A two-parameter family of Hollander–Proschan–type tests against NBUE alternatives

Juan Ding; Chi Zhou; Anqi Xia; Wenxin Zhou; Wenjun Xiong; Juan Ding; Chi Zhou; Anqi Xia; Wenxin Zhou; Wenjun Xiong

doi:10.3934/math.2026558

AIMS Mathematics

2026, Volume 11, Issue 5: 13567-13588. doi: 10.3934/math.2026558

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Research article

A two-parameter family of Hollander–Proschan–type tests against NBUE alternatives

1.
School of Mathematics, Hohai University, Nanjing 211100, China
2.
Laboratory of Mathematical Modeling and Intelligent Computing for Water Systems, Hohai University, Nanjing 211100, China
3.
School of Mathematics and Statistics, Guangxi Normal University, Guilin 541004, China

Received: 04 February 2026 Revised: 19 April 2026 Accepted: 27 April 2026 Published: 15 May 2026
MSC : 62N05, 62G10

Testing for aging behavior is a fundamental problem in reliability analysis. We study tests of exponentiality against new better than used in expectation (NBUE) alternatives. Many Hollander–Proschan type procedures can be interpreted as weighted functionals of a common mean residual life gap, but are often presented in isolated forms, making it difficult to understand how weighting affects sensitivity to different alternatives. We develop a unified weighted-functional framework in which several classical procedures arise as special or limiting cases. Within this framework, we construct a centered and studentized statistic with an asymptotic normal null distribution and an explicit centering term. Pitman efficiency analysis and Monte Carlo experiments illustrate how weighting influences sensitivity and finite-sample performance, while real-data applications demonstrate practical interpretability.
- NBUE,
- Hollander–Proschan test,
- Pitman asymptotic efficiency,
- reliability,
- mean residual life
Citation: Juan Ding, Chi Zhou, Anqi Xia, Wenxin Zhou, Wenjun Xiong. A two-parameter family of Hollander–Proschan–type tests against NBUE alternatives[J]. AIMS Mathematics, 2026, 11(5): 13567-13588. doi: 10.3934/math.2026558

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Abstract

Testing for aging behavior is a fundamental problem in reliability analysis. We study tests of exponentiality against new better than used in expectation (NBUE) alternatives. Many Hollander–Proschan type procedures can be interpreted as weighted functionals of a common mean residual life gap, but are often presented in isolated forms, making it difficult to understand how weighting affects sensitivity to different alternatives. We develop a unified weighted-functional framework in which several classical procedures arise as special or limiting cases. Within this framework, we construct a centered and studentized statistic with an asymptotic normal null distribution and an explicit centering term. Pitman efficiency analysis and Monte Carlo experiments illustrate how weighting influences sensitivity and finite-sample performance, while real-data applications demonstrate practical interpretability.

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