In this paper, a new class of life distribution is proposed, named renewal new better
than used in Laplace transform order (RNBUL). The test statistic for testing
exponentiality versus (RNBUL) based on U-statistic is proposed. Pitman's asymptotic
efficiencies of the test and Pitman's asymptotic relative efficiencies (PARE) are
calculated. The percentiles of this test statistic are tabulated, and the powers of this
test are estimated for some famous alternatives distributions in reliability such as
Weibull, linear failure rate (LFR), and Gamma distributions. The problem regarding the
right-censored data is also handled. Finally, some applications to elucidate the
usefulness of the proposed test in reliability analysis are discussed.