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In this paper, a new hypothesis test is constructed to test exponentiality against Renewal New Better than Used in moment- generating function 􏰀RNBUmg f 􏰁 based on Laplace transform order and another test based on goodness of fit approach follows as a special case. Pitman Asymptotic Efficiency (PAE) are studied, the critical values of the tests are tabulated for sample sizes n = 5(5)50, and the power estimates are calculated to assess the performance of the tests. Also a test of exponentiality versus 􏰀RNBUmg f 􏰁 for right censored data is considered. The power estimates of the tests are simulated for some commonly used distributions in reliability. Finally, sets of real data are used as examples to elucidate the use of the proposed test statistic for practical problems in case of complete and uncomplete data in the reliability analysis.

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