Modeling Joint Actuarial Costs Using a Bivariate Lognormal-Pareto Distribution

We propose a bivariate two-spliced distribution built from the bivariate lognormal distribution for joint smaller values and from the bivariate Pareto for the joint largest ones. For this bivariate composite distribution, we present some properties and we propose an estimation method based on the maximization of the log-likelihood function. Using simulation, we analyze the properties of this estimation method for finite sample size. We also present two applications on bivariate costs data from car insurance with the aim to estimate the risk using the Value-at-Risk measure. We compare our bivariate distribution to the Gumbel copula with log-logistic marginal distributions, which was previously used with the same type of bivariate cost data with positive upper tail dependence.