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

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.