Weighted likelihood estimation of multivariate location and scatter

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 univariate distribution of the Mahalanobis distances rather than the multivariate distribution of the data at the assumed model. This strategy allows to avoid the curse of dimensionality affecting multivariate non-parametric density estimation, that is involved in the construction of the weights through the Pearson residuals. Asymptotic properties of the proposed weighted likelihood estimator are also discussed. 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.