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

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.