Nonparametric Depth-Based Multivariate Outlier Identifiers, and Robustness Properties

In extending univariate outlier detection methods to higher dimension, various special issues arise, such as limitations of visualization methods, inadequacy of marginal methods, lack of a natural order, limited scope of parametric modeling, and restriction to ellipsoidal contours when using Mahalanobis distance methods. Here we pass beyond these limitations via an approach based on depth functions, which order multidimensional data points by “outlyingness” measures and generate contours following the shape of the data set. This approach to multivariate outlier detection is nonparametric and, with typical choices of depth function, robust. For depth-based outlier identifiers, we define masking and swamping breakdown points, adapting ideas of Davies and Gather (1993) and Becker and Gather (1999) given for parametric outlier identification with normal models. The values of these robustness measures are established for three important depth functions, the spatial, the projection, and a generalized Tukey. Comparison is made with procedures of Maronna and Yohai (1995) and Becker and Gather (1999) using a 5-dimensional data set treated by them. Our three depth-based procedures, and one of Maronna and Yohai (1995) using the StahelDonoho multivariate location and dispersion estimators, are closely competitive and superior to the others, with perhaps the spatial and projection depth procedures having a slight edge. AMS 2000 Subject Classification: Primary 62G10 Secondary 62H99.