Comparisons of the Expectations of System and Component Lifetimes in the Failure Dependent Proportional Hazard Model

In the failure dependent proportional hazard model, it is assumed that identical components work jointly in a system. At the moments of consecutive component failures the hazard rates of still operating components can change abruptly due to a change of the load acting on each component. The modification of the hazard rate consists in multiplying the original rate by a positive constant factor. Under the knowledge of the system structure and parameters of the failure dependent proportional hazard model, we determine tight lower and upper bounds on the expected differences between the system and component lifetimes, measured in various scale units based on the central absolute moments of the component lifetime. The results are specified for the systems with unimodal Samaniego signatures.


Introduction
We consider an arbitrarily fixed coherent system composed of n elements with a structure function ϕ : {0, 1} n → {0, 1}. It has the Samaniego structural signature s = (s 1 , . . . , s n ) ∈ [0, 1] n whose coordinates are determined as follows . . x n ), i = 1, . . . , n. (1) The notion of the signature was introduced in Samaniego (1985), and formula (1) is due to Boland (2001) (see also Marichal et al. 2011). We assume that nonnegative random variables T 1 , . . . , T n are the lifetimes of the system components, and their joint distribution satisfies conditions of the failure dependent proportional hazard model (see, e.g., Hollander and Peña 1995;Aki and Hirano 1997;Burkschat 2009;Navarro and Burkschat 2011). It follows that T 1 , . . . , T n are exchangeable and the respective order statistics T 1:n , . . . , T n:n satisfy assumptions of the generalized order statistics model proposed by Kamps (1995a) (see also Kamps 1995bKamps , 2016 with some baseline distribution function F and a vector of positive parameters γ = (γ 1 , . . . , γ n ). Note that we obtain the same joint distribution of T 1:n , . . . , T n:n if we replace F and γ = (γ 1 , . . . , γ n ) with F α = 1 − (1 − F ) α and γ α = γ 1 α , . . . , γ n α for some α > 0. For convenience of interpretation we choose the model parameters so that γ 1 = n. Then the baseline distribution function F represents the lifetime distribution of a component which operates under optimal conditions without outer stress acting on it. The stress that working components undergo after consecutive failures of other components is described by means of the parameters γ 1 , . . . , γ n (see, e.g., Kamps 1995a;Cramer and Kamps 2001;Cramer 2016).
The marginal distribution function of consecutive component failure times T r:n can be written as G γ ,r • F , r = 1, . . . , n, where G γ ,r is the distribution function of the rth generalized order statistic with the standard uniform baseline distribution. It depends on the first r coordinates of the parameter vector γ by means of a complicated formula which strongly depends on the multiplicities of parameter values. Respective expressions can be found in Cramer and Kamps (2003). Exchangeability of the component lifetimes in the failure dependent proportional hazard model implies that the common distribution function of every component is the uniform mixture of the distribution functions of order statistics (see Rychlik 1993). By the same reason, the distribution function of the system lifetime is the convex combination of order statistics distribution functions where the occurring coefficients coincide with the elements of the Samaniego signature vector (see Navarro et al. 2008, andMarichal et al. 2011). Formula (3) is called Samaniego representation, because it was first presented in Samaniego (1985) under the restriction to i.i.d. component lifetimes with a continuous parent distribution.
The purpose of our paper is to evaluate the expected lifetime of the system ET with fixed structure ϕ working in given circumstances represented by the parameter vector γ = (γ 1 , . . . , γ n ) when the baseline distribution function F of the component lifetime is unknown. In the following, we always assume that F and therefore also H 1 is nondegenerate. Moreover, for any distribution function F we denote by F −1 the right-continuos version of the corresponding quantile function. We compare ET with the mean value of the single component lifetime μ 1 = ET 1 . The difference is gauged in the scale units based on central absolute moments of component lifetime σ 1 (p) = (E|T 1 − ET 1 | p ) 1/p of various orders p ≥ 1 under the conditions that they are finite. Moreover, distributions attaining derived bounds, with a support in the non-negative real numbers, are given. For the parent distributions with finite support, we also consider the scale measure σ 1 (∞) = max{H −1 1 (1) − μ 1 , μ 1 − H −1 1 (0)} being the distance of the mean from the furthest point of the support. For the comparison of ET with the mean of the underlying distribution function F in the case of k-out-of-n systems, which reduces to the derivation of bounds for single generalized order statistics, the reader is referred to Cramer et al. (2002Cramer et al. ( , 2004 and Goroncy (2014).
The paper is organized as follows. In Section 2 we describe general results. In Section 3, we deliver more specific evaluations for the systems with unimodal signatures, and present exemplary numerical evaluation for k-out-of-n systems working under a practically justified load-sharing regime.

Main Results
We observe that every distribution function G γ ,r is a differentiable, strictly increasing function on [0, 1] that maps the interval onto itself. The properties are shared by the convex combinationsG γ and G γ ,s whose derivatives we denote byg γ and g γ ,s , respectively. Obviously, if g γ ,r denotes the density corresponding to G γ ,r , then where μ 1 = ET 1 and σ 1 (p) = (E|T 1 − ET 1 | p ) 1/p , and c p , c p ∈ (0, 1) are the solutions to equations , respectively), for some 0 ≤ x ≤ 1, then the lower (upper, respectively) bound is attained by the distribution functions H 1 (H 1 , respectively) satisfying provided that respectively.
Proof The actual component lifetime distribution function under the failure dependent proportional hazard regime has representation (2), where F is the baseline distribution function of the generalized order statistics model. Accordingly, the expectation and the pth absolute central moment take on the forms respectively. By Eq. 3, the expectation of the system lifetime for the system with signature s can be written as Since is the density function of distribution function G γ ,s (G −1 γ (x)) supported on [0, 1], we have clearly Also, by Eq. 9, for any c ∈ R Combining (11), (12) and (13), we obtain for arbitrary real c. We now use formula (14) for proving the upper bound in Eq. 5 and conditions of its attainability. Since the first factor in the integral of Eq. 14 is non-decreasing by assumption, we can write (see, e.g., Moriguti 1953;Rychlik 2001). By definition and monotonicity of G γ ,s (G −1 γ (x)), the derivative h γ ,s (x) of the greatest convex minorant is non-decreasing and non-negative. It satisfies ) on some subintervals of [0, 1], and is linear on the others, but each linear part is tangent to G γ ,s (G −1 γ (x)) at the respective interval end-points. This implies that the derivative h γ ,s (x) is continuous. We finally check that it is bounded as well. Note that the original density function is bounded, because by Eq. 4 we have The property is shared by h γ ,s (x), because it is equal to there. Observe that the left-hand side of Eq. 6 is continuous in c, and monotonously nondecreasing from 0 at 0 to a finite value at 1. The right-hand side is continuously non-increasing from 1 0 h 1/(p−1) γ ,s (x)dx < ∞ at c = 0 to 0 at c = 1. Accordingly, there exists a non-empty, possibly degenerate interval consisting of solutions to Eq. 6. Plugging c = h γ ,s (c p ) with any c p that solves (6) into (15), and then applying the Hölder inequality we get which is equivalent to the upper evaluation in Eq. 5 by Eq. 10.
For some parameters γ and signatures s, it may happen that G γ ,s (x) ≥G γ (x) and so (6) is satisfied for any 0 ≤ c ≤ 1, and, in consequence, B γ ,s (p) = 0 for every 1 < p < ∞. We do not analyze here if these zero bounds are optimal. Now we check that Eq. 7 with (8) determine the conditions for attainability of the bound Moriguti 1953;or Rychlik 2001, Lemma 3, p. 34). The minorant is linear on these intervals, and its derivative h γ ,s (x) is constant there. Relation (7) guarantees constancy of H −1 1 (x) on these intervals. The equality in the Hölder inequality (16) holds when proportional with a nonnegative proportionality coefficient α, and here α = σ 1 (p) > 0, as desired. It remains to show that Eqs. 7 and 8 define the quantile function of a lifetime distribution with assumed moments. Firstly, H −1 1 is a nondecreasing function by Eq. 7 and non-negative by Eq. 8. The respective distribution function has the expectation equal to μ 1 , when (7) integrates over (0, 1) to 0, which is guaranteed by condition (6). It has the pth absolute central moment σ 1 (p) which follows from the equations The proof for the lower bounds is analogous.
For p = 2, relations h γ ,s (c 2 ) = h γ ,s (c 2 ) = 1 hold, and all the formulas of Theorem 1 simplify significantly. In particular, we have The extreme cases p = 1 and p = ∞ are treated separately.
Theorem 2 Under the assumptions and notation of Theorem 1, for p = 1 and every distribution function of component lifetime T 1 with finite expectation μ 1 , we get where σ 1 (1) denotes the mean absolute deviation from the mean of T 1 .
≥ 0, respectively. If any of these sets has the Lebesgue measure zero, then the bound is attained in the limit when we shift probability mass ε > 0 to all corresponding probabilities, and let ε tend to 0.
Observe that in cases when h −1 γ ,s ({0}) and h −1 γ ,s ({0}) have Lebesgue measures zero, the approximations proposed in the last statement of Theorem 2 impose increasing restrictions σ 1 (1) ≤ 2μ 1 ε on the relation between the dispersion and the mean of T 1 .
Proof We focus on upper bounds, because arguments for deriving lower ones are similar. Recalling (15), we get for every c ∈ R. Since Determining the attainability conditions, we omit the case when G γ ,s (x) ≥G γ (x) for all 0 ≤ x ≤ 1 which leads to B γ ,s (1) = 0. In the opposite case, we have h γ ,s (0) < h γ ,s (1). Suppose first that h γ ,s (x) = h γ ,s (0) for some x > 0 and h γ ,s (x) = h γ ,s (1) for some x < 1. Then we get the equality in the latter inequality of Eq. 19 with c For h γ ,s (x) = h γ ,s (0) and h γ ,s (x) = h γ ,s (1) on two non-degenerate intervals, it is possible that H γ ,s (x) < G γ ,s (G −1 γ (x)) on their interiors. In order to get the equality in the first inequality of Eq. 19, we require that H −1 1 is constant there. The respective distribution is non-degenerate with mean μ 1 if the two constants are non-zero. Relations for some a, b > 0 combined with the moment constraints imply that a = . The component lifetime is non-negative if μ 1 ≥ a.
The attainability proof in the cases when h γ ,s is constant on either of the neighborhoods of 0 and 1, and is strictly increasing on the other is much the same, and therefore we omit it here.  (1), then the equality in the latter inequality in Eq. 20 holds when

Theorem 3 Under the assumptions and notation of Theorem 1, for p = ∞ and all bounded component lifetime distributions we have
consists only of the single point 1 2 , then the denominator in the formula describing the middle support point vanishes, but so do the numerator and the probability value, and this line can be simply dropped). If h γ ,s (0) = h γ ,s 1 2 < h γ ,s (1), then the equality holds if (1), then the equality is attained when The conditions for nonnegativity of component lifetimes T 1 defined above are μ 1 ≥ s (0) in case (22), and μ 1 ≥ σ 1 (∞) for Eqs. 21 and 23.
In order to describe the lower bound attainability conditions, it suffices to replace h γ ,s by h γ ,s , and reverse the inequalities between them in the previous paragraph.
Proof Again, we confine ourselves to considering positive upper bounds only under the condition G γ ,s and the integral in the second line is minimized at c = h γ ,s We get the equality in Eq. 24 if H −1 The attainability conditions in Theorems 1-3 can be also expressed in terms of distribution functions F of the nominal component lifetime. To this end it suffices to apply transformation F =G −1 γ • H 1 . We also notice that we obtain zero upper bounds for all 1 ≤ p ≤ ∞ when G γ ,s ≥G γ . This means that for any nominal distribution function F of the component lifetime, the system lifetime T is stochastically less than the actual component lifetime, and inequality ET ≤ ET 1 is evident. Similarly G γ ,s ≤G γ always implies ET ≥ ET 1 . Even a minor violation of these stochastic orderings between G γ ,s andG γ implies that we obtain nontrivial positive upper bounds, and negative lower ones, respectively.

Unimodal Signatures
In order to evaluate the bounds in Theorems 1, 2 and 3, derivatives of the smallest concave majorant and the greatest convex minorant of the distribution function G γ ,s •G −1 γ should be determined. For corresponding unimodal density functions, there is a well-known procedure to obtain these derivatives (see, e.g., Moriguti 1953;Rychlik 2001). Namely, if g : [0, 1] → [0, ∞) is any unimodal density function with the corresponding distribution function G, then the derivative of the greatest convex minorant is obtained as follows. If g is decreasing or g(0) ≥ 1, then g(x) = 1 for x ∈ [0, 1]. Otherwise, the derivative is given by If g is increasing, then u * = 1. Otherwise, 0 < u * < 1 is the unique solution to the so-called Moriguti equation The derivative of the smallest concave majorant can be derived similarly. If g is increasing or g(1) ≥ 1, then g(x) = 1 for x ∈ [0, 1]. Otherwise, the derivative has the form If g is decreasing, then u * = 0. Otherwise, u * coincides with the unique solution in (0, 1) to The next lemma shows that unimodality of the signature is sufficient to get unimodality of the density. The assumption is not very restrictive, because coherent systems with non-unimodal signatures are very rare. An example of a system of size 5 with bimodal signature was presented in Jasiński et al. (2009), and the construction was extended to higher dimensions by Bieniek and Burkschat (2018). In the proof, we apply a characterization of unimodality based on sign change behavior (cf., e.g., Marshall and Olkin 2007, proof of Proposition B.2., p. 99).
Theorem 4 If the signature s = (s 1 , . . . , s n ) is unimodal, i.e., there exists 1 ≤ k ≤ n such that s 1 ≤ · · · ≤ s k , s k ≥ · · · ≥ s n , then the density function Proof Recall that the density function is positive and continuous on (0, 1) (cf. Cramer et al. 2004). Moreover, the density is a bounded function on (0, 1) (see the proof of Theorem 1). Therefore, defines a non-negative, continuous (and bounded) function on the interval [0, 1]. Clearly, it is unimodal iff the function g = Observe that g is unimodal on [0, 1] iff A c is an interval (possibly empty or degenerate) in where g γ ,r denotes the density function of the rth uniform generalized order statistic. Since the signature s is unimodal, for every c ≥ 0 there are at most two sign changes in the sequence (s 1 − c n , . . . , s n − c n ) and, if there are exactly two changes, then the pattern is −+−. Thus, the variation diminishing property of the densities of uniform generalized order statistics (see Bieniek 2007) yields that condition (27) is satisfied. Consequently, g = g γ ,s g γ is unimodal.
Proof The statement can be proven analogously to Theorem 4 by noting that the function g = g γ ,s g γ is increasing (decreasing) iff the following condition holds: g(x) − c changes sign at most once for every c ∈ [0, ∞) and, if there is a sign change, then the sequence must be − + (+−).
For coherent systems with two or more components, the density function g γ ,s •G −1 γ g γ •G −1 γ cannot be a constant function on an interval. This follows from the next lemma, because every coherent system with n ≥ 2 components has a signature vector with s 1 = 0 or s n = 0 (see Miziuła and Rychlik 2015, Remark 3).