A saddlepoint approximation for testing exponentiality against some increasing failure rate alternatives

In this article we discuss uniformly most powerful unbiased tests for testing exponentiality against a specific class of two-parameter exponential models with increasing failure rate. We show that the optimal test statistic for this problem admits an alternative representation in terms of a spacings statistic. Using the conditional saddlepoint approximation proposed by Gatto and Jammalamadaka (J. Amer. Statist. Assoc. 94 (1999) 533), we provide highly accurate approximations for the significance values. The test procedure is illustrated with two practical examples from reliability and survival analysis. We also obtain the asymptotic distribution of the test statistic under a sequence of converging alternatives, which allows for the computation of asymptotic relative efficiency.