Finite-sample Rousseeuw-Croux scale estimators

The paper is based on a series of my research notes:

Abstract

The Rousseeuw-Croux Sn, Qn scale estimators and the median absolute deviation 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, Sn and Qn are much more efficient than MAD_n: their asymptotic Gaussian efficiency values are 58% and 82% respectively compared to 37% for MAD_n. Although these values look impressive, they are only asymptotic values. The actual Gaussian efficiency of Sn and Qn for small sample sizes is noticeably 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 Sn,Qn 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 (n100) and prediction equations for samples of larger sizes.

Reference

Andrey Akinshin “Finite-sample Rousseeuw-Croux scale estimators” (2022) arXiv:2209.12268

@Article{akinshin2022frc,
  title = {Finite-sample Rousseeuw-Croux scale estimators},
  author = {Akinshin, Andrey},
  year = {2022},
  month = {9},
  arxiv = {2209.12268},
  abstract = {The Rousseeuw-Croux $S_n$, $Q_n$ scale estimators and the median absolute deviation 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 MAD_n: their asymptotic Gaussian efficiency values are $58\%$ and $82\%$ respectively compared to $37\%$ for 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 noticeably 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.}
}