Approximating Symmetrized Estimators of Scatter via Balanced Incomplete U-Statistics

We derive limiting distributions of symmetrized estimators of scatter, where instead of all $n(n-1)\mathrm{/}2$ pairs of the $n$ observations we only consider $nd$ suitably chosen pairs, $1\le d<\lfloor n\mathrm{/}2\rfloor$. It turns out that the resulting estimators are asymptotically equivalent to the original one whenever $d=d(n)\to \mathrm{\infty }$ 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,say, $10$ and $20$ yield already estimators which are computationally feasible and rather close to the original ones.