A special Tweedie sub-family with application to loss reserving prediction error

Abstract

The paper is concerned with a particular sub-family of the Tweedie family of distributions characterized by a constant mean-to-variance ratio (“CMVR”). The properties of the CMVR sub-family are explored. Some of these, concerned with addition of CMVR variates, are well adapted to the treatment of insurance losses and loss reserves. The tricky issue of parameter estimation within the CMVR Tweedie family is investigated. This family is applied to the estimation of the prediction error associated with a loss reserve, especially the model distribution error component of the prediction error. The model distribution error is the error in the loss reserve that arises from the wrong choice of distribution of observations. This is considered within the Tweedie family of distributions, examining the prediction error that occurs when one value of the Tweedie dispersion parameter p is correct, but a different one is assumed in the modelling a claim array. The study is carried out under the CMVR condition across all cells of the array, a condition that is found commonly compatible with real data sets. The main result of the paper is that, when cells have relatively large coefficients of variation under the CMVR condition, and the array can be modelled with a GLM, the MSEP of the loss reserve is relatively insensitive to the value of p. This implies that model distribution error can often be dismissed as small in many situations where MSEP of loss reserve is the measure of interest.