The Statistical Distribution of Incurred Losses and Its Evolution Over Time I: Non-Parametric Models

Abstract

The distribution of the total incurred losses of an accident year (or underwriting year) is considered. Before commencement of the accident year, there is a prior on this quantity. The distribution may eveolve over time according to Bayesian revision which takes account of the accumulation of data with time.The distribution in question can be made subject to various assumptions and restrictions. The different forms of these are explored in a sequence of papers that include the present one.A previoius paper examined the situation in which no restrictions were imposed. The resulting models were refered to as non-parametric.The present paper considers the case in which the posterior-to-data estimates of the subject distribution are restricted to a specific parametric family. These models are referred to as parametric. The subject distribution evolves with the evolution of its parameters under Bayesian revision. The credibility approximations to this revision are worked out in general (Section 4) and for the special case of normally distributed data (Section 5).The results are illustrated by application to a very simply example (Section 6). They are illustrated further by application to a more extensive example involving real data (Section 7). The examples use the same respective data sets as in the previous paper.All models to this point represent the loss experience of each development year in terms of a separate set of parameters applicable to just that year. Section 9 analyses the case in which all of these parameters are expressed as functions of a reduced parameter set. Again, a numerical example is given.