Generalized Post-Widder inversion formula with application to statistics

TL;DR

This paper introduces a generalized inversion formula for Laplace transforms applicable to complex domain curves, with convergence analysis and a statistical application to nonparametric estimation in mixture models.

Contribution

It extends the classical Post-Widder formula to a broader setting and applies it to develop a new estimator for mixing densities in variance-mean mixture models.

Findings

Convergence of the generalized inversion method is established.

Derived bounds for the estimator's root mean square error.

Numerical example demonstrating the estimator's performance.

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.