  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.
