A class of consistent tests for exponentiality based on the empirical Laplace transform

The Laplace transform (t=Eexp(–tX)]) of a random variable with exponential density exp(–x), x0, satisfies the differential equation (+t)(t)+(t=0, t0). We study the behaviour of a class of consistent (omnibus) tests for exponentiality based on a suitably weighted integral of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9qqVa0l% b9peea0lb9sq-JfrVkFHe9peea0dXdarVe0Fb9pgea0xa9pue9Fve9% Ffc8meGabaqaciGacaGaaeqabaWaaeaaeaaakeaacaGGBbGaaiikai% qbeU7aSzaajaWaaSbaaSqaaGqaciaa-5gaaeqaaOGaey4kaSIaamiD% aiaacMcacqaHipqEcaWFNaWaaSbaaSqaaiaad6gaaeqaaOGaaiikai% aadshacaGGPaGaey4kaSIaeqiYdK3aaSbaaSqaaiaad6gaaeqaaOGa% aiikaiaadshacaGGPaGaaiyxamaaCaaaleqabaGaaGOmaaaaaaa!4C69!\(\hat \lambda _n + t)\psi '_n (t) + \psi _n (t)]^2 \], where % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9qqVa0l% b9peea0lb9sq-JfrVkFHe9peea0dXdarVe0Fb9pgea0xa9pue9Fve9% Ffc8meGabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaH7oaBgaqcam% aaBaaaleaaieGacaWFUbaabeaaaaa!3A66!\\hat \lambda _n \] is the maximum-likelihood-estimate of and _n is the empirical Laplace transform, each based on an i.i.d. sample X ₁,...,X _n .