An adaptive omnibus test for exponentiality

Let X$_1$,..., X$_n$ be independent and identically distributed observations on a non-negative random variable X. If X has the exponential density $\lambda$exp(-$\lambda$x), x $\geq$ 0, its Laplace transform $\psi$(t) = E[exp(-tX)] satisfies the differential equation ($\lambda$+t)$\psi$'(t) + $\psi$(t) = 0,t $\geq$ 0. We show that a strong omnibus test for exponentiality may be based on max$_{a >0}$ $\int_0^\infty$ [($\lambda_n$+t)$\psi'_n$(t) + $\psi_n$(t)]$^2$ aX$_n$ exp(-aX$_n$t)dt, where $\lambda$ = X$_n^{-1}$ is the Maximum-Likelihood- estimate of λ and Ψ$_n$ is the empirical Laplace transform, each based on X$_1$,..., X$_n$.