A new flexible class of omnibus tests for exponentiality

Abstract

Let X₁,…,X_n be independent and identically distributed non-negative random variables, and let Y_j = X_jX_n (j= 1,…,n), where If X₁ has the exponential density λexp(−λx),x>0, for some λ>0, the empirical Laplace transform of Y₁,…,Y_n should be close to (1 + t)⁻¹ which is the Laplace transform of the unit exponential distribution. We study the properties of as a statistic for testing for exponentiality. The limiting null distribution of T_n,a is found, and it is shown that the test rejecting the hypothesis of exponentiality for large values of T_n,a is consistent against each fixed alternative distribution. This new class of tests offers great flexibility in that the parameter a may be chosen so as to yield high power against specific alternatives. It is also possible to let a depend on X₁,…, X_n. Power performance of the new tests for finite samples is assessed in a Monte Carlo study.

Keywords:

• test for exponentiality
• consistency
• empirical Laplace transform
• goodness-of-fit test