Approximating Symmetrized Estimators of Scatter via Balanced Incomplete U-Statistics

TL;DR

This paper analyzes the asymptotic behavior of symmetrized scatter estimators using balanced incomplete U-statistics, showing they are effective and computationally feasible with moderate fixed pair selections.

Contribution

It introduces a method to approximate symmetrized scatter estimators using balanced incomplete U-statistics, providing theoretical and numerical insights into their asymptotic properties.

Findings

Estimators are asymptotically equivalent when the number of pairs grows slowly.

Moderate fixed values of d (10-20) yield practical and close estimators.

Numerical examples support the effectiveness of the proposed approximation.

Abstract

We derive limiting distributions of symmetrized estimators of scatter, where instead of all $n(n-1)/2$ pairs of the $n$ observations we only consider $nd$ suitably chosen pairs, $1 \le d < \lfloor n/2\rfloor$. It turns out that the resulting estimators are asymptotically equivalent to the original one whenever $d = d(n) \to \infty$ 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.