2. CHARACTERIZATIONS OF RHR AND MIT ORDERS

First, we give a characterization of the RHR order by means of the Laplace transform order. Let X be a nonnegative random variable with distribution function F; the Laplace–Stieltjes transform of F is given by

Given two random variables X and Y, X is said to be smaller than Y in the Laplace transform order (denoted by X ≤_Lt Y) if

Interpretations, properties, and applications of this order in reliability theory and actuarial science can be found in Alzaid, Kim, and Proschan [3], Klefsjo [20], and Denuit [14], among others. To state and prove Theorem 2.1, we need the following result.

Proposition 2.1: Let X and Y be two continuous nonnegative random variables with distribution functions F and G, respectively; then for all t ≥ 0 and s > 0,

Proof: First let us observe that, for all t ≥ 0,

Therefore, given s > 0,

and this completes the proof. █

Theorem 2.1: Let X and Y be two continuous nonnegative random variables. Then for all t ≥ 0,

Proof: The implication X ≤_RHR Y ⇒ X_(t) ≥_Lt Y_(t) follows from Theorem 3.1 of Nanda et al. [28] together with Theorem 3.B.6 of Shaked and Shanthikumar [35]. We now need to prove the implication X_(t) ≥_Lt Y_(t) ⇒ X ≤_RHR Y. Observing that, for fixed s ≥ 0 and t ≥ 0, we have

then

Using the dominated convergence theorem together with (2.1) and following the proof of Proposition 2.2 of Belzunce, Gao, Hu, and Pellerey [9], the theorem follows. █

Let us now denote the expected values of the random variables X_(t) and Y_(t) by μ(t) and β(t), respectively, where

The reversed hazard order implies the mean inactivity time order, but the converse is not necessarily true (see Nanda et al. [28]). The next result, however, gives a condition under which X ≤_MIT Y if and only if X ≤_RHR Y.

Theorem 2.2: Let X and Y be two nonnegative continuous random variables with differentiable MIT functions μ and β, respectively. Suppose that μ(t)/β(t) is increasing in t. Then

Proof: Let X have a RHR function r(t) = f (t)/F(t) and MIT function μ(t). Similarly, define q(t) and β(t) to be the analogous functions for Y. First, note that the mean inactivity time function is differentiable over {t : P(X < t)} and

where μ′ denotes the derivative of μ. Similarly,

Using the monotonicity property of μ(t)/β(t), together with (1.1), implies that

that is, X ≤_RHR Y. █

In many instances in the analysis of system reliability, for the purpose of comparison of two distribution functions, one is often concerned with properties of a system life distribution, which can be guaranteed from properties of component life distributions; general closure properties of stochastic orders with respect to reliability operations have been studied extensively. In addition to the theoretical interest in the problem, closure properties furnish both rules to choose among various types of replacement and maintenance policies and useful bounds when the exact distribution of the system is not available in closed form. Actually, it is a well-known fact that some stochastic orders are preserved under the formation of a k-out-of-n system of independent and identically distributed (i.i.d.) functions, whereas others are preserved under the formation of simple parallel or series systems. In the following example, we show that the MIT order is not preserved by a k-out-of-n system of i.i.d. functions.

Example 2.1: Let X and Y have distribution functions as follows:

It is easily seen that

implying that X ≤_MIT Y.

For n = 2,

and

Therefore, from (2.2) and (2.3), it follows that

thus implying that X_(1:2) is not less than Y_(1:2) in the MIT ordering.

As shown in Example 2.1, in general, the MIT ordering is not preserved by a k-out-of-n system of i.i.d. functions. However, under an additional stronger assumption on the variables involved (i.e., X ≤_RHR Y), such closure can be satisfied.

3. CHARACTERIZATIONS OF DRHR AND IMIT CLASSES

Many technical systems or subsystems have a k-out-of-n structure. These so-called k-out-of-n systems consist of n components of the same kind. The entire system is working if at least k of its n components are operating. It fails if n − k + 1 or more components fail. Hence, a k-out-of-n system breaks down at the time of the (n − k + 1)st component failure. Important particular cases of k-out-of-n systems are parallel and series systems corresponding to k = 1 and k = n, respectively. Practical examples of k-out-of-n systems are an aircraft with four engines that will not crash if at least 2 out of its 4 engines remain functioning or a satellite that will have enough power to send signals if not more than 4 out of its 10 batteries are discharged.

In reliability theory, the lifetime of a k-out-of-n system is usually described by the (n − k + 1)st-order statistic X_n−k+1,n from the sample X₁,…,X_n, where the random variable X_i represents the lifetime or failure time of the ith component of the system, 1 ≤ i ≤ n. Many authors have paid attention to behaviors of nonparametric life classes based on the parallel (series) systems and the k-out-of-n systems (see Barlow and Proschan [6], Belzunce, Franco, and Ruiz [8], Li and Zuo [25], Pellerey and Petakos [33], and Li and Chen [22]).

Assume now that the component with lifetime X_n−k+1,n fails and the system can be considered as a black box in the sense that the exact failure time of X_n−k+1,n in it is unknown. Motivated by this, we assume that at time t the system is not working and in fact it has failed at time t or sometimes before time t. In fact, it is of special interest to estimate the time that has elapsed since the (n − k + 1)st component failure. This is quite useful for estimation of the survival function for left-censored data. As a result, some authors centered their attention on investigating the behavior of some nonparametric life classes based on the inactivity time of k-out-of-n systems given that the time of the (n − k + 1)st failure occurs at time t ≥ 0. These characterizations are meaningful because they help in the understanding of the intrinsic meaning of reliability properties that are involved (see, e.g., Andersen, Borgan, Gill, and Keiding [4], Kalbfleisch and Lawless [18], Xu and Li [36], and Asadi [5]).

The inactivity time of a k-out-of-n system given that the (n − k + 1)st failure occurs at or some time before time t ≥ 0 is represented by the following conditional random variable:

and the inactivity time of the unit that fails at first is

where (X₁)_(t),…,(X_n−k+1)_(t) are i.i.d. copies of X_(t).

In this way, using stochastic comparisons, we will characterize the DRHR and the IMIT classes of life distributions by the stochastic ordering of the inactivity time of the k-out-of-n system, given that the (n − k + 1)st failure has occurred at different times. The following results give such characterizations in terms of the inactivity time of the first failure component as well as the inactivity time of k-out-of-n systems.

Theorem 3.1: Let X be a continuous nonnegative random variable with finite left end point of the support l_X and F(l_X) = 0.

Proof: According to Theorem 3.1 in Ahmad et al. [1], for i = 1,2,…,n, X_i is IMIT if and only if

Theorem 3.2:

Proof: First, note that the survival function of ITS_{n−k+1,n,(t)} and the distribution function of the ITP_1,n,(t) can be represented respectively as

Since X is DRHR if and only if

hence, from (1.2) and (1.3), we obtain part (a). Similarly, part (b) can be established. █

Suppose now that the system is composed of independent but not identical components. In this case, the survival function of ITS_n−k+1,n,t can be expressed as

In the next result, we investigate the inactivity time of the k-out-of-n systems given that the (n − k + 1)st failure occurs at time t ≥ 0.

Proposition 3.1: For any integer 1 ≤ k < n, if X_i′s, i = 1,2,…,n, are all DRHR, then

Proof: From the DRHR property, we have, for all t > s ≥ 0,

Thus, for all X > 0 and t ≥ s ≥ 0,

Hence,

This is just (3.2). █

In the rest of this section, we establish some useful inequalities and relations for weighted distributions related to RHR and MIT functions. Let X be a nonnegative random variable with distribution function F and probability density function (pdf) f. The weighted distribution of X or the pdf of the weighted random variable X_w is given by

where 0 < E(w(x)) < ∞.

The distribution given in (3.3) arises naturally when one subsamples from the original distribution with large sample size, say N, given a chance proportional to w(x) to observation x. The weighted distribution with weight function w(x) = x is called the length-biased distribution. Statistical applications of weighted distributions, especially to the analysis of data relating to human populations and ecology, can be found in Patil and Rao [31,32]. Pakes, Sapatinas, and Fosam [30] studied relations between length-biased distributions and infinite divisibility. Jain, Singh, and Bagai [17], Nanda and Jain [27], Bartoszewicz and Skolimowska [7], and Belzunce, Navarro, Ruiz, and del Aguila [10] studied relations of weighted distributions with classes of life distributions. Navarro, del Aguila, and Ruiz [29] developed characterizations through reliability measures from weighted distributions. Some known and important distributions in statistics and applied probability may be expressed as weighted distributions (e.g., truncated distributions, the equilibrium renewal distribution, distributions of order statistics, distributions in proportional hazards, and proportional reversed hazards models); for more details, see Broderick and Olusegun [12] and Bartoszewicz and Skolimowska [7].

The weighted distribution function can be expressed as

where

Note that if w′(x) > 0, then M_F(x) ≥ 0 for all x ≥ 0. The reversed hazard function of the weighted distribution F_w is given by

where r_F(x) is the reversed hazard rate function of X. Bartoszewicz and Skolimowska [7] obtained the following result.

Proposition 3.2: Let w be a monotone left continuous function. If w(x)r_F(x) is decreasing, then F_w(x) is DRHR.

The next result provides inequalities for the lower bound of the inactivity time distributions for large values x from the use of the information about the reversed hazard function of the weighted distribution function.

Theorem 3.3: Let F_w be a weighted distribution function with increasing weight function w(x) and pdf f_w > 0 for x ≥ x₀ and let X be the original random variable. If the reversed hazard function is such that r_Fw(x) ≥ c/x for all x ≥ x₀, where c is a real positive number, then

for all t < 0 and x ≥ x₀.

Proof: Let w(x) be increasing in x; then the reversed hazard function of the distribution function F of X satisfies

Now, for x ≥ x₀ and t < 0,

The last inequality follows from algebraic arguments; that is, ln a ≥ 1 − a⁻¹ for all a > 0. █

Proposition 3.3: Suppose that, for all t > 0,

then for all x ∈ [a,b) such that x − t > a,

Proof: First, note that the MIT function of the weighted distribution function F_w can be expressed as

The result follows from the observation that

where the second inequality follows from the assumption and, hence, the proof is completed. █