Numerical Approximation of Probability Mass Functions Via the Inverse Discrete Fourier Transform

TL;DR

This paper presents a fast and stable numerical method for approximating probability mass functions of lattice distributions using the inverse discrete Fourier transform, with proven error bounds and practical implementation in R.

Contribution

It introduces an efficient Fourier-based approach for computing first passage distributions of semi-Markov processes, including error bounds and implementation details.

Findings

Fast computation using inverse discrete Fourier transform

Proven bounds for numerical inversion error

Practical R implementation included

Abstract

First passage distributions of semi-Markov processes are of interest in fields such as reliability, survival analysis, and many others. The problem of finding or computing first passage distributions is, in general, quite challenging. We take the approach of using characteristic functions (or Fourier transforms) and inverting them, to numerically calculate the first passage distribution. Numerical inversion of characteristic functions can be numerically unstable for a general probability measure, however, we show for lattice distributions they can be quickly calculated using the inverse discrete Fourier transform. Using the fast Fourier transform algorithm these computations can be extremely fast. In addition to the speed of this approach, we are able to prove a few useful bounds for the numerical inversion error of the characteristic functions. These error bounds rely on the existence of a first or second moment of the distribution, or on an eventual monotonicity condition. We demonstrate these techniques in an example and include R-code.