Robust multivariate estimation based on statistical depth filters

In the classical contamination models, such as the gross-error (Huber and Tukey contamination model or case-wise contamination), observations are considered as the units to be identified as outliers or not. This model is very useful when the number of considered variables is moderately small. Alqallaf et al. (Ann Stat 37(1):311–331, 2009) show the limits of this approach for a larger number of variables and introduced the independent contamination model (cell-wise contamination) where now the cells are the units to be identified as outliers or not. One approach to deal, at the same time, with both type of contamination is filter out the contaminated cells from the data set and then apply a robust procedure able to handle case-wise outliers and missing values. Here, we develop a general framework to build filters in any dimension based on statistical data depth functions. We show that previous approaches, e.g., Agostinelli et al. (TEST 24(3):441–461, 2015b) and Leung et al. (Comput Stat Data Anal 111:59–76, 2017), are special cases. We illustrate our method by using the half-space depth.