Inverse functions for Monte Carlo simulations with applications to hitting time distributions

Abstract

Random sampling is a ubiquitous tool in simulations and modeling in a variety of applications. There are efficient algorithms for several known distributions, but in general, one must resort to computing or approximating the inverse of the distributions to generate random samples. In certain physical and biomedical applications with which we have been particularly concerned, it has proven to be more efficient to provide random times for a walk of a fixed length, rather than the conventional random step lengths in a given time step for the walker. For these, the hitting‐time distributions must be sampled, but the problem is that the distributions contain complicated expressions for which no efficient method exists to compute an inverse. In this article, we explore a well‐known probability (the F‐ratio distribution)—whose inverse is efficiently computable—as an alternative to generating look‐up tables and interpolations to obtain the required stochastic time samples. We transform the two‐parameter F‐distribution into a four‐parameter distribution and we match the first four moments of the distribution to the moments of the exact hitting‐time distribution. We find that this simple procedure approximates the hitting‐time distribution well and produces very small errors. Future Monte Carlo simulations in a number of fields of applications may benefit from our method.