Estimating extreme tail risk measures with generalized Pareto distribution

• A new GPD parameter estimator is proposed.
• It is based on a nonlinear weighted least squares method.
• Under the POT framework, we estimate tail risk measures. Extensive simulation studies show the new method works well.

The generalized Pareto distribution (GPD) has been widely used in modelling heavy tail phenomena in many applications. The standard practice is to fit the tail region of the dataset to the GPD separately, a framework known as the peaks-over-threshold (POT) in the extreme value literature. In this paper we propose a new GPD parameter estimator, under the POT framework, to estimate common tail risk measures, the Value-at-Risk (VaR) and Conditional Tail Expectation (also known as Tail-VaR) for heavy-tailed losses. The proposed estimator is based on a nonlinear weighted least squares method that minimizes the sum of squared deviations between the empirical distribution function and the theoretical GPD for the data exceeding the tail threshold. The proposed method properly addresses a caveat of a similar estimator previously advocated, and further improves the performance by introducing appropriate weights in the optimization procedure. Using various simulation studies and a realistic heavy-tailed model, we compare alternative estimators and show that the new estimator is highly competitive, especially when the tail risk measures are concerned with extreme confidence levels.