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Revista Colombiana de Estadística

Print version ISSN 0120-1751

Rev.Colomb.Estad. vol.42 no.2 Bogotá July/Dec. 2019

http://dx.doi.org/10.15446/rce.v42n2.70271 

Artículos originales de investigación

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

Teoría de valores extremos aplicada a las r estadísticas de orden superior desde el punto de vista

Renato Santos da Silvaa  , Fernando Ferraz do Nascimentob 

aInstituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, Brazil. E-mail: renatoifpi@gmail.com

bDepartamento de Estatística, Universidade Federal do Piauí, Teresina, Brazilecife Military School, Recife, Brazil. E-mail: ernandofn@ufpi.edu.br

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.

Key words: Markov chain monte carlo; Extreme value; Bayesian inference

Resumen

La 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 simulaciones 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 proporciona inferencias para estos parámetros y niveles de retorno. También, proponemos un procedimiento para seleccionar el r-óptimo de la distribución de las r estadísticas de orden superior basadas en el en foque bayesiano y aplicando el método de Monte Carlo para cadenas de Markov (MCMC). Los resultados de la simulación muestran que el enfoque bayesiano produce un rendimiento similar al de la estimación de máxima verosemelianza. Finalmente, las aplicaciones desarrolladas utilizando el enfoque bayesiano mostraron una ganancia en la precisión en comparación con otros estimadores.

Palabras-clave: Monte Carlo para cadena de Markov; Valores extremos; Inferencia bayesiana

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