Mixed Size-Biased Log-Normal Distribution with Truncated Normal Prior and Its Application in Insurance Ratemaking

Abstract

In the insurance literature, accurately predicting extreme losses has been a persistent and important problem. Recently, under the modelling framework of weighted distributions, several finite-mixture size-biased distributions, including size-biased Weibull and size-biased truncated log-normal distributions, have gained popularity for modelling heavy-tailed insurance claim data. In this study, unlike existing models, we explicitly account for the individual heterogeneity commonly observed in insurance claims by treating the order of size-biased weighting as a continuous latent variable, thereby constructing a mixed size-biased distribution. In particular, we study the various distributional properties of the mixed log-normal distribution with a truncated normal prior, which serves as a conjugate prior for the size-biased log-normal model. For applications in non-life insurance, we discuss the Bayesian credibility premium and present an estimation of a regression model via the EM algorithm. We further conduct a real-data analysis using insurance loss data, comparing goodness-of-fit and tail risk measures with those of standard heavy-tailed distributions. © 2026 by the authors.