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 \(d \ge 1\) is substantially smaller than n. It turns out that the resulting estimators are asymptotically equivalent to the original one whenever \(d = d(n) \rightarrow \infty\) 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.