A New Notion of Data Depth Based on Goodness-of-Fit Tests

Abstract

Data depth provides a natural means to rank multivariate vectors with respect to an underlying multivariate distribution. Most existing depth functions emphasize a center-outward ordering of data points, which may fail to capture other important distributional features such as multimodality of concern to certain statistical applications. Such inadequacy motivates us to develop a novel notion of data depth with emphasis on 'representativeness' rather than 'centrality.' Derived essentially from a choice of goodness-of-fit test statistic, the new notion calls for a new interpretation of 'depth' more akin to the concept of density instead of location. It copes well with multivariate data exhibiting multimodality in particular. The new definition also extends naturally to a depth measure for patterns of points as well as for singletons.