Recently, I published a preprint of a paper ‘Finite-sample Rousseeuw-Croux scale estimators’. It’s based on a series of my research notes.

The paper preprint is available on arXiv: arXiv:2209.12268 [stat.ME]. The paper source code is available on GitHub: AndreyAkinshin/paper-frc. You can cite it as follows:

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

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 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.

### Relevant blog posts

Here is the full list of the relevant blog posts:

- Finite-sample efficiency of the Rousseeuw-Croux estimators
*(August 9, 2022)* - Finite-sample bias correction factors for Rousseeuw-Croux scale estimators
*(September 6, 2022)* - Preprint announcement: 'Finite-sample Rousseeuw-Croux scale estimators'
*(October 18, 2022)*

### BibTeX reference

```
@article{akinshin2022frc,
title = {Finite-sample Rousseeuw-Croux scale estimators},
author = {Akinshin, Andrey},
year = {2022},
month = {9},
publisher = {arXiv},
doi = {10.48550/arXiv.2209.12268},
url = {https://arxiv.org/abs/2209.12268}
}
```