论文标题
有限样本的Rousseeuw-Croux量表估计器
Finite-sample Rousseeuw-Croux scale estimators
论文作者
论文摘要
Rousseeuw-croux $ s_n $，$ q_n $比例估计器和中值绝对偏差$ \ permatatorName {mad} _n $可以用作正常性下标准偏差的一致估计器。所有这些都非常健壮：所有三个估计器的分解点均为$ 50 \％$。但是，$ s_n $和$ q_n $比\ $ \ operatatorName {mad} _n $要高得多：它们的渐近高斯效率值分别为$ 58 \％$ $和$ 82 \％$，而$ 37 \％\％\％\％\％$ for \ $ \ $ \ $ \ $ \ operatateRonname {mad} _n $。尽管这些值看起来令人印象深刻，但它们只是渐近值。小样本量的实际高斯效率为$ s_n $和$ q_n $，明显低于渐近案例。 Rousseeuw and Croux（1993）的原始作品仅提供了$ s_n $，$ q_n $的有限样本偏差校正因子的粗略近似，并提供了有关其有限样本效率值的简短说明。在本文中，我们进行了广泛的蒙特卡罗模拟，以获得Rousseeuw-Croux量表估计器的有限样本特性的精制值。我们提出了偏置校正因子的准确值和小样本（$ n \ leq 100 $）的高斯效率和较大尺寸样品的预测方程。
The Rousseeuw-Croux $S_n$, $Q_n$ scale estimators and the median absolute deviation $\operatorname{MAD}_n$ can be used as consistent estimators for the standard deviation under normality. All of them are highly robust: the breakdown point of all three estimators is $50\%$. However, $S_n$ and $Q_n$ are much more efficient than\ $\operatorname{MAD}_n$: their asymptotic Gaussian efficiency values are $58\%$ and $82\%$ respectively compared to $37\%$ for\ $\operatorname{MAD}_n$. Although these values look impressive, they are only asymptotic values. The actual Gaussian efficiency of $S_n$ and $Q_n$ for small sample sizes is noticeable lower than in the asymptotic case. The original work by Rousseeuw and Croux (1993) provides only rough approximations of the finite-sample bias-correction factors for $S_n$, $Q_n$ and brief notes on their finite-sample efficiency values. In this paper, we perform extensive Monte-Carlo simulations in order to obtain refined values of the finite-sample properties of the Rousseeuw-Croux scale estimators. We present accurate values of the bias-correction factors and Gaussian efficiency for small samples ($n \leq 100$) and prediction equations for samples of larger sizes.
