Abstract

SANTOS DA SILVA, Renato  and  FERRAZ DO NASCIMENTO, Fernando. Extreme Value Theory Applied to r Largest Order Statistics Under the Bayesian Approach. Rev.Colomb.Estad. [online]. 2019, vol.42, n.2, pp.143-166. ISSN 0120-1751.  https://doi.org/10.15446/rce.v42n2.70271.

Extreme value theory (EVT) is an important tool for predicting efficient gains and losses in economic and environmental domains. Moreover, EVT was initially developed for use with normal and gamma parametric distribution patterns. However, economic and environmental data present a heavy-tailed distribution in most cases, which is in contrast with the above patterns. Thus, the framing of extreme events using EVT presented great difficulties. Furthermore, it is nearly impossible to use conventional models to make predictions about non-observed events that exceeded the maximum number of observations. In some situations, EVT is used to analyze only the maximum values of a given dataset, which provides few observations. In such cases, it is more effective to use the r largest order statistics. This study proposes Bayesian estimators for the parameters of the r largest order statistics. We use a Monte Carlo simulation to analyze the experimental data and observe certain estimator properties, such as mean, confiance interval, credibility interval, bias, and root mean square error (RMSE); estimation provided inferences for these parameters and return levels. In addition, this study proposes a procedure for selecting the r-optimal of the r largest order statistics based on the Bayesian approach and applying the Markov chains Monte Carlo (MCMC) method. Simulation results reveal that the Bayesian approach produced performance similar to that of the maximum likelihood estimation. Finally, the applications developed using the Bayesian approach showed a gain in accuracy compared with other estimators.

Keywords : Markov chain monte carlo; Extreme value; Bayesian inference.

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