Bahadur efficiency and local optimality of a test for the exponentiality based on the Moran statistics

The scale-free test for exponentiality introduced by Moran is under study. It had been constructed as an optimal test of exponentiality against the gamma alternatives but may also be used to test exponentiality against IFR and DFR classes of alternatives. We obtain the rough large deviations asymptotics of the test under the null hypothesis and find its Bahadur efficiency values for most commonly used alternatives to the exponentiality. We also describe the local Bahadur optimality domain of the test. The large deviations theorem describes as well the asymptotic behaviour of a spacings-based Darling test for the uniformity under the null hypothesis.