Tests of exponentiality based on Arnold-Villasenor characterization, and their efficiencies

TL;DR

This paper introduces new exponentiality tests based on Arnold and Villasenor's characterization, demonstrating their high efficiency and optimality under certain alternatives, with practical recommendations based on power analysis.

Contribution

The paper develops two families of exponentiality tests using U-empirical distribution function functionals, showing their asymptotic normality and high local Bahadur efficiency, and identifies conditions for their optimality.

Findings

Integral statistics can be reduced to V- or U-statistics with simple kernels.

The tests are asymptotically normal and highly efficient under common alternatives.

Kolmogorov type tests show low efficiency and moderate power.

Abstract

We propose two families of scale-free exponentiality tests based on the recent characterization of exponentiality by Arnold and Villasenor. The test statistics are based on suitable functionals of U-empirical distribution functions. The family of integral statistics can be reduced to V- or U-statistics with relatively simple non-degenerate kernels. They are asymptotically normal and have reasonably high local Bahadur efficiency under common alternatives. This efficiency is compared with simulated powers of new tests. On the other hand, the Kolmogorov type tests demonstrate very low local Bahadur efficiency and rather moderate power for common alternatives,and can hardly be recommended to practitioners. We also explore the conditions of local asymptotic optimality of new tests and describe for both families special "most favorable" alternatives for which the tests are fully efficient.