Abstract
We develop Bayesian reference analyses for linear regression models when the errors follow an exponential power distribution. Specifically, we obtain explicit expressions for reference priors for all the six possible orderings of the model parameters and show that, associated with these six parameters orderings, there are only two reference priors. Further, we show that both of these reference priors lead to proper posterior distributions. Furthermore, we show that the proposed reference Bayesian analyses compare favorably to an analysis based on a competing noninformative prior. Finally, we illustrate these Bayesian reference analyses for exponential power regression models with applications to two datasets. The first application analyzes per capita spending in public schools in the United States. The second application studies the relationship between sold home videos versus profits at the box office.MSC62F15; 62F35; 62J05
Highlights
A flexible way to deal with outliers in linear regression is to assume that the errors follow an exponential power (EP) distribution
In what follows we find that the reference priors for the EP regression model are related to the independence Jeffreys priors given in Equations (5) and (6)
We have developed three reference priors that lead to useful proper posterior distributions
Summary
A flexible way to deal with outliers in linear regression is to assume that the errors follow an exponential power (EP) distribution. Assuming an EP distribution decreases the influence of outliers and, as a result, increases the robustness of the analysis (Box and Tiao 1962; Liang et al 2007; Salazar et al 2012; West 1984). The EP distribution may have tails either lighter (platykurtic) or heavier (leptokurtic) than Gaussian. Platykurtic distributions may be a result of truncation, whereas leptokurtic distributions provide protection against outliers. Salazar et al (2012) have developed three types of Jeffreys priors for linear regression models with independent EP errors. Two of those priors lead to useless improper posterior distributions and only one leads to a proper posterior distribution. We develop explicit expressions for reference priors for all the six possible orderings of the model parameters
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