Results on relative mean residual Life and relative cumulative residual entropy

Wei [21] has proposed the relative mean residual life function for comparing two lifetime distributions and studied its properties and relationship with other stochastic orders. In this paper, we obtain some new results on the relative mean residual life function and give a characterization result for a relative ordering based on this function. Motivated by this notion, we also introduce two notions of the dynamic relative cumulative residual entropy functions. Their properties and relationship with other relative orderings are investigated.


Introduction
In reliability and survival analysis and other life sciences, it is of interest to compare between lifetime distributions. This comparison is usually done based on partial orderings between their corresponding reliability measures and ageing functions. The relative ageing is another comparison method between lifetime distributions which has been studied in the literature. For example, Kalashnikov and Rachev [9] have studied the comparison of life distributions through a partial ordering which is equivalent to a monotone hazard ratio. Wei [21] has introduced the relative mean residual life (RMRL) function and investigated its relationship with other orderings. Sengupta and Deshpande [18] have considered two other models of relative ageing which include an increasing cumulative hazard ratio. Finkelstein [6] has studied the relative characteristics of the mean residual life functions. Hazra and Nanda [8] have introduced some new generalized stochastic orderings in the spirit of relative ageing. Li and Li [12] have studied the relative ageing order on series and parallel systems with mutually statistically independent and heterogeneous component lifetimes. Misra et al. [13] have introduced a new notion of relative aging based on the ratio of the mean inactivity time functions.
Let X be a non-negative random variable with distribution function F , survival functionF = 1 − F and cumulative hazard functions Λ = − logF . The uncertainty contained in random variable X, since its first mathematical formulation by Shannon in 1948, has been defined in several formula. Recently, Rao et al. [16] defined a new uncertainty measure, the cumulative residual entropy (CRE), through

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Properties of the CRE can be found in [15], [5], and [14]. Asadi and Zohrevand [1] have considered the corresponding dynamic measure, the CRE corresponding to the residual lifetime variable: ∫ ∞ tF (x)dx is the mean residual life (MRL) function. They have also obtained results about comparing between lifetime variables based on E(X; t). In this paper, we introduce a definition of dynamic relative cumulative residual entropy (RCRE) using E(X; t).
Wei [21] has defined the relative mean residual of F 1 with respect to F 2 through provided thatF 2 (t) > 0. Motivated by this, we define the dynamic relative cumulative residual entropy of F 1 with respect to F 2 by The aim of this paper is twofold. In the first part, we give some new results on RMRL function and related orders. In the Second part, we study properties of E 1.2 (t) along with introducing and investigating properties of another dynamic RCRE function. The rest of the paper is organized as follows. In Section 2, we provide various standard definitions and notations that are used later in the paper. In Section 3 we obtain some new results about the RMRL function. In Section 4 we investigate properties of the dynamic RCRE. Section 5 is devoted to the definition and properties of the other dynamic relative cumulative residual entropy and finally, some concluding remarks are given in section 6.

Some Preliminaries
Before proceeding to give the main results of the paper, we overview some preliminary concepts of ageing and standard partial orderings describing relative ageing. (For more details of these concepts see, for example, [18], [6], [19], [10], [13]).
The following definitions pertaining to ageing classes and stochastic orders have been taken from [19].
Definition 1(i) The random variable X is said to have increasing (decreasing) failure rate, IFR (DFR), if X t = X − t | X > t is stochastically decreasing (increasing) in t ≥ 0. If X has a density f this is equivalent to say that the failure rate (ii) The random variable X is said to be new better (worse) than used, NBU(NWU), for all x, t ≥ 0.
(vi) The random variable X is said to be increasing (decreasing) mean residual life (IMRL(DMRL)) if m(t) is increasing (decreasing) in t.
(vii) The random variable X 1 is said to be smaller than X 2 in the usual stochastic order (denoted by (viii) The random variable X 1 is said to be smaller than X 2 in the hazard rate ordering (denoted by (ix) The random variable X 1 is said to be smaller than X 2 in the mean residual life ordering (denoted by The random variable X 1 is said to be smaller than X 2 in the convex order (denoted by (xi) The random variable X 1 is said to be smaller than X 2 in the dilation order (denoted by

Definition 2
The random variable X 1 is said to be aging faster than X 2 in the has a decreasing failure rate (DFR) distribution (equivalently, [18]), has a decreasing failure rate average (DFRA) distribution ( [18]), has a new better than used (NBU) distribution ( [18]), In contrast to Definition 2 (iv), Finkelstein [6] defined the random variable X 1 to be aging faster than As Misra et al. [13] have mentioned, since increasing mean residual life describes negative aging, we believe that the correct way of defining relative aging in terms of mean residual function is as given in Definition 2 (iv).

Remark 1
As a special case of Proposition 2.1 (iii) in [8] (for s = 1), one can see easily that where X e is the equilibrium random variable (see, for example, [7], and references therein) corresponding to X with survival function
As it is clear from the complete chain of implications among the positive ageing criteria in [4], one can see that the NBU class has two sperate ways of generalization. The one is toward the classes NBUFR and NBUFRA and the other one is toward the classes NBU(2) and NBUE. Here, we follow [8] and define two generalized relative ordering classes. Let X 1s and X 2s be random variables as defined in [8].

Definition 3
For any positive integer s, X 1 is said to be more The properties of these classes can be obtained in similar way as those of the other classes in [8]. For s = 1, we denote the s-NBUE(R) ordering between the random variables X 1 and X 2 by X 1 ≺ rm X 2 which is equivalent to that Z = Λ 1 (X 2 ) has a NWUE distribution. It follows from Proposition 6.3 in [4] that Z is NWUE if and only if Z e is NWUFR, where Z e is the equilibrium random variable corresponding to Z. The following theorem gives another characterization of this ordering based on the RMRL function.

Theorem 1
Suppose that F 1 is continuous and strictly increasing.
Proof: Using the integration by part, it follows from the equation (3) that The result now is clear.

Remark 2
It follows form the proof of the above theorem that m 1.2 (t) is an increasing function if and only if Z = Λ 1 (X 2 ) has an IMRL distribution.
The following theorem gives a bound for m 1.2 (t) and shows that the behavior of m 1.2 (t) is related to the behavior of the hazard rate function of X 2 .
It is worth to mention that if X 1 is distributed as exponential with E(X 1 ) = E(X 2 ), then m 1.
Hence, m 1.2 (t) ≥ 1 implies that X 2 is NWUE. The next result gives a bound for m 1.2 (t), in terms of the survival functions, when X 1 and X 2 are stochastically ordered and gives conditions under which m 1.2 (t) ≥ 1 and X 1 ≤ mrl X 2 follow each other.

RESULTS ON RELATIVE MEAN RESIDUAL LIFE AND RELATIVE CUMULATIVE RESIDUAL ENTROPY
Proof: (a) If X 1 ≤ st X 2 , thenF 1 (t) ≤F 2 (t). Now, the result follows from the fact that To prove part (b), first note that the assumption that X 1 is IFR implies that Λ 1 (t) is a convex function. On the other hand, it is easy to see that the random variable U = Λ 1 (X 1 ) is distributed according as standard exponential distribution. It follows now from the proof of Theorem 1 and the Theorem 2.A.19 in [19], p. 92 that This completes the proof of (b). The proof of part (c) is similar.

Remark 3
Let X 1 , X 2 and X be three independent random variables. Assume that X 1 and X are IFR and that X 1 ≤ mrl X 2 . Lemma 2.A.8. in [19], p. 86, follows that X 1 + X ≤ mrl X 2 + X. The assumption also implies that X 1 + X is IFR (cf. [2], p. 100). Then, it follows from part (b) of Theorem 3 that m X 1.2 (t) ≥ 1, where m X 1.2 (t) represents the RMRL of X 1 + X with respect to X 2 + X.
Let X k:n be the kth smallest order statistic of random variables X 1 , . . . , X n , k = 1, . . . , n. In reliability theory, X k:n indicates the lifetime of a n − k + 1-out-of-n system, which works if at least n − k + 1 of the n components function normally. In particular, X 1:n and X n:n represent lifetimes of the series and parallel systems, respectively. Let also X 11 , . . . , X 1n be i.i.d. random lifetimes of type X 1 and X 21 , . . . , X 2n be i.i.d. ones of type X 2 with X 1 k:n and X 2 k:n as their corresponding kth smallest order statistics.
The following examples give some applications of the above theorem.

Example 1
Let X 1 and X 2 have Pareto distributions with survival functions For a < b, we haveF 1 (t) ≤F 2 (t), i.e. X 1 ≤ st X 2 . Thus from part (a) of Theorem 3 it follows that On the other hand, it follows form Theorem 1.A.23. (a) in [19], p. 13 that X 1 k:n ≤ st X 2 k:n for k = 1, . . . , n. Then, using again the part (a) of Theorem 3 we get that where m k 1.2 (t) stands for the RMRL of X 1 k:n with respect to X 2 k:n , and I p (a, b) = B(a, b) is the incomplete beta function.

Example 2
Unloaded redundancy. Consider the unloaded redundancy when one of the identical components starts operating and the other n − 1 are in stand by. As the operating one fails, it is immediately replaced by the stand by one etc. The system fails when the last component fails. We compare the RMRL for two objects with exponentially distributed components lifetimes and different levels of redundancy: m < n. For a component with exponential distribution,F (t) = e −λt , the lifetime distribution of the object having n components is the Gamma (or the Erlangian) distribution with parameters n and λ. Let X 1 , X 2 be the lifetimes of the objects with m and n components, respectively. For 1 < m < n, X 1 is IFR and X 1 ≤ mrl X 2 . Then, if m 1.2 (t) is the RMRL of X 1 with respect to X 2 , it follows from part (b) of Theorem 3 that m 1.2 (t) ≥ 1.

Properties of E 1.2 (t)
In this section we investigate some properties of the dynamic RCRE. First, note that E 1.2 (t) can also be given by Asadi and Zohrevand [1] have shown that where m(t) is the MRL function. The following theorem gives the direct relation between the RMRL and dynamic RCRE functions.

Theorem 4
Let X 1 and X 2 be two non-negative random variables with distribution functions F 1 and F 2 , respectively, and finite dynamic RCRE function E 1.2 (t). Then, This completes the proof.
The following theorem shows that the behavior of E 1.2 (t) is closely related to the behavior of m 1.2 (t) and the ratio of the hazard rates of X 1 and X 2 . In particular, the theorem proves that the constancy of E 1.2 (t) gives a characterization of proportional hazards model. Proportional hazards model, introduced by Cox [3], plays an important role in reliability and survival analysis. Two random variables X 1 and X 2 with survival functionsF 1 and F 2 are said to have proportional hazards if there exists θ > 0 such thatF 1 (t) =F θ 2 (t) for all t ≥ 0. If the hazard rates of X 1 and X 2 exist, i.e. the survival functions are absolutely continuous, this is equivalent to say that for all t > 0, λ 1 (t) = θλ 2 (t).

RESULTS ON RELATIVE MEAN RESIDUAL LIFE AND RELATIVE CUMULATIVE RESIDUAL ENTROPY
Derivation from both side of the above equation with respect to t implies that m 1.2 (t) = θ. On the other hand m 1.2 (t) can also be written as Derivation again follows that λ 1.2 (t) = θ. This proves part (a) of the theorem. To prove part (b), it is easy to see that the derivative of E 1.2 (t) is equal to The if part now follows from the definition of E 1.2 (t). For the only if part, note that if The result now follows from the integral comparison lemma (cf. [17], p. 336) and the fact that m ′ 1.
Proposition 2.2 in [21] gives that if λ 1.2 (t) is increasing (decreasing) function then m 1.2 (t) is also increasing (decreasing) function. However, the reverse is not true. For a counterexample, let F 1 be the standard exponential distribution and F 2 the distribution given in Example 4.9 in [11], p. 114.
Wei [21] has defined another relative mean residual life of F 1 with respect to F 2 as m * 1.
The reverse is not necessarily true(cf. [19], p. 83). The following example shows that the same is true for the relation between λ 1.2 (t), m 1.2 (t) and E 1.

Example 4
Let X be a continuous non-negative random variable with survival functionF and a finite mean µ. It can be easily seen that the RMRL and RCRE of F with respect to F e are given as follows respectively, which are both less (greater) than one if X is DMRL (IMRL).

Another relative dynamic cumulative residual entropy
It is easy to show that the cumulative residual entropy (1) and the dynamic cumulative residual entropy (2) can also be given by E(X) = Cov(X, Λ(X)), E(X; t) = Cov(X t , Λ(X t )).
Motivated by this, we define the relative cumulative residual entropy of F 1 with respect to F 2 by Equivalently, we define the another dynamic relative cumulative residual entropy of F 1 with respect to F 2 by Fi (t) and F −1 it are the distribution and quantile functions corresponding to residual lifetime variable X it = X i − t|X i > t, i = 1, 2, respectively. Note that if F 2 is an exponential distribution with parameter 1, then , and E * (X 1 , X 2 ) and E * 1.2 (t) are reduced to E(X 1 ) and E(X 1 ; t), respectively. Example 5 Let X 1 and X 2 be uniformly distributed on intervals (0, b 1 ) and (0, b 2 ), respectively. Then The following theorem gives the relation between the dilation order and the above RCRE.
The above proof implies that (7) can also given by

Conclusion
Models of relative aging are tools for comparing two lifetime distributions. In this paper, we obtained some new properties of the relative mean residual life (RMRL) and obtained a characterization result for a special case of s-NBUE(R) ordering based on the RMRL. We also introduced two definitions of the dynamic relative cumulative residual entropy (RCRE) for comparing the uncertainty within two lifetime random variables. It was shown that the behavior of the dynamic RCRE has close relation with the behavior of the RMRL and the ratio of the hazard functions. Some results obtained to explore the relation between these new notions and some well-known stochastic orderings such as dilation order.