On the inversion of the Laplace transform (In Memory of Dimitris Gatzouras)

The Laplace transform is a useful and powerful analytic tool with applications to several areas of applied mathematics, including differential equations, probability and statistics. Similarly to the inversion of the Fourier transform, inversion formulae for the Laplace transform are of central importance; such formulae are old and well-known (Fourier-Mellin or Bromwich integral, Post-Widder inversion). The present work is motivated from an elementary statistical problem, namely, the unbiased estimation of a parametric function of the scale in the basic model of a random sample from exponential distribution. The form of the uniformly minimum variance unbiased estimator of a parametric function $h(\lambda)$, as well as its variance, are obtained as series in Laguerre polynomials and the corresponding Fourier coefficients, and a particular application of this result yields a novel inversion formula for the Laplace transform. MSC: Primary 44A10, 62F10. Key words and phrases: Exponential Distribution, Unbiased Estimation; Fourier-Laguerre Series; Inverse Laplace Transform; Laguerre Polynomials.