Abstract

Let Xn = fx 1 ; : : : ; xn g, be an i.i.d. sample having multivariate distribution P . We derive a.s. limits for the power weighted edge weight function of greedy approximations to a class of minimal graphs spanning k of the n samples. The class includes minimal k-point graphs constructed by the partitioning method of Ravi, Sundaram, Marathe, Rosenkrantz and Ravi [43] where the edge weight function satises the quasi-additive property of Redmond and Yukich [45]. In particular this includes greedy approximations to the k-point minimal spanning tree (k-MST), Steiner tree (k-ST), and the traveling salesman problem (k-TSP). An expression for the inuence function of the minimal weight function is given which characterizes the asymptotic sensitivity of the graph weight to perturbations in the underlying distribution. The inuence function takes a form which indicates that the k-point minimal graph in d > 1 dimensions has robustness properties in IR d which are analogous to those of rank order statistics in one dimension. A direct result of our theory is that the log-weight of the k-point minimal graph is a consistent nonparametric estimate of the Renyi entropy of the distribution P . Possible applications of this work include: analysis of random communication network topologies, estimation of the mixing coecient in -contaminated mixture models, outlier discrimination and rejection, clustering and pattern recognition, robust non-parametric regression, two sample matching and image registration.

Citations