The sensitivity of posterior inferences to model specification can be considered as an indicator of the presence of outliers, that are to be considered as highly unlikely values under the assumed model. The occurrence of anomalous values can seriously alter the shape of the likelihood function and lead to posterior distributions far from those one would obtain without these data inadequacies. In order to deal with these hindrances, a robust approach is discussed, which allows us to obtain outliers’ resistant posterior distributions with properties similar to those of a proper posterior distribution. The methodology is based on the replacement of the genuine likelihood by a weighted likelihood function in the Bayes’ formula
A novel approach to obtain weighted likelihood estimates of multivariate location and scatter is discussed. A weighting scheme is proposed that is based on the distribution of the Mahalanobis distances rather than the distribution of the data at the assumed model. This strategy allows to avoid the curse of dimensionality affecting nonparametric density estimation, that is involved in the construction of the weights through the Pearson residuals ]. Then, weighted likelihood based outlier detection rules and robust dimensionality reduction techniques are developed. The effectiveness of the methodology is illustrated through some numerical studies and real data examples.
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