On a test for exponentiality against Laplace order dominance

In a recent paper, Basu and Mitra (2002) introduced a class of tests for exponentiality against the non-parametric class L of life distributions. The test statistics are fractional sample moments. In a simulation study, the tests show a remarkable behavior: when performed on a nominal level of 5%, each of the tests always accepted the null hypothesis of exponentiality if the underlying data came from an exponential distribution. Furthermore, power of the tests was very low compared to other procedures designed for the same testing situation. It is the aim of this paper to show that the reasons for this behavior are an incorrectly stated asymptotic distribution, a slow convergence of the finite sample distributions of the test statistics to their limit distribution, and, to a minor extent, the dependence of the proposed tests on the choice of parameter values. To this end, we derive the limit distributions of the test statistics in case of a general underlying distribution and the local approximate Bahadur efficiency of the tests against several parametric families of alternatives to exponentiality. Additionally, the finite sample behavior of the tests is examined by means of a simulation study. Further, we enlarge the proposed class of tests by extending some characterization of exponentiality within the L-class.