Depth Functions, Multidimensional Medians and Tests of Uniformity on Proximity Graphs Open Access
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We represent the d-dimensional random vectors as vertices of a complete weighted graph and propose depth functions that are applicable to distributions in d-dimensional spaces and data on graphs. We explore the proximity graphs, examine their connection to existing depth functions, define a family of depth functions on the Beta-skeleton graph, suggest four types of depth functions on the minimal spanning tree (MST) and define depth functions including path depth, path depth of path length at most Delta, all paths probability depth, eccentricity depth, peeling depth and RUNT depth. We study their properties, including affine invariance, maximality at the center, monotonicity and vanishing at infinity. We show that the Beta-skeleton depth is a family of statistical depth functions and define the sample Beta-skeleton depth function. We show that it has desirable asymptotic properties, including uniform consistency and asymptotic normality. We consider the corresponding multidimensional medians, investigate their robustness, computational complexity, compare them in a simulation study to find the multivariate medians under different distributions and sample sizes and explore the asymptotic properties of $beta$-skeleton median. We generalize the univariate Greenwood statistic and Hegazy-Green statistic using depth induced order statistics and propose two new test statistics based on normal copula and interpoint distances for testing multivariate uniformity. We generalize the path depth under two-sample setting and propose a new multivariate equality of DF test. We study their empirical power against several copula and multivariate Beta alternatives. The topic is complemented with a discussion on the distribution and moments of the interpoint distances (ID) between bivariate uniform random vectors and the IDs between FGM copula random vectors.