Simultaneous Inference for Centre-Outward Rankings of Random Objects

Abstract: In many applications one object is chosen to represent a sample and the least typical objects are set aside, usually without any measure of uncertainty. We measure how central an object is by its reference depth, the probability that a random reference point is closer to it than to a random opponent. Using only pairwise dissimilarities, we construct simultaneous intervals for all ranks, a centre set that contains the most central observation under stated conditions, and outlier sets that contain only objects among the least central ones, each with asymptotic probability at least a prescribed level. These guarantees hold even though the ranked objects are themselves random. For symmetric distributions on the real line, the centre set has the smallest possible order of size up to a logarithmic factor, and in dimension three or more it reduces to the most central observation with probability tending to one. For typical meteorological years the centre set usually contains several candidate years, so the data do not determine the year on which building-energy simulations rest; for three collections of gene trees it contains one tree, whose features can be reported as those of the most central tree.