Abstract
In several applications, ultimately at the largest data, truncation effects can be observed when analysing tail characteristics of statistical distributions. In some cases truncation effects are forecasted through physical models such as the Gutenberg-Richter relation in geophysics, while at other instances the nature of the measurement process itself may cause under recovery of large values, for instance due to flooding in river discharge readings. Recently Beirlant et al. (2016) discussed tail fitting for truncated Pareto-type distributions. Using examples from earthquake analysis, hydrology and diamond valuation we demonstrate the need for a unified treatment of extreme value analysis for truncated heavy and light tails. We generalise the classical Peaks over Threshold approach for the different max-domains of attraction with shape parameter $\xi>-1/2$ to allow for truncation effects. We use a pseudo-maximum likelihood approach to estimate the model parameters and consider extreme quantile estimation and reconstruction of quantile levels before truncation whenever appropriate. We report on some simulation experiments and provide some basic asymptotic results.
Highlights
Modelling extreme events has recently received a lot of interest
Condition (1) is equivalent to the convergence of the distribution of excesses over high thresholds t to the generalised Pareto distribution (GPD): as t tends to the endpoint of the distribution of Y, with Fthe right tail function (RTF) of a given distribution, P
In the case of Pareto-type tails with ξ > 0 the proposed methods should be compared with the methods which have been developed for that sub-case. To this purpose we extend the classical peaks over threshold (POT) technique with maximum likelihood estimation of the GPD parameters ξ and σ
Summary
Modelling extreme events has recently received a lot of interest. Assessing the risk of rare events through estimation of extreme quantiles or corresponding return periods has been developed extensively and was applied to a wide variety of fields such as meteorology, finance, insurance and geology, among others. The data are peaks over threshold values taken from a complete series of hourly flow measurements which was filtered in order to satisfy hydrological independence as discussed in Willems (2009). This river is prone to flooding at high flow levels and the. This corresponds to situations where the deviation from the Pareto behaviour due to truncation at a high value T will be visible in the data from t on, and the approximation of the POT distribution using the limit distribution in (13) appears more appropriate than with a simple GPD. These derivations will motivate the proposed estimators of DT and extreme quantiles QT (1 − p)
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