Efficiency of exponentiality tests based on a special property of exponential distribution

Abstract

New goodness-of-fit tests for exponentiality based on a particular property of exponential law are constructed. Test statistics are functionals of U -empirical processes. The first of these statistics is of integral type, the second one is a Kolmogorov type statistic. We show that the kernels corresponding to our statistics are nondegenerate. The limiting distributions and large deviations of new statistics under the null hypothesis are described. Their local Bahadur efficiency for various parametric alternatives is calculated and is compared with simulated powers of new tests. Conditions of local optimality of new statistics in Bahadur sense are discussed and examples of "most favorable" alternatives are given. New tests are applied to reject the hypothesis of exponentiality for the length of reigns of Roman emperors which was intensively discussed in recent years.

New $L^2$-type exponentiality tests

“…Such tests have become very popular in recent times, as they are proven to be rather efficient. Tests that use U-empirical and V-empirical distribution functions, of integral-type (integrated difference) and supremum-type, can be found in [28], [33], [15], [23], [21], [25]. A class of weighted integral-type tests that uses U-empirical Laplace transforms is presented in [22].…”