Tests of fit for exponentiality based on the empirical Laplace transform

The Laplace transform $\psi(t)$=E[exp(-tX)] of a random variable X with exponential density $\lambda$exp(-$\lambda$x), x $\geq$ 0, satisfies the equation ($\lambda$+t)$\psi(t)$-$\lambda$-0 , t $\geq$ 0. We study the behavior of a class of consistent tests for exponentiality based on a suitably weighted integral of [($\lambda$$_n$+t)$\psi_n$(t)-$\lambda$$_n$]$^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$_n$. As the decay of the weight function tends to infinity, the test statistic approaches the square of the first nonzero component of Neyman's smooth test for exponentiality. The new tests are compared with other omnibus tests for exponentiality.