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

The Laplace transform psi(t) = E[exp(-tX)] of a random variable with exponential density lambda exp(-lambda-x), x greater-than-or-equal-to 0, satisfies the differential equation (lambda + t)psi'(t) + psi(t) = 0, t greater-than-or-equal-to 0. We study the behaviour of a class of consistent ("omnibus") tests for exponentiality based on a suitably weighted integral of [(lambda-n + t)psi'n(t) + psi-n(t)]2, where lambda-n is the maximum-likelihood-estimate of lambda and psi-n is the empirical Laplace transform, each based on an i.i.d. sample X1, ..., X(n).