Prediction intervals for future Pareto record claims

Abstract

Stochastic models and methods for quantifying extreme events are of interest in numerous disciplines. Within this paper, statistical prediction of extreme claims or losses in insurance industry is considered based on upper record values, which describe successively largest observations in a sequence of data over time. The problem of predicting a future record value (here in particular, a future record claim) based on a sequence of previously observed record values (here, past record claims) is addressed by means of prediction intervals. For an underlying Pareto distribution, respective exact and approximate intervals from the literature are summarized and modified and new ones are developed. In a simulation study, these prediction intervals are evaluated and compared regarding coverage frequency and length. The impact of the number of observed record values as well as the choice of the Pareto distribution is discussed. In the case of a small number of record values, the use of k-th record values is considered as an option for statistical analyses to predict, e.g., second largest record claims. Selected prediction methods are applied to several real data sets, which turn out to perform well and to be able to capture the magnitude of future record claims, even for fairly small numbers of record observations. For comparison, generalized Pareto distributions are fitted to real data sets and a corresponding point predictor as well as a respective upper prediction interval for the next record to appear are derived and evaluated.