Optimal Thresholds for a Mixture of Lognormal Distributions as the Continuous Part of the Mixed Distribution

Abstract

The paper considers a mixture of two and three lognormal distributions as the continuous part of the mixed distribution, which consists of a positive, continuous, skewed distribution with discrete mass at the origin. A mixture of two lognormal distributions with constant shape parameter is unimodal. A mixture of three lognormal distributions with constant shape parameter is not always unimodal. A sufficient condition for unimodality is provided. Expressions produced by fitting the first and second moments of a single lognormal distribution to those of lognormal mixtures are treated. Empirical optimal thresholds for estimating area rain-rate first and second moments are compared with theoretical optimal thresholds chosen by the maximum likelihood approach and by the distance function approach. Theoretical optimal thresholds for estimating the area rain-rate first moment are insensitive to distribution variations relative to those for the second moment.