Univariate and multivariate mixtures of exponential distributions, with applications in risk modeling

Abstract

AbstractMixed exponential distributions are frequently used in actuarial risk modeling. Distributions obtained through mixtures allow greater flexibility in the modeling of nonlife insurance loss amounts. Several research works have studied mixed exponential distributions in univariate and multivariate settings. The present article highlights the usefulness of such distributions and lays the story of the mixing technique behind them. It also explains the underlying link between all these works. We study in detail three univariate and multivariate mixed exponential distributions defined with a discrete mixing random variable (rv). For each of these three univariate (multivariate) mixed exponential distributions and using an appropriate scaling, we identify the continuous mixing rv to which converges in distribution the discrete mixing rv and the corresponding univariate (multivariate) mixed exponential distribution. In a multivariate setting, we show that these three choices of discrete mixing distributions lead us to known Archimedean copulas constructed with continuous mixing rvs. Applications in actuarial science of these distributions are presented throughout the article highlighting their many uses and useful properties.