Approximating symmetrized estimators of scatter via balanced incomplete U-statistics

Abstract

We derive limiting distributions of symmetrized estimators of scatter. Instead of considering all n(n−1)/2 pairs of the n observations, we only use nd suitably chosen pairs, where d≥1 is substantially smaller than n. It turns out that the resulting estimators are asymptotically equivalent to the original one whenever d=d(n)→∞ at arbitrarily slow speed. We also investigate the asymptotic properties for arbitrary fixed d. These considerations and numerical examples indicate that for practical purposes, moderate fixed values of d between 10 and 20 yield already estimators which are computationally feasible and rather close to the original ones.