All in order: Distribution of serially correlated order statistics with applications to hydrological extremes

Abstract

Classic extreme value theory provides asymptotic distributions of block maxima (BM) Y or peaks over threshold (POT). However, as BM and POT are relatively small subsets of the values assumed by the parent process Z, alternative approaches have been proposed to make inferences on extremes by using intermediate values and non-asymptotic models. In this study, we investigate the finite sample theory of extremes based on order statistics, and present a set of results enabling the analysis of the properties of non-asymptotic distributions of BM in finite-size blocks of data under the assumption that the parent process Z has stationary temporal dependence. In particular, we suggest the beta-binomial distribution (FβB) as a suitable approximation of the marginal distribution of order statistics and BM under dependence, thus generalizing the theoretical results available under independence. We demonstrate the usefulness of the FβB distribution in three conceptual applications of hydrological interest. Firstly, we review the so-called Complete Time-series Analysis (CTA) framework, showing that the differences between FY and FZ are due to the inherent theoretical nature of the processes Y and Z and the different probabilistic failure scenarios described by FY and FZ. Secondly, we provide a theoretical rule to select the number of the largest maxima (LM) and over-threshold exceedances (OT) required to approximate the desired portion of the upper tail of FZ with specified accuracy. Finally, we discuss how the FβB distribution offers an interpretation and generalization of the so-called metastatistical extreme value (MEV) framework and its simplified version (SMEV), avoiding the use of high-dimensional joint distributions and preliminary data thresholding and declustering. All methodological results are validated by Monte Carlo simulations involving widely used stochastic processes with persistence. Real-world stream flow data are also analyzed as proof of concept.