A multivariate unsupervised lognormal-GPD mixture for dependent losses

Abstract

We propose a multivariate two-population mixture distribution for heavy-tailed and positively skewed non-negative data. The first mixture population is a multivariate lognormal, while the second is a distribution obtained by joining univariate zero-mean generalized Pareto densities via a Gumbel copula. The model is unsupervised, since no threshold has to be set or estimated. We derive some properties of the distribution and develop an Expectation-Conditional Maximization algorithm for maximum likelihood estimation, which is found to perform well in multivariate contexts beyond the bivariate case. The finite-sample properties of the estimators are assessed by means of simulation experiments, and an empirical application to aggregated commercial property-casualty insurance claims illustrates the impact of dependence on risk estimation. All the methods are implemented in the R package MultLognGPD, available on Github.