Bias Reduced Peaks over Threshold Tail Estimation

Abstract

In recent years several attempts have been made to extend tail modelling towards the modal part of the data. Frigessi et al. (2002) introduced dynamic mixtures of two components with a weight function {\pi} = {\pi}(x) smoothly connecting the bulk and the tail of the distribution. Recently, Naveau et al. (2016) reviewed this topic, and, continuing on the work by Papastathopoulos and Tawn (2013), proposed a statistical model which is in compliance with extreme value theory and allows for a smooth transition between the modal and tail part. Incorporating second order rates of convergence for distributions of peaks over thresholds (POT), Beirlant et al. (2002, 2009) constructed models that can be viewed as special cases from both approaches discussed above. When fitting such second order models it turns out that the bias of the resulting extreme value estimators is significantly reduced compared to the classical tail fits using only the first order tail component based on the Pareto or generalized Pareto fits to peaks over threshold distributions. In this paper we provide novel bias reduced tail fitting techniques, improving upon the classical generalized Pareto (GP) approximation for POTs using the flexible semiparametric GP modelling introduced in Tencaliec et al. (2018). We also revisit and extend the secondorder refined POT approach started in Beirlant et al. (2009) to all max-domains of attraction using flexible semiparametric modelling of the second order component. In this way we relax the classical second order regular variation assumptions.