Abstract
We observe a sample of independent and identically distributed random elements of a Polish space, and measure the cost of representing one point by another with a nonnegative kernel on pairs of points. The polarization measure of the sample is the minimal average cost of representing it by a single consensus point, and we study its large-sample behaviour. Using epi-convergence, we prove strong consistency of the measure, under an inf-compactness condition on the kernel. Using empirical process theory, we derive the limiting distribution of the centred and rescaled measure, namely the infimum of a centred Gaussian process indexed by the set of population minimizers, and we characterize when this limit is Gaussian. Finally, we show that the nonparametric bootstrap is consistent when the population minimizer is essentially unique and inconsistent otherwise, and we provide a delta method based on an estimated directional derivative to address the inconsistency.
Raffaello Seri
Professor of Econometrics
My research interests include statistics, numerical analysis, operations research, psychology, economics and management.