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...
The present study aimed to characterize common bean cultivars (Phaseolus vulgaris L.) in the Aquidauana-MS region based on the DHE test (Distinction, Homogeneity and Stability). The experiment was carried out at the State University of Mato Grosso do Sul. The design used was completely randomized blocks, with four replications, each containing a sample consisting of 25 seeds. Ten common bean cultivars were used, BRS Radiante, Pérola, Madre Pérola, BRS Estilo, BRS Esplendor, Jalo Precoce, BRS Campeiro, BRS Amethyst, BRSMG Highce and BRS Pitanga. The characterization of the genotypes was based on the morphological descriptors for the crop, standardized according to the Minimum Descriptors to Characterize Common Bean Cultivars / Varieties. To describe the qualitative factors, we used: Ribs in the seed, absent or present; Seed color, uniform or uneven; Seed halo, absent or present; Seed halo color, same seed color or different from seed; Commercial Group to which it belongs, white, carioca, jalo, mulatinho, black, rosinha, purple or special; Seed shape, spherical; elliptical: oblong / short reniform; medium oblong / reniform; long oblong / reniform; Flatness degree, flattened; semi-full; full. While the quantitative descriptors were measured with the aid of a caliper and an analytical digital scale, being evaluated: mass of one hundred grains (g), with average humidity between 12 to 14%; seed length, width and thickness (mm). The use of qualitative or quantitative descriptors considerably modifies the grouping of common bean genotypes, the Gower algorithm being effective in grouping common bean genotypes based on the descriptors.
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