On Conditions for Mixtures of Increasing Failure Rate Distributions To Have an Increasing Failure Rate

Abstract

Consider a family of distributions with survival distributions that are log concave and stochastically increasing in a parameter over which it will be mixed. It is shown that a necessary and sufficient condition for a mixture over any such family to have an increasing failure rate is that the mixing distribution have an increasing failure rate. Some observations are given on generalizing this result as well as weakening the conditions on Prekopa's Theorem.

Failure Rate Modelling for Reliability and Risk

“…By contrast, as was already discussed in this chapter, the mixture failure rate ) (t m λ can have a different pattern: it can ultimately decrease, for instance, or preserve the property that it is increasing in t as in Lynch (1999). There is even a possibility of a number of oscillations (Block et al, 2003).…”

“…The Gompertz-gamma mixture, as will be shown later in this chapter, is also IFR for certain values of parameters. Lynch (1999) had derived rather restrictive conditions for the preservation of the IFR property: the mixture failure rate…”

“…On the other hand, mixtures of increasing failure rate (IFR) distributions can decrease, at least in some intervals of time, which means that the IFR class of distributions is not closed under the operation of mixing (Lynch, 1999). IFR distributions usually model lifetimes governed by ageing processes, which means that the operation of mixing can dramatically change the pattern of ageing, e.g., from positive ageing (IFR) to negative ageing (DFR) ( Example 3.2).…”

Asymptotic baseline of the hazard rate function of mixtures

“…For example, Lynch (1999) found conditions under which the increasing failure rate (IFR) class is preserved. Block et al (2003b) and Li (2005) explored conditions for the preservation of some other classes.…”

On the behavior and shape of mixture failure rates from a family of IFR Weibull distributions

“…Recently, Lynch [12] has given conditions under which the mixture of lifetimes with increasing failure rate has an increasing failure rate. According to his result, if the survival functions F (t, ) of lifetime random variables involved in the mixture are log concave in (t, ) and increasing with respect to the parameter , then a necessary and sufficient condition for the mixture failure rate of such a family to be increasing is that the mixing distribution has an increasing failure rate function.…”