4TheoremIf X1,…,Xn are i.d. with copula C, then the following conditions are equivalent:i) The IFR (DFR) class is preserved by M=max(X1,…,Xn); ii) The IFR (DFR) class is preserved by M=max(X1,…,Xn) for the standard exponential distribution; iii) The function α(u)=uδ′(1−u)/(1−δ(1−u)) is decreasing (increasing) in (0,1), where δ(u)=C(u,…,u) is the diagonal section of C.ProofThe distribution function of M is FM(t)=Pr(X1≤t,…,Xn≤t)=C(F(t),…,F(t))=δ(F(t)),where F is the common distribution function of X1,…,Xn. Hence, its survival function is F‾M(t)=1−δ(F(t))=1−δ(1−F‾(t))=q‾(F‾(t))for q‾(u)=1−δ(1−u). Then the proof is completed by applying Proposition 4.It is well known that the IFR property is preserved in maxima of i.i.d. random variables (see Esary and Proschan31), that is, the property in (iii) above holds for the product copula. This property can be extended for k‐out‐of‐n systems (order statistics) with i.i.d. components (see Esary and Proschan31). It is also true for other copulas (see next example). However, it can be easily verified that this preservation property is not always true.5ExampleLet M=max(X1,X2) with (X1,X2) having the same marginal distribution function F and the following Clayton‐Oakes copula11C(u,v)=uvu+v−uvfor u,v∈[0,1]. Its diagonal section is δ(u)=C(u,u)=u2−ufor u∈[0,1]. Hence the dual distortion function of M is q‾(u)=1−δ(1−u)=1−1−u1+u=2u1+uand the associated α function is α(u)=11+ufor u∈[0,1]. As α is strictly decreasing, the IFR class is preserved by M but the DFR class is not. Then the NBU and IFRA classes are preserved as well. Moreover, the limiting behavior of the hazard rate of M can be determined as limt→∞rM(t)r(t)=limu→0+α(u)=limu→0+11+u=1.Also note that Xi≤HRM holds for any F (since α(u)≤1 for all u∈[0,1]). Even more, as q‾′ is decreasing, then Xi≤LRM holds for any F from Proposition 8, (iv).It is easy to verify that results similar to those described in Example 5 can be provided considering the FGM copula as the connecting copula of (X1,X2) instead of the Clayton‐Oakes copula.For non i.d. components we have the following result. The proof is immediate from proposition 2.5 in Navarro et al..2 The similar results for other aging classes such as DMRL/IMRL, NBU/NWU, and IFRA/DFRA can be obtained from the results for distortions given in Navarro30 and Navarro et al..2 Results for NBUE/NWUE aging classes are still an open question.5TheoremIf X1,…,Xn have distribution functions F1,…,Fn and copula C, and if the following conditions hold:i) X1,…,Xn are IFR (DFR); ii) The functionsαi(u1,…,un)=ui∂iC(1−u1,…,1−un)1−C(1−u1,…,1−un)are decreasing (increasing) in (0,1)n for i=1,…,n;then M=max(X1,…,Xn) is IFR (DFR).For the ILR class we have the following result, that follows from Proposition 5.6TheoremLet (X1,…,Xn) be a random vector with copula C and ILR margins Xi having the same distribution F. Let δ(u)=C(u,…,u) be the diagonal section of C. If δ′ preserves logconcave distribution functions, then M=max(X1,…,Xn) is ILR.ProofObserve that the distribution function of M can be written as FM(t)=δ(F(t)), for all t. Thus its density fM can be written as fM(t)=δ′(F(t))f(t), so that logfM(t)=log(δ′(F(t)))+logf(t)for all t. The term logf is concave by assumption (being F ILR). Moreover, since logconcavity of f implies logconcavity of F (see, e.g., Barlow and Proschan1(p. 77)), if δ′ preserves logconcave distribution functions, then the first term is also concave, and the assertion follows.Note that non‐negativity of X1,…,Xn is not required in Theorem 6. Unfortunately, as stated already above, it is not easy to check if a function preserves logconcavity. However, Theorem 6 provides an immediate tool to check when the maximum of dependent and ILR variables is still ILR, knowing the copula C and the distribution F. For example, if C is the Clayton copula with parameter θ∈(0,∞), so that δ(u)=(2u−θ−1)−1/θ for u∈[0,1], and the Xi are exponentially distributed with the same rate λ, then one can easily verify that the ratio δ′(F(t+s))δ′(F(t))is decreasing in t≥0 for all s≥0, that is, that δ′(F(t)) is logconcave. Here δ′(u)=2u−1−θ(2u−θ−1)−(1+θ−1) for u∈[0,1]. Thus, the maximum is ILR.Similarly, if the connecting copula C is an AMH copula (see (4.2.3) in Nelsen15(p. 116)) with parameter θ∈[−1,1), so that δ(u)=u2/(1−θ(1−u)2), and the Xi are exponentially distributed with the same rate, then one can easily verify that the ratio above is decreasing in t≥0 for all s≥0, that is, that δ′(F(t)) is logconcave. Here δ′(u)=(2u(1−θ(1−u)2)−2θ(1−u)t2)/(1−θ(1−u)2)4. Thus, again the maximum is ILR.The preceding theorem can also be applied to copulas having diagonal sections δ(u)=uc for 1<c≤2 (since, as mentioned above, the functions q‾′(u)=δ′(u)=uc−1 preserve logconcave functions for c>1). For example, this property holds for the Cuadras‐Augé family of copulas, see Nelsen.15(p. 15)Load‐sharing models with time‐homogeneous failure ratesHere we consider a particular model dealing with the maximum of dependent variables. In the case of absolute continuity, the joint probability law of n non‐negative random variables X1,…,Xn can be described by means of the family of its Multivariate Conditional Hazard Rates (MCHR) functions. Such a description is alternative but mathematically equivalent to the one expressed in terms of the joint density function of (X1,…,Xn) (see, e.g., the review paper Shaked and Shanthikumar32 and the references cited therein).In a sense, the MCHR functions arise as direct extensions of the univariate concept of hazard rate function for a single non‐negative random variable X. In the bivariate case the family of the MCHR functions is ℒ={λ10(t),λ20(t),λ11(t,x),λ21(t,x)}, where λ10(t)=limΔt→0+Pr[X1>t+Δt|X1>t,X2>t]Δt,λ20(t)=limΔt→0+Pr[X2>t+Δt|X1>t,X2>t]Δt,and, for 0<x<t, λ11(t,x)=limΔt→0+Pr[X1>t+Δt|X1>t,X2=x]Δt,λ21(t,x)=limΔt→0+Pr[X2>t+Δt|X2>t,X1=x]Δt.The functions λ10(t),λ20(t),λ11(t,x), and λ21(t,x) belonging to ℒ can be obtained in terms of the joint density function fX1,X2 of (X1,X2). One can also check that, vice versa, fX1,X2 can be recovered from the knowledge of the family ℒ.In particular, special models arise from the condition that λ11(t,x) and λ21(t,x) do not depend on x. The corresponding models were called Load‐Sharing models, see, for example, Spizzichino33 and references cited therein for more details and further remarks. They were also called models in which the failure rates depend on the working set by some authors, such as in Ross.34 The term load‐sharing model is inspired by the model in which L(t) has the meaning of a level of load to be shared by the surviving components at time t>0, and by letting L(t)=λ10(t)+λ20(t) if both components are alive at time t, or L(t)=λi1(t) if only component i is alive at time t.Note that if X1 and X2 are exchangeable, so that λ10(t)=λ20(t)=λ0(t) and λ11(t)=λ21(t)=λ1(t), then, in agreement to what is stated above, it must be λ1(t)=2λ0(t). However, we will consider here the more general case where λ1(t)=θλ0(t) for some θ>0.Let us thus consider a vector (X1,X2) with distribution defined as in the model above. One can be interested in the distribution of the sum S=X1+X2, having survival function H‾θ depending on the parameter θ. In this case, with straightforward calculations, one can verify that H‾θ(t)=F‾(t)+∫0tF‾θ(t)F‾θ(x)f(x)dxfor t≥0, where F‾(t) is the survival function corresponding to the hazard rate λ0(t), while F‾θ(t) is the survival function corresponding to the hazard rate λ1(t). Thus, the study of the sum of two lifetimes in a load‐sharing model reduces to the case of minimal repair models (see Section 5.1).However, dealing with these models, one is mainly interested in the total time the components can deal with the load to be shared, that is, in M=max(X1,X2). Letting H‾θ be the survival function of M, one has H‾θ(t)=Pr(X1>t,X2>t)+∫0tPr(X1=x,X2>t)dx+∫0tPr(X1>t,X2=x)dxthat is, H‾θ(t)=F‾2(t)(1−2logF‾(t))ifθ=2,and H‾θ(t)=12−θ2F‾θ(t)−θF‾2(t)ifθ≠2for t≥0. In both cases we get distorted distributions.Thus, for θ=2, one has q‾2(u)=u2(1−2logu) and q‾2′(u)=−4ulogu. It follows that α2(u)=−4logu1−2loguwhich is decreasing in (0,1). Therefore, by Proposition 4, if F‾ is IFR (i.e., if λ0(t) is increasing), then M is IFR as well.For θ≠2 one has q‾θ(u)=12−θ(2uθ−θu2) and q‾θ′(u)=12−θ(2θuθ−1−2θu). It follows that αθ(u)=2θuθ−2θu22uθ−θu2,which is also decreasing in (0,1) for every strictly positive θ, with θ≠2. Therefore, again by Proposition 4, if F‾ is IFR (i.e., if λ0(t) is increasing), then M is IFR as well.It is interesting to observe that this property is satisfied also for θ∈(0,1) even if in this case the marginal distributions of X1 and X2 can be not IFR (for example, if λ0(t)=λ∈ℝ+, then at the failure of one component the hazard of the remaining one reduces to θλ, which is smaller than λ).For some specific choices of λ0(t), and therefore of F‾, one can verify that M is also ILR. For example, if θ=2 and λ0(t)=bktk−1 for t≥0 (which is the case of a Weibull distribution), then logq‾2′(F‾(t))=log(4btkexp(−btk))=log(4b)+klogt−btkfor t≥0. Since this function is concave for any positive b and k then, by Proposition 5, one has that M is also ILR.ACKNOWLEDGMENTSThe authors would like to thank the editors and the anonymous reviewers for several helpful suggestions and tasks. Jorge Navarro is supported in part by Ministerio de Ciencia e Innovación of Spain under Grant PID2019‐103971GB‐I00/AEI/10.13039/501100011033 and Franco Pellerey by the GNAMPA research group of INdAM (Istituto Nazionale Di Alta Matematica), Italy, and by the Grant Progetto di Eccellenza, CUP: E11G18000350001, Italy.DATA AVAILABILITY STATEMENTData sharing not applicable to this article as no datasets were generated or analyzed during the current study.REFERENCESBarlow RE, Proschan F. Statistical Theory of Reliability and Life Testing: Probability Models. Holt, Rinehart and Winston, INC; 1975.Navarro J, del Águila Y, Sordo MA, Suárez‐Llorens A. Preservation of reliability classes under the formation of coherent systems. Appl Stoch Model Bus Ind. 2014;30:444‐454.Navarro J, Shaked M. Some properties of the minimum and the maximum of random variables with joint logconcave distributions. Metrika. 2010;71:313‐317.Karlin S, Proschan F. Pólya type distributions of convolutions. Ann Math Stat. 1960;31:721‐736.Belzunce F, Martínez‐Riquelme C, Mulero J. An Introduction to Stochastic Orders. Academic Press; 2015.Alimohammadi M, Alamatsaz MH, Cramer E. Convolutions and generalization of logconcavity: implications and applications. Naval Res Log. 2016;63:109‐123.Pellerey F, Navarro J. Stochastic monotonicity of dependent variables given their sum. Test. 2021;1‐19. doi:10.1007/s11749-021-00789-5Navarro J, Sarabia JM. Copula representations for the sums of dependent risks: models and comparisons. Probab Eng Inf Sci. 2020;1‐21. doi:10.1017/S0269964820000649Shaked M, Shanthikumar JG. Stochastic Orders. Springer Series in Statistics. Springer; 2007.Pellerey F, Shaked M, Zinn J. Nonhomogeneous Poisson processes and logconcavity. Probab Eng Infor Sci. 2000;14:353‐373.Walther G. Inference and modeling with log‐concave distributions. Stat Sci. 2009;24:319‐327.Block H, Langberg N, Savits T. The limiting failure rate for a convolution of life distributions. J Appl Probab. 2015;52:894‐898.Cherubini U, Mulinacci S, Romagnoli S. A copula‐based model of speculative price dynamics in discrete time. J Mult Anal. 2011;102:1047‐1063.Durante F, Sempi C. Principles of Copula Theory. CRC/Chapman & Hall; 2016.Nelsen RB. An Introduction to Copulas. 2nd ed. Springer; 2006.Yaari ME. The dual theory of choice under risk. Econometrica. 1987;55:95‐115.Navarro J, del Águila Y, Sordo MA, Suárez‐Llorens A. Stochastic ordering properties for systems with dependent identically distributed components. Appl Stoch Model Bus Ind. 2013;29:264‐278. doi:10.1002/asmb.1917Prékopa A. A note on logarithmic concave measures. Acta Sci Math. 1973;34:335‐343.Hu T, Li Y. Increasing failure rate and decreasing reversed hazard rate properties of the minimum and maximum of multivariate distributions with log‐concave densities. Metrika. 2007;65:325‐330.An MY. Logconcavity versus logconvexity: a complete characterization. J Eco Theory. 1998;80:350‐369.Cramer E, Kamps U. Marginal distributions of sequential and generalized order statistics. Metrika. 2003;58:293‐310.Chen H, Xie H, Hu T. Log‐concavity of generalized order statistics. Stat Probab Lett. 2009;79:396‐399.Cramer E. Logconcavity and unimodality of progressively censored order statistics. Stat Probab Lett. 2004;68:83‐90.Krakowski M. The relevation transform and a generalization of the Gamma distribution function. Revue Française D'Automatique, Informatique et Recherche Opérationnelle Mai. 1973;V‐2:107‐120.Belzunce F, Martínez‐Riquelme C, Mercader JA, Ruiz JM. Comparisons of policies based on relevation and replacement by a new one unit in reliability. Test. 2021;30:211‐227.Brown M, Proschan F. Imperfect repair. J Appl Probab. 1983;20:851‐859.Bhattacharjee MC. New results for the Brown‐Proschan model of imperfect repair. J Stat Plann Infer. 1987;16:305‐316.Burkschat M, Navarro J. Stochastic comparisons of systems based on sequential order statistics via properties of distorted distributions. Probab Eng Infor Sci. 2018;32:246‐274.Abouammoh A, El‐Neweihi E. Closure of NBUE and DMRL under the formation of parallel systems. Stat Probab Lett. 1986;4:223‐225.Navarro J. Preservation of DMRL and IMRL aging classes under the formation of order statistics and coherent systems. Stat Probab Lett. 2018;137:264‐268.Esary J, Proschan F. Relationship between system failure rate and component failure rates. Technometrics. 1963;5:183‐189.Shaked M, Shanthikumar JG. Multivariate conditional hazard rate functions ‐ an overview. Appl Stoch Model Bus Ind. 2015;31:285‐296.Spizzichino F. Reliability, signature, and relative quality functions of systems under time‐homogeneous load‐sharing models. Appl Stoch Model Bus Ind. 2019;35:158‐176.Ross SM. A model in which component failure rates depend on the working set. Naval Res Log. 1984;31:297‐300.

Journal

Applied Stochastic Models in Business and IndustryWiley

Published: Mar 1, 2022

Keywords: C‐convolutions; copulas; load sharing models; relevation transforms

There are no references for this article.