Discrete Distributions

We will briefly comment on bivariate distributions. Results for the case of more than two discrete variates can be found in SHAKED et al. (1995). Let X = (X1, X2) be a random vector with

support in N20and denote its joint PMF by

Pr(x1, x2) = Pr(X1= x1, X2= x2), (x1, x2) ∈ N20. (1.88a)

Remember, that in the discrete case the hazard rat is a conditional probability. The discrete multivariate conditional hazard rate functions of (X1, X2) are defined as

λ1(x) = Pr(X1 = x, X2 > x | X1 ≥ x1, X2 ≥ x), x ∈ N0, (1.88b)

λ2(x) = Pr(X2 = x, X1 > x | X1 ≥ x1, X2 ≥ x), x ∈ N0, (1.88c)

λ12(x) = Pr(X1 = x, X2 = x | X1 ≥ x1, X2 ≥ x), x ∈ N0, (1.88d)

λ1(x | x2) = Pr(X1 ≥ x | X2 ≥ x, X2 = x2), x > x2, (x1, x2) ∈ N20, (1.88e)

λ2(x | x1) = Pr(X2 ≥ x | X1 = x1, X2 ≥ x), x > x1, (x1, x2) ∈ N20, (1.88f)

provided the conditions in the above conditional probabilities have positive probabilities. Other- wise, these functions are set to 1.

The meaning of these functions is as follows. The functions λ1(x), λ2(x) and λ12(x) describe

the initial hazard rates, i.e., the hazard rates before a failure of any component. Given that no failure has occurred before time x, then, at time x one of the following four events must occur:

1. only component 1 fails, the probability being λ1(x),

2. only component 1 fails, the probability being λ2(x),

3. both components fail, the probability being λ12(x),

4. no component fails, the probability being 1 − λ1(x) − λ2(x) − λ12(x).

Now suppose that one component failed at x1 (or x2) and that the other component stayed alive

at that time. Then, conditional on X1 = x1 (or X2 = x2), the hazard rate of the live component

at time x > x1(or x > x2) is given by λ2(x | x1) [or λ1(x | x2)].

The hazard rates given in (1.88b–f) are the discrete analogs of the bivariate hazard rate functions described in COX (1972), see (1.87a–c) for the absolute continuous case. But in COX there is no analog of (1.88d) since absolute continuity of the distribution of (X1, X2) is assumed there.

Failure of both components at the same time has zero probability in the absolute continuous case, but in the discrete case it may be positive and is given by λ12(x).

From (1.88b–f) we see that the joint distribution of X1and X2determines the conditional hazard

rate functions. But also the converse is true, i.e., (1.88b–f) determine Pr(x1, x2) of (1.88a). For

more details see SHAKED et al. (1995) who also give necessary and sufficient conditions on the functions (1.88b–f) which ensure that they are hazard rate functions of some random vector (X1, X2).

2 Aging Criteria and Classes of

Univariate Lifetime Distributions

1

It is quite natural and obvious to classify lifetime distributions by using so–called aging criteria. In the context of lifetime analysis aging does not mean that a statistical unit becomes older in the sense of chronological calendarian time, rather aging is a notion pertaining to the behavior of residual life. Aging is thus the phenomenon that a chronological older unit has a shorter residual life in some statistical sense than a newer or chronological younger unit. We may distinguish between

• positive (true or adverse) aging indicating a decline — in some way or the other — of residual life with growing age x.

• negative (inverse or beneficial) aging when residual life is increasing with x in some way or the other.

Lifetime distributions are mostly characterized with respect to aging by the behavior of • their hazard rate h(x) or

• their mean residual life µ(x).

Hazard rate classes will be discussed in Sections 2.1 and 2.2. Mean residual life classes are the topic of Section 2.3. But there are more statistical concepts used in classifying lifetime distribu- tions. These will be presented in Section 2.4 where we will also show how all the aging criteria are linked.

Classes of lifetime distributions based on notions of aging afford statisticians an opportunity to consider problems of a character somewhat different from the usual. Instead of assuming that he knows nothing about the underlying lifetime distribution, the statistician assumes that he does not know the parametric form of the distribution, but that he does know, for example, that the hazard rate is increasing. More generally, he knows that some type of aging property holds for the lifetime distribution; this aging property give rise to a corresponding geometric property of the distribution. Knowing that a lifetime distribution belongs to a certain class, it is possible by using certain additional information to give approximations and bounds of the percentiles, moments and survival probabilities of this distribution. Of course, it is possible to test whether certain hypotheses on aging hold or not, see Sect. 10.3.

This chapter only present results for univariate distributions. Readers interested in aging criteria for multivariate distributions are referred to BLOCK/ SAVITS (1982, 1988), HARRIS (1970) or

SHAKED/SHANTHIKUMAR(1987, 1988).

2.1

Monotone Hazard Rate Distributions

Since most materials, structures and devices wear out with time, the class of failure distribu- tions for which the hazard rate is increasing is one of special interest. The phenomenon of work hardening of certain materials and the debugging of complex systems make the class of failure

1

distributions with decreasing hazard rate also of some interest. Here, the terms ‘increasing’ and ‘decreasing’ are not used in the strict sense, but increasing (decreasing) stands for non–decreasing (non–increasing). Note that with this convention the continuous exponential distribution and the discrete geometric distribution with constant hazard rates belong to both classes. There are, of course, examples such as dynamic loading of structures, where a non–monotonic hazard rate function would be appropriate. Structures undergoing adjustment and modification also tend to have a non–monotonic hazard rate.

The assumption that a lifetime distribution has a monotone hazard rate is quite strong as we shall show, but such distributions possess many useful and interesting properties. Most results on monotone hazard rates hold for the continuous as well as for the discrete case, but there are some differences, especially in the way how to detect whether the distribution’s hazard rate is increasing or decreasing. So we have decided to present the continuous and the discrete cases in two separate Sections 2.1.1 and 2.1.2. In Section 2.1.3 we will introduce the related concept of the hazard rate average and see when this is increasing or decreasing.