Preservation Properties for the Discrete Mean Residual Life Ordering

Discrete lifetimes usually arise through grouping or finite-precision measurement of continuous time phenomena. They may also be found natural choice where failure may occur only due to incoming shocks. Parametric models for discrete life distributions may be found in Bain (1991), Adams and Watson (1989) and Xekalaki (1983). Nonparametric families of discrete life distributions has been considered in the reliability literature mainly in connection with shock models leading to various continuous-time ageing families see Barlow and Proschan, (1975), Among others, have studied interrelations and closure properties of some nonparametric ageing families of distributions having a finite support. Related partial orders have been considered by Abouammoh (1990).


Introduction and Motivation
Discrete lifetimes usually arise through grouping or finite-precision measurement of continuous time phenomena.They may also be found natural choice where failure may occur only due to incoming shocks.Parametric models for discrete life distributions may be found in Bain (1991), Adams and Watson (1989) and Xekalaki (1983).Nonparametric families of discrete life distributions has been considered in the reliability literature mainly in connection with shock models leading to various continuous-time ageing families see Barlow and Proschan, (1975), Among others, have studied interrelations and closure properties of some nonparametric ageing families of distributions having a finite support.Related partial orders have been considered by Abouammoh (1990).
Recently, Nanda and Sengupta (2005) have discussed reversed hazard rate in discrete setup and obtained several interesting results.
There is an abundance of literature on continuous life distributions used in modeling failure data.However, very little has appeared in the literature for discrete failure models.Discrete failure data arise in various common situations.Consider the following examples: a) A device can be monitored only once per time period and the observation are taken as the number of time periods successfully completed prior to the failure of the device.b) A piece of equipment operates in cycles.In this case the random variable of interest is the successful number of cycles before the failure.For instance, the number of flashes in a car flasher prior to failure of the device.c) In some situations the experimenter groups or finite precision measurement of continuous time phenomena.Shaked et al. (1994Shaked et al. ( , 1995) ) state that discrete failure rates arise in several common situations in reliability theory where clock time is not the best scale on which to describe lifetime.For example, in weapons reliability, the number of rounds fixed until failure is more important than age in failure.They also showed the usefulness of these functions for modeling imperfect repair and for characterizing ageing in the discrete setting.For more applications of discrete models in reliability and survival analysis, see Padgett and Spurrier (1985) and Ebrahimi (1986).More precise concepts of discrete reliability theory have been settled by Salvia and Bollinger (1982), Roy and Gupta (1992) examined classification of discrete life distributions and they introduced the concepts of second rate of failure to maintain analogy with the continuous aging classes.Salvia (1996) presents some results on discrete mean residual life.
Similar to continuous distributions, discrete distributions can also be classified by the properties of the failure rates, the mean residual lifetimes, and survival functions of

Preliminaries
In this section we present definitions, notation, and basic facts used throughout the paper.We use "increasing" in place of "non-decreasing" and "decreasing" in place of "nonincreasing".Let  and  be two non-negative random variables with F and G as their respective distribution functions.Let  ̅ () = 1 − (), and  ̅ () = 1 − ().We will assume that  ̅ (0) =  ̅ (0) = 1 in all cases.

Definition 2.1
The random variable  is said to have a smaller discrete mean residual lifetime than that of , written  ≤ (−)  , if Note that, (2.1) is equivalent to saying is non-increasing (nondecreasing) for  ∈ ₊ and () + () > 0.
We notice that the discrete decreasing failure rate life distributions govern, a) In the grouped data case, the number of periods until failure of a device governed by a DFR life distribution.b) The number of seasons a TV show is run before being cancelled.Thus the d-DFD life distributions are of great significance in spite of their relative neglect in the reliability literature.

Definition 2.4
The distribution F is called discrete new better than used in expectation or  −  (discrete new worse than used in expectation or  − ) if , for all  ∈ .

Definition 2.5
The distribution F is called discrete decreasing mean residual lifetime or  −  if its mean residual lifetime   () = ( ); for  ∈  is decreasing in  ∈ .The following two definitions will be used in sequel.

Definition 2.7
A function : ² → [0, ∞) is said to be log-concave if The reminder of this paper is organized as follows.In section 3 we present the main results concerning the statistical properties of the discrete mean residual life ordering, such as convolution, mixtures and convergence in distributions.Section 4 contains two results concerning discrete renewal process in connection with the orders in the paper.

Main Results
In this section we present preservation results for the discrete mean residual life ordering.We point out that similar results hold for both the hazard rate ordering and the likelihood ratio ordering.We begin by showing that the discrete mean residual life ordering is preserved under weak limits in distributions.

Theorem 3.1
The discrete mean residual life ordering (≤ − ) preserves the weak convergence property.

Proof
, for all 0 ≤  ≤ . (3.1) Or equivalently, Next, by the well known basic composition formula (Karlin, 1968, p.17, the left side of (3.2) is equal to the conclusion now follows if we note that the first determinant is non-negative since g is log-concave, and that the second determinant is nonnegative since  1 ≤ −  2 .

Corollary 3.3
If  1 ≤ −  1 and  2 ≤ −  2 where  1 is independent of  2 and  1 independent of  2 then the following statements hold: a) If  1 and  2 have log-concave probability mass functions, then b) If  2 and  1 have log-concave probability mass functions, then

Proof
The following chain of inequalities establish (a): Let () be a non-negative discrete random variable having distribution function   and let   be a random variable having distribution   ( = 1; 2) and support ⁺.The following theorem shows that the  −  ordering is preserved under mixtures.

Proof
Let   be the distribution function of (  ) with  = 1, 2. We know that In the light of 2.2 we shall prove that By assumption ( 1 ) ≤ − ( 2 ) whenever we have  1 ≤  2 , we have (, ) is  2 in (, ), while the assumption  1 ≤   2 implies that   () is  2 in (, ).Thus the assertion follows from the basic composition formula (see, Karlin, 1968).

Proof
We need to establish , for all 0 ≤  ≤ . (3.4) Multiplying by the denominators and canceling out equal terms shows that (3.
which is nonnegative because both terms are nonnegative by assumption.This completes the proof.
In any attempt to construct new discrete mean residual life ordered random variables from known ones, the following theorem might be used.

Proof
We shall prove the theorem by induction.Clearly, the result is true for  = 1.Assume that the result is true for  =  − 1, this means that Note that each of the two sides of (3.8) has log-concave probability mass function (see, e.g., Karlin, 1968, p.128).Appealing to Corollary 3.3, the result follows.

Remark 3.7
Similar results hold if the discrete mean residual life ordering is replaced by the discrete hazard rate ordering in Theorem 3.2 and its corollary, Theorem 3.4 and Theorem 3.5.To demonstrate the usefulness of the above results in recognizing discrete mean residual life ordered random variables, we consider the following.

6 )
Now, for each fixed pair (, ) with  <  we have    ∑  + Suppose {  } and {  } gconverge weakly to  and  and that   ≤ −   .Then if  is a continuity point of both and , it follows that   () ≤   ().Thus,   () >   () is possible only if  is a discontinuity point of either  or .Such discontinuity points are at most countable, so there exist continuity points   of  and  for which   ↓  as  → ∞.Consequently, appealing to the right-continuity property of distribution functions Let  1 ,  2 and  be three nonnegative discrete random variables, where  is independent of both  1 ,  2 , also let  have a probability mass function .Then  1 ≤ −  2 and  is log-concave imply that  1 +  ≤ −  2 + For next results we shall use the notation   and   to replace () and () respectively.