Nonparametric tests for detecting greater residual life times

A nonparametric procedure is proposed for testing the exponentiality against the hypothesis that one life distribution has a greater residual life time than the other life distribution. Such a hypothesis turns out to be equivalent to the one that one failure rate is greater than the other and so the proposed test works as a competitor to more IFR tests given by Kochar [Kochar, S. C. (1979). Distribution-free comparison of two probability distributions with reference to their hazard rate. Biometrika, 66, 437-441; Kochar, S. C. (1981). A new distribution-free test for the equality of two failure rates. Biometrika, 68, 423-426] and Cheng [Cheng, K. F. (1985). Tests for the equality of failure rates. Biometrika, 72, 211-215]. Our test statistic utilizes the U-statistics theory and a large sample nonparametric test is established. The power of the proposed test is discussed by calculating the Pitman asymptotic relative efficiencies against several alternative hypotheses. A numerical example is presented to exemplify the proposed test.