Applications of an algorithm for solving Fredholm equations of the first kind

TL;DR

This paper explores an iterative EM-based algorithm for solving Fredholm equations of the first kind, demonstrating its application in estimating densities and extending it to non-negative functions with proven convergence.

Contribution

It introduces an extension of the EM algorithm for Fredholm equations to handle non-negative and general functions, with convergence proofs and practical examples.

Findings

Successful estimation of mixing and passage time densities

Extension of the algorithm to non-negative functions

Proof of convergence for all algorithm variants

Abstract

In this paper we use an iterative algorithm for solving Fredholm equations of the first kind. The basic algorithm is known and is based on an EM algorithm when involved functions are non-negative and integrable. With this algorithm we demonstrate two examples involving the estimation of a mixing density and a first passage time density function involving Brownian motion. We also develop the basic algorithm to include functions which are not necessarily non-negative and again present illustrations under this scenario. A self contained proof of convergence of all the algorithms employed is presented.