A Bayesian-Frequentists approach for detecting outliers in a one-way variance components model

The most common Bayesian approach for detecting outliers is to assume that outliers are observations which have been generated by contaminating models. An alternative idea was applied by Zellner (1975) and Chaloner (1994). They studied the properties of realized regression error terms. Posterior distributions for individual realized errors, and for linear and quadratic combinations of them, were derived. In this note, the theory and results derived by Chaloner (1994) are extended. Since it is not clear to us what the frequentist properties of the Bayesian procedures of Chaloner and Zellner are (i.e., what the size of the Type I error is or the power of their tests are), a Bayesian-Frequentist approach is used for detecting outliers in a one-way random effects model. For illustration purposes, the Sharples (1990) contaminated data are used as our first example. It is concluded that the Bayesian-Frequentist approach seems to be more conservative than Chaloner's method. In the second example, the Bayesian-Frequentist method is applied to O'Hagan's (2003) artificial dataset and compared with the partial posterior predictive measures derived by Bayarri and Castellanos (2007).