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 $\geq$ 0, satisfies the differential equation ($\lambda$ + t)$\psi$'(t) + $\psi$(t) - 0, t $\geq$ 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 X$_1$,...,X$_n$.