Finite-sample Rousseeuw-Croux scale estimators

Abstract: 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.