Wishart-Gamma mixtures for multiperil experience ratemaking, frequency-severity experience rating and micro-loss reserving

This paper studies multivariate mixtures with Wishart-Gamma mixing distribution. Af- ter having recalled the definition and main properties of Wishart distributions for random symmetric positive definite matrices, it is shown how they can be used to extend Gamma distributions to the multivariate case, by considering the joint distribution of the diagonal terms. The resulting distribution, which we call Wishart-Gamma distribution, appears to be particularly useful to model correlated random effects in multivariate frequency, severity and duration models, leading to closed form likelihood function and posterior ratemak- ing formula. Three main applications are discussed to demonstrate the versatility of the Wishart-Gamma mixture models: (i) experience rating with several policies or guarantees per policyholder, (ii) experience rating taking into account the correlation between claim fre- quency and severity components, and (iii) dependence modeling between time-to-payment and amount of payment in micro-loss reserving when the ultimate payment is subject to censoring. Besides introducing the Wishart and Wishart-Gamma distributions, we are also among one of the first to employ the techniques such as fractional integral and symbolic calculation in the non-life actuarial literature.