Approximating symmetrized estimators of scatter via balanced incomplete U-statistics

We derive limiting distributions of symmetrized estimators of scatter. Instead of considering all $$n(n-1)/2$$ n ( n - 1 ) / 2 pairs of the n observations, we only use nd suitably chosen pairs, where $$d \ge 1$$ d ≥ 1 is substantially smaller than n. It turns out that the resulting estimators are asymptotically equivalent to the original one whenever $$d = d(n) \rightarrow \infty$$ 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.