Some hydrological applications of small samples estimators of Generalized Pareto and Generalized Extreme Value distributions

The Generalized Pareto (GP) and Generalized Extreme Value (GEV) distributions have been widely applied in the frequency analysis of numerous meteorological and hydrological events. There are several techniques for the estimation of the parameters, which use the total sample as a source of information. In this paper, we show how valuable estimates are also possible considering only a proper subset of the sample, and we identify the portion of the sample containing the most relevant information for estimating a given parameter. In turn, this may prevent the use of anomalous values, which may adversely affect standard techniques. Here, we illustrate original techniques (based on linear combinations of ‘selected’ order statistics) to estimate the position parameter, the scale parameter, the quantiles, and the possible scaling behavior of the GP and GEV distributions with negative shape parameters. These estimators are generally unbiased and Mean-Square-Error-consistent. In addition, weakly consistent estimators of quantiles are introduced, the calculation of which does not require the knowledge of any parameter. Some case studies illustrate the applicability of the new techniques in hydrologic practice, and comparisons with standard methods are presented. The new estimators proposed may provide a reasonable alternative to standard methods, and may serve, at least, as a methodology to cross-check the estimates resulting from the application of other techniques.