The Langevin function and truncated exponential distributions

TL;DR

This paper explores the relationship between the Langevin function and truncated exponential distributions, providing methods to estimate distribution parameters via inverse Langevin functions and developing practical approximations.

Contribution

It introduces a novel connection between the Langevin function and truncated exponential distributions, offering new approximation techniques for parameter estimation.

Findings

Parameter estimation can be expressed using the inverse Langevin function.

Developed practical approximations for the transcendental equation.

Provides insights into the behavior of truncated exponential distributions.

Abstract

Let K be a random variable following a truncated exponential distribution. Such distributions are described by a single parameter here denoted by $\gamma$. The determination of $\gamma$ by Maximum Likelihood methods leads to a transcendental equation. We note that this can be solved in terms of the inverse Langevin function. We develop approximations to this guided by work of Suehrcke and McCormick.