Return level estimation: a theoretical comparison between the peak-over-threshold approach and the block maxima method

Two fundamental approaches in extreme value theory are the peakover- threshold (POT) approach and the block maxima (BM) method. We provide a theoretical comparison between these methods for estimating the return level. The two methods are studied both for independent and identically distributed observations and for serial dependent observations. For both dependence structures and both methods, we propose an estimator for the return level and prove its asymptotic normality. Explicit calculations for specific time series models show that the BM method outperforms the POT approach in terms of having a lower asymptotic variance.