Abstract
In this paper, the problem of testing exponentiality against new better (worse) than renewal used in expectation is investigated and similarly for the case of nuharmonic new better than renewal used in expectation. For each of these two aging properties, a nonparametric procedure (U-statistic) is presented. Selected critical values are tabulated for sample sizes n = 5(1)30(10)50. The Pitman asymptotic relative efficiency to the test relative to other classes are studied. A real example is given to elucidate the use of the proposed test statistics for the reliability analysis.