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A Note on NBUE and NWBUE Classes of Life Distributions

M. Z. Anis

TL;DR

This work clarifies the NBUE and NWBUE classes of life distributions, establishing robust moment inequalities and weak convergence for NBUE while demonstrating that NWBUE does not inherit these NBUE moment bounds. It also corrects a longstanding misconnection between the IDMRL class and NWBUE, providing precise conditions under which an IDMRL distribution is NWBUE or NWUE. By presenting counterexamples and corrected NWBUE instances, the paper sharpens theoretical understanding of non-monotonic ageing and guides reliability modelling and testing against NBUE/NWBUE alternatives. Overall, the results deepen the theoretical foundation for non-monotonic ageing notions and correct previous misconceptions in the literature.

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.

A Note on NBUE and NWBUE Classes of Life Distributions

TL;DR

This work clarifies the NBUE and NWBUE classes of life distributions, establishing robust moment inequalities and weak convergence for NBUE while demonstrating that NWBUE does not inherit these NBUE moment bounds. It also corrects a longstanding misconnection between the IDMRL class and NWBUE, providing precise conditions under which an IDMRL distribution is NWBUE or NWUE. By presenting counterexamples and corrected NWBUE instances, the paper sharpens theoretical understanding of non-monotonic ageing and guides reliability modelling and testing against NBUE/NWBUE alternatives. Overall, the results deepen the theoretical foundation for non-monotonic ageing notions and correct previous misconceptions in the literature.

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.
Paper Structure (9 sections, 8 theorems, 26 equations)

This paper contains 9 sections, 8 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Results on the NBUE Class
  3. Moment Inequalities and Related Results
  4. Weak Convergence within the NBUE Class
  5. Results on the NWBUE Class
  6. Moment Bounds for NWBUE Distributions
  7. Relationship with IDMRL Class
  8. A Correct Example
  9. Conclusion

Key Result

Lemma 2.1

(Marshall and Proschan [40]) If $F$ is NBUE with finite mean $\mu,$ then $\int_{x}^{\infty} \bar{F}(t) dt\leqslant \mu e^{-x/\mu}, x\geqslant 0,$ where $\bar{F}=1-F.$

Theorems & Definitions (14)

  • Lemma 2.1
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • proof
  • Lemma 2.5
  • proof
  • Theorem 2.6
  • ...and 4 more