On a new class of data depths for measuring representativeness

Abstract

Data depth provides a natural means to rank multivariate vectors with respect to an underlying multivariate distribution. The conventional notion of a depth function emphasizes a centre-outward ordering of data points. While useful for certain statistical applications, such emphasis has rendered most classical data depths insensitive to some distributional features, such as multimodality, of concern to other statistical applications. To get around the problem we introduce a new notion of data depth which seeks to rank data points according to their representativeness, rather than centrality, with respect to an underlying distribution of interest. We propose a general device for defining such depth functions, based essentially on a choice of goodness-of-fit test statistic. Our device calls for a new interpretation of depth more akin to the concept of density than location. It copes particularly well with multivariate data exhibiting multimodality. In addition to providing depth values for individual data points, the new class of depth functions derived from goodness-of-fit tests also extends naturally to provide depth values for subsets of data points, a concept new to the data-depth literature. Applications of the new depth functions are demonstrated with both simulated and real data.