ABSTRACT In this paper we explore a Bayesian approach for stable distributions in presence of covariates. This class of distribution has great flexibility for fitting asymmetric and heavy-tailed empirical data. These models are commonly used for data sets in finance and insurance. In this paper we show that these distributions can also be used to fit clinical data. Since there is not an analytical form for the density probability function which implies in serious difficulties to obtain the maximum likelihood estimators for the parameters, we use Bayesian methods with data augmentation techniques to get the inferences of interest. In this study we also discuss the choice of different prior distributions for the parameters considering regression models for the location and scale parameters of the stable distribution. We use MCMC (Markov Chain Monte Carlo) algorithms to generate samples from the posterior distributions in order to evaluate the point and interval estimators. A great simplification is obtained using the OpenBugs software. Two real data examples illustrate the applicability of the proposed modeling approach.

Abstract This paper takes into account the estimation for the unknown parameters of the Gompertz distribution from the frequentist and Bayesian view points by using both objective and subjective prior distributions. We first derive non-informative priors using formal rules, such as Jefreys prior and maximal data information prior (MDIP), based on Fisher information and entropy, respectively. We also propose a prior distribution that incorporate the expert’s knowledge about the issue under study. In this regard, we assume two independent gamma distributions for the parameters of the Gompertz distribution and it is employed for an elicitation process based on the predictive prior distribution by using Laplace approximation for integrals. We suppose that an expert can summarize his/her knowledge about the reliability of an item through statements of percentiles. We also present a set of priors proposed by Singpurwala assuming a truncated normal prior distribution for the median of distribution and a gamma prior for the scale parameter. Next, we investigate the effects of these priors in the posterior estimates of the parameters of the Gompertz distribution. The Bayes estimates are computed using Markov Chain Monte Carlo (MCMC) algorithm. An extensive numerical simulation is carried out to evaluate the performance of the maximum likelihood estimates and Bayes estimates based on bias, mean-squared error and coverage probabilities. Finally, a real data set have been analyzed for illustrative purposes.

Considerando a importância sócio-econômica da região de Presidente Prudente, este estudo teve como objetivo estimar a precipitação pluvial máxima esperada para diferentes níveis de probabilidade e verificar o grau de ajuste dos dados ao modelo Gumbel, com as estimativas dos parâmetros obtidas pelo método de máxima verossimilhança. Pelos resultados, o teste de Kolmogorov-Sminorv (K-S) mostrou que a distribuição Gumbel testada se ajustou com p-valor maior que 0.28 para todos os períodos de tempo considerados, comprovando que a distribuição Gumbel apresenta um bom ajustamento aos dados observados para representar as precipitações pluviais máximas. As estimativas de precipitação obtidas pelo método de máxima verossimilhança são consistentes, conseguindo reproduzir com bastante fidelidade o regime de chuvas da região de Presidente Prudente. Assim, o conhecimento da distribuição da precipitação pluvial máxima mensal e das estimativas das precipitações diárias máximas esperadas, possibilita um planejamento estratégico melhor, minimizando assim o risco de ocorrência de perdas econômicas para essa região.

Considering the socioeconomic importance of Presidente Prudente area, this study aimed at to estimate the maximum pluvial precipitation expected for different levels of probability and to check the fitting degree of the data to the Gumbel model, with the parameter estimation obtained by the maximum likelihood approach. The Kolmogorov-Sminorv (K-S) test showed that the Gumbel distribution has fitted with p-value larger than 0.28 for all of the time periods considered, showing that the Gumbel distribution presents a good fitting to the observed data representing the maximum pluvial precipitations values. The precipitation estimates obtained by the maximum likelihood approach are consistent allowing to reproduce with plenty reliability the rainfall regime for Presidente Prudente region. Therefore, the knowledge of the monthly maximum pluvial precipitation distribution and the expected maximum daily precipitation estimates permits a better strategic planning thus minimizing the risk of economical losses for this region.