A Note on NBUE and NWBUE Classes of Life Distributions

TL;DR

This paper explores non-monotonic ageing classes of life distributions, specifically NBUE and NWBUE, providing new results, counterexamples, and corrections to existing notions in reliability theory.

Contribution

It extends the understanding of NBUE and NWBUE classes by presenting new properties, counterexamples, and corrections for these non-monotonic ageing notions.

Findings

Some properties of NBUE carry over to NWBUE

Counterexamples show moment bounds do not transfer

Corrective results for NWBUE notions are provided

Abstract

Non-monotonic ageing notions are looked upon as an extension of the corresponding monotonic ageing notions in this work. In particular, the New Better than Used in Expectation (NBUE) and the corresponding non-monotonic analogue New Worse then Better than Used in Expectation (NWBUE) classes of life distributions is considered. Some additional results for the NBUE class are obtained. While many properties of the NBUE class carry over in an analogous way to the NWBUE class, it is shown by means of counterexamples that the moment bounds do not. Some corrective results with respect to popular notions of the NWBUE class are also presented.