Accounting for threshold uncertainty in extreme value estimation

Abstract

Tail data are often modelled by fitting a generalized Pareto distribution (GPD) to the exceedances over high thresholds. In practice, a threshold $u$ is fixed and a GPD is fitted to the data exceeding $u$. A difficulty in this approach is the selection of the threshold above which the GPD assumption is appropriate. Moreover the estimates of the parameters of the GPD may depend significantly on the choice of the threshold selected. Sensitivity with respect to the threshold choice is normally studied but typically its effects on the properties of estimators are not accounted for. In this paper, to overcome the difficulties of the fixed-threshold approach, we propose to model extreme and non-extreme data with a distribution composed of a piecewise constant density from a low threshold up to an unknown end point $alpha$ and a GPD with threshold $alpha$ for the remaining tail part. Since we estimate the threshold together with the other parameters of the GPD we take naturally into account the threshold uncertainty. We will discuss this model from a Bayesian point of view and the method will be illustrated using simulated data and a real data set.