Objective Bayesian analysis for exponential power regression models
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Objective Bayesian analysis for exponential power regression models Esther Salazar Duke University, Durham, USA
Marco A.R. Ferreira University of Missouri, Columbia, USA
Helio S. Migon Federal University of Rio de Janeiro, Rio de Janeiro, Brazil
Abstract We develop objective Bayesian analysis for the linear regression model with random errors distributed according to the exponential power distribution. More specifically, we derive explicit expressions for three different Jeffreys priors for the model parameters. We show that only one of these Jeffreys priors leads to a proper posterior distribution. In addition, we develop fast posterior analysis based on Laplace approximations. Moreover, we show that our proposed Bayesian analysis compares favorably to a posterior analysis based on a competing noninformative prior. Finally, we illustrate our methodology with applications of the exponential power regression model to two different datasets. AMS (2000) subject classification. Primary 62F15; Secondary 62J05, 62F35. Keywords and phrases. Bayesian inference, exponential power errors, frequentist properties, Jeffreys prior, robustness.
1 Introduction The exponential power is a flexible distribution for errors of regression models that may have tails either lighter (platykurtic) or heavier (leptokurtic) than Gaussian. In addition, the use of the exponential power distribution reduces the influence of outliers and consequently increases the robustness of the analysis (Box and Tiao, 1962; West, 1984; Liang, Liu and Wang, 2007). While leptokurtic distributions automatically protect against outliers, platykurtic distributions may occur as a result of truncation. Finally, the EP distribution is especially attractive because it includes the normal distribution as a special case and allows continuous variation from
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E. Salazar, M.A.R. Ferreira and H.S. Migon
normality to nonnormality. Despite the importance of the EP distribution, there is no literature on objective priors for regression models with EP errors. In this work we develop objective Bayesian analysis for regression models with independent EP errors. More specifically, we derive explicit expressions for three different Jeffreys priors and we show that only one of these Jeffreys priors leads to a proper posterior distribution. Moreover, a Monte Carlo study shows that our proposed Bayesian approach compares favorably to a posterior analysis based on a competing noninformative prior in terms of coverage of credible intervals and relative mean squared error. This good frequentist behavior of our objective Bayesian procedure has also been found in objective Bayesian analyses for other models such as, for example, in the analysis of elapsed times in continuous-time Markov chains (Ferreira and Suchard, 2008). The EP distribution has been studied and popularized by Box and Tiao (1992) in the context of robustness studies. The EP density is given by −1 1 y − μ p 1 1/p 2p Γ(1 + 1/p) , −∞ < y < ∞, exp − f (y|μ, ξ, p) = ξ p ξ (1.1) where p > 1, −∞ < μ < ∞ and ξ >
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