In this paper, we introduce a new age-dependent reliability class, termed the new better (worse) than renewal used in Laplace transform order after age t0 (NBRUL*t0). This class extends existing aging notions by characterizing lifetime distributions through renewal-based Laplace transform ordering beyond a specified age threshold. Several theoretical properties of the proposed class are established, including its relationships with classical aging classes. A goodness-of-fit test for exponentiality against the NBRUL*t0 alternative is developed using the framework of U-statistics, yielding a scale-invariant test statistic with a tractable asymptotic distribution. The asymptotic normality and Pitman asymptotic efficiency of the proposed test are derived, demonstrating superior efficiency relative to several existing nonparametric competitors. Extensive Monte Carlo simulations are conducted to obtain critical values and to assess the power performance of the test under both complete and randomly right-censored samples. The results indicate that the proposed test exhibits high power and robustness, particularly in the presence of aging effects and censoring. Applications to real engineering and medical datasets illustrate the practical relevance of the NBRUL*t0 class in reliability analysis and survival studies.
Ramadan et al. (Mon,) studied this question.