A comparative simulation study of the annual maxima and the peaks-over-threshold methods

Abstract

In order to assess their relative merits in the context of the determinalion of Hydraulic Boundary Conditions (HBC), the Annual Maxima / Generalized Extreme Value distribution (AM/GEV) and Peaks over Threshold / Generalized Pareto Oislribution (POT/GPO) approaches are compared in terms of Iheir accuracy, as measured by mean square errors, in estimating exceedance probabiliLies on the basis of time series with various lengths and with characteristics, that mimic those of real time series, such as non-stationarity and serial dependence. Two types of simulation studies were carried out. Both studies took into account Ihe characterislics of the dala currenlly available on the so-called basic variables. The firsl sludy focused on Ihe lintte-sample properties of the estimators of the GEV and GPO models based on independent and identically distributed dala, both the Maximum Ukelihood (ML) and Probability Weighted Momenls (PWM) eslimalion methods having been considered. The second study focused on Ihe finite-sample properties of the AMlGEV and Ihe POT/GPO approaches applied to non-stationary and dependenl dala and using the method of PWM. The conclusions of the firsl study were Ihat wilh POT samples having an average of two or more observations per year, the GPO estimates are more accurate than the corresponding GEV estimates, and thai wilh more than 200 years of data the accuracies of the two approaches are similar and rather good. Furthermore, it was concluded that with less than 50 years of dala the method of PWM should, on the basis of its error characteristics and robustness, be preferred 10 Ihe ML method, and thai with longer dala sets Ihe two estimation methods have comparable accuracies. In Ihe second study, based on non-stationary and serially dependent dala, it was concluded that Ihe POT/GPO estimates of the shape parameter are more accurate than those of Ihe AMlGEV approach with time series less than 100 years long. Wtth 10Q.year long time series the performance of Ihe two approaches is comparable, the accuracy of the AMlGEV approach being slighUy greater with lighter tails and that of the POT/GPO approach being slightiy greater wtth heavier tails. In terms of return valUe estimates (namely of the 4,000-yr and 10,000 return values), the POT/GPO approach is significantly more accurate. Only for time series of 200 years or more do Ihe two approaches yield comparably small mean square errors. Still, even with 200-year long time series the relative root-mean square errors of the POT/GPO approach are about 213 of those of Ihe AMlGEV approach whenever Ihe underlying tail index exceeds -0.1. A noteworthy aspect of this second sludy is that the choice of Ihe threshold in the POT/GPO approach has been chosen automatically; if visual inspection or a more sophisticated and theoretically grounded melhod is used to choose the threshold, Ihe POT/GPO approach is expected to perform even better. Based on Ihe results of this study, we recommend Ihat, irrespective of the variable of interest, Ihe POT/GPO approach be used for the extreme value analyses of the data required for the computation of HBC. Furthermore. we recommend that the parameters of the GPO be estimated using the method of PWM.