The Langevin function and 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.