Efficiency of Exponentiality Tests Based on a Special Property of Exponential Distribution

TL;DR

This paper introduces new goodness-of-fit tests for exponentiality based on a unique property of the exponential distribution, utilizing U-empirical processes, and evaluates their statistical efficiency and practical application.

Contribution

The paper develops novel exponentiality tests using U-empirical processes, analyzes their limiting behavior, and assesses their efficiency and optimality compared to existing methods.

Findings

New tests have nondegenerate kernels.

Limiting distributions and large deviations are characterized.

Tests show high power and efficiency in simulations.

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.