On the combination of depth-based ranks

Abstract

The depth of a multivariate observation assesses its degree of centrality with respect to a probability distribution, and thus it can be interpreted as a measurement of the fit of the observation wrt the distribution. If such depth is transformed into a (depth-based) rank, then we obtain a kind of p-value of a goodness-of-fit test run on a single observation. For a sample of observations, the goal is to combine their ranks in order to decide whether they were taken from some prescribed distribution. From the meta-analysis literature, it is well known that there does not exist a combination procedure for such p-values (or ranks) that outperforms the remaining ones in all possible scenarios. Here we explore several combination procedures of the depth-based ranks and analyse their behaviour in the detection of some given shifts from a prescribed distribution.