Generalized Post–Widder inversion formula with application to statistics

Abstract

In this work we derive an inversion formula for the Laplace transform of a density observed on a curve in the complex domain, which generalizes the well known Post–Widder formula. We establish convergence of our inversion method and derive the corresponding convergence rates for the case of a Laplace transform of a smooth density. As an application we consider the problem of statistical inference for variance-mean mixture models. We construct a nonparametric estimator for the mixing density based on the generalized Post–Widder formula, derive bounds for its root mean square error and give a brief numerical example.