Algorithm for the maximum likelihood estimation of the parameters of the truncated normal and lognormal distributions

TL;DR

This paper introduces a straightforward maximum likelihood estimation method for univariate truncated normal and lognormal distributions, utilizing a reparameterization that simplifies estimation especially near power law cases.

Contribution

It presents a novel reparameterization of the lognormal distribution that improves parameter estimation near power law limits and clarifies the relationship between the distributions.

Findings

New parameters quantify the distance from a power law.

Parameters simplify estimation near power law cases.

Method improves estimation stability in limiting cases.

Abstract

This paper describes a simple procedure to estimate the parameters of the univariate truncated normal and lognormal distributions by maximum likelihood. It starts from a reparameterization of the lognormal that was previously introduced by the author and is especially useful when the lognormal is close to a power law, which is a limiting case of the first distribution. One of the new parameters quantifies the distance from the power law, and vanishes when the power law gives a sufficient description of the data. At this point, the other parameter equals the exponent of the power law. In contrast, when using the standard parameterization, the parameters of the lognormal diverge in the neighborhood of the power law. Whether or not we are in this neighborhood, the new parameters have properties that ease the process of estimation.