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.

我们推导了散点的对称估计量的极限分布。我们不考虑n个观测值的所有\(n(n-1)/2\)对，而是仅使用nd 个适当选择的对，其中\(d \ge 1\)远小于n。事实证明，只要\(d = d(n) \rightarrow \infty\)以任意慢的速度，所得的估计量就渐近等价于原始估计量。我们还研究了任意固定d的渐近性质。这些考虑因素和数值示例表明，出于实际目的，d的适度固定值10 到 20 之间已经产生了估计器，这些估计器在计算上是可行的并且相当接近原始估计器。