Fitting mixed-effects models when data are left truncated

Damage sizes, i.e. all damages occurring to a policy and not only those that are reported to an insurance company, are modelled as a linear mixed model. Only those damages that are larger than their deductibles are reported to the company, and this fact should be taken into account when analyzing such data. In statistical terms, the problem is to make inference in a linear mixed model with left truncated data. Estimation methods based on a Monte Carlo simulation of the likelihood are proposed, and extensive simulations to evaluate the quality of the methods are reported. The proposed methods are then used to analyze claimsizes for some marine insurance data, where shipowners represent random effects and technical data about the ships represent fixed effects.