On the Exact Distribution of Correlated Extremes in Hydrology

The analysis of hydrological hazards usually relies on asymptotic results of extreme value theory, which commonly deals with block maxima or peaks over threshold (POT) data series. However, data quality and quantity of block maxima and POT hydrological records do not usually fulfill the basic requirements of extreme value theory, thus making its application questionable and results prone to high uncertainty and low reliability. An alternative approach to better exploit the available information of continuous time series and nonextreme records is to build the exact distribution of maxima (i.e., nonasymptotic extreme value distributions) from a sequence of low-threshold POT. Practical closed-form results for this approach do exist only for independent high-threshold POT series with Poisson occurrences. This study introduces new closed-form equations of the exact distribution of maxima taken from low-threshold POT with magnitudes characterized by an arbitrary marginal distribution and first-order Markovian dependence, and negative binomial occurrences. The proposed model encompasses and generalizes the independent-Poisson model and allows for analyses relying on significantly larger samples of low-threshold POT values exhibiting dependence, temporal clustering, and overdispersion. To check the analytical results, we also introduce a new generator (called Gen2Mp) of proper first-order Markov chains with arbitrary marginal distributions. An illustrative application to long-term rainfall and streamflow data series shows that our model for the distribution of extreme maxima under dependence takes a step forward in developing more reliable data-rich-based analyses of extreme values.