The lOOα -percentile (0 < α < 1) residual life function at time x is defined to be the lOOα -percentile of the remaining life given survival up to time x. Joe and Proschan (1982b) develop tests for testing the alternatives representing decreasing 100α-percentile residual life (DPRL-α ) and the property 'new better than used with respect to the lOOα -percentile' (NBUP-α ). In this paper, tests are developed for DPRL[α, l) and NBUP[α, l) alternatives, where DPRL[α, l) is the class of life distributions which are DPRL-β distributions for all a ≤ β < 1 if 0 ≤ α < 1 and for all 0 < β < 1 if α = 0, and NBUP[α, l) is similarly defined. When α = 0, the DPRL[α, l) class of life distributions corresponds to the increasing failure rate class and the NBUP[α, l) class of life distributions corresponds to the new better than used class, and the test statistics are respectively asymptotically equivalent to the Hollander and Proschan (1975) test statistics for decreasing mean residual life and new better than used in expectation alternatives.
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