A Prediction Error Criterion for Choosing the Lower Quantile in Pareto Index Estimation

Successful estimation of the Pareto tail index from extreme order statistics relies heavily on the procedure used to determine the number of extreme order statistics that are used for the estimation. Most of the known procedures are based on the minimization of (an estimate of) the asymptotic mean square error of the Hill estimator for the Pareto tail index. The principal drawback of these approaches is that they involve the estimation of nuisance parameters, and therefore lead to complicated selection procedures. Instead, we propose to use Pareto quantile plots and build a prediction error estimator. The latter depends on the quantile at which the data are truncated, and it is minimized to find the optimal number of extreme order statistics used for estimating the Pareto tail index. The main advantages of the new approach are computational simplicity and the absence of estimation of nuisance parameters. The prediction error estimator is actually a generalization of Mallows' Cp to non-normal regression models. Through a simulation study involving several data generating models, we show that our prediction error criterion performs very well in terms of MSE compared to other procedures. The proposed method is also successfully applied to alluvial diamond deposits, finance,income, and insurance data.