A CLASS OF CONSISTENT TESTS FOR EXPONENTIALITY BASED ON THE EMPIRICAL LAPLACE TRANSFORM

LUDWIG BARINGHAUS AND NORBERT HENZE

Institut für Mathematische Stochastik, Universität Hannover, Welfengarten 1, D-3000 Hannover 1, FRG

(Received November 6, 1989; revised May 14, 1990)

Abstract. The Laplace transform ψ(t) = E[exp(−tX)] of a random variable with exponential density λ exp(−λx), x ≥ 0, satisfies the differential equation (λ + t)ψ′(t) + ψ(t) = 0, t ≥ 0. We study the behaviour of a class of consistent (“omnibus”) tests for exponentiality based on a suitably weighted integral of [(λ̂_{n} + t)ψ′_{n}(t) + ψ_{n}(t)]^{2}, where λ̂_{n} is the maximum-likelihood-estimate of λ and ψ_{n} is the empirical Laplace transform, each based on an i.i.d. sample X_{1}, …, X_{n}.

Key words and phrases: Exponential distribution, goodness-of-fit test, empirical Laplace transform, consistency.