Extreme Value Theory Applied to r Largest Order Statistics Under the Bayesian Approach

Abstract

Extreme Value Theory Applied to r r r Largest Order Statistics Under the Bayesian ApproachTeoría de valores extremos aplicada a las r r r estadísticas de orden superior desde el punto de vista bayesiano Abstract 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. ResumenLa teoría de valores extremos (EVT) es una herramienta importante para predecir ganancias y pérdidas eficientes en ambientes económicos y ambientales. Además, la EVT se desarrolló inicialmente para uso con patrones de distribución paramétricos normales y gamma. Sin embargo, los datos económicos y ambientales presentan una distribución de cola pesada en la mayoría de los casos, lo que contrasta con los patrones anteriores. Así, la formulación de eventos extremos con EVT presenta grandes dificultades. Además, es casi imposible usar modelos convencionales para obtener predicciones sobre eventos no observados que excedieron el número máximo de observaciones. En algunas situaciones, EVT es utilizado para analizar solamente los valores máximos de un conjunto de datos dado, que proporcionan poca información. En tales casos, es más eficiente usar las r estadísticas de orden superior. Este trabajo propone estimadores bayesianos para los parámetros de las r estadísticas de orden superior. Utilizamos simulaciónes de Monte Carlo para analizar los datos experimentales y observar ciertas propiedades del estimador como: media, intervalos de confianza y credibilidad, sesgo y error cuadrático medio (RMSE). Este tipo de estimación p...

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