Accounting for threshold uncertainty in extreme value estimation.

Tail data are often modelled by fitting a generalized Pareto distribution (GPD) to the exceedances over high thresholds. In practice, a threshold υ is fixed and a GPD is fitted to the data exceeding υ. 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 this does not fully take account of threshold uncertainty. We propose to model extreme and non extreme data with a distribution composed by a piecewise constant density up to an unknown end point α and by a GPD with threshold α 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 two real data sets.