Generalizing the Log-Moyal Distribution and Regression Models for Heavy-Tailed Loss Data

Abstract

Catastrophic loss data are known to be heavy-tailed. Practitioners then need models that are able to capture both tail and modal parts of claim data. To this purpose, a new parametric family of loss distributions is proposed as a gamma mixture of the generalized log-Moyal distribution from Bhati and Ravi (2018), termed the generalized log-Moyal gamma (GLMGA) distribution. While the GLMGA distribution is a special case of the GB2 distribution, we show that this simpler model is effective in regression modeling of large and modal loss data. Regression modeling and applications to risk measurement are illustrated using a detailed analysis of a Chinese earthquake loss data set, comparing with the results of competing models from the literature. To this end, we discuss the probabilistic characteristics of the GLMGA and statistical estimation of the parameters through maximum likelihood. Further illustrations of the applicability of the new class of distributions are provided with the fire claim data set reported in Cummins et al. (1990) and a Norwegian fire losses data set discussed recently in Bhati and Ravi (2018).

An Expectation-Maximization Algorithm for the Exponential-Generalized Inverse Gaussian Regression Model with Varying Dispersion and Shape for Modelling the Aggregate Claim Amount

“…Additionally, the use of continuous mixture distributions has been proposed as a way to capture the heavy-tailed behavior of insurance losses. In Li et al (2020), the authors considered a new parametric family of loss distributions, termed the Generalized Log-Moyal Gamma distribution (GLMGA), which can be derived as a Gamma mixture of the Generalized Log-Moyal distribution; see Bhati and Ravi (2018). While the GLMGA distribution that they presented is a special case of the GB2 distribution, they demonstrated that it is effective in the regression modelling of large and modal loss data.…”

A new class of copula regression models for modelling multivariate heavy-tailed data

“…While univariate risk models based on heavy tailed distributions are well developed (see e.g. Beirlant and Goegebeur (2003), Li et al (2016), Leppisaari (2016), and Li et al (2021)), predicting extreme loss through multivariate models has received much less attention, especially in the presence of additional covariate information.…”

“…The main contribution of this article is to propose a new class of beta-type copulas, the MGL copula family, where the dependence function is extracted from a new multi-dimensional version of the univariate GLMGA distribution which was proposed in Li et al (2021). We provide some important characteristics of this class and obtain the corresponding extreme-value copula (the MGL-EV copula).…”

“…In the second application we model the dependence between a continuous and a semi-continuous variable in a Chinese earthquake data set. This data set was already analyzed in Li et al (2021) concerning the univariate losses. In both case studies we consider the evolution of the dependence as a function of time.…”

“…The GLMGA three-parameter distribution model as proposed in Li et al (2021), is obtained by mixing a generalized log-Moyal distribution (GlogM) (Bhati and Ravi, 2018) with the gamma distribution. The density function of the GLMGA distribution is then given by f (y) = (2b) a σB(a, 1 2 )…”