Suggested in shamos1976 (page 260), a robust measure of scale/spread.

For a sample $\mathbf{x} = \{ x_1, x_2, \ldots, x_n \}$, it is defined as follows:

$$ \operatorname{Shamos}_n = C_n \cdot \underset{i < j}{\operatorname{median}} (|x_i - x_j|), $$where $\operatorname{median}$ is a median estimator, $C_n$ is a scale factor, which is usually used to make the estimator consistent for the standard deviation under the normal distribution. The asymptotic consistency factor: $C_\infty \approx 1.048358$. The asymptotic Gaussian efficiency is of $\approx 86\%$; the asymptotic breakdown point is of $\approx 29\%$. The finite-sample consistency factor and efficiency values can be found in park2020.

In rousseeuw1993, it is claimed that the Rousseeuw-Croux estimator is a good alternative with much higher breakdown point of $50\%$ and slightly decorated statistical efficiency (the asymptotic value is of $\approx 82%$). However, for small samples the efficiency gap is huge, so I prefer the Shamos estimator.

Posts (2) Papers (3)

## Posts (2)

- Finite-sample Gaussian efficiency: Shamos vs. Rousseeuw-Croux Qn scale estimators (2023-12-19) 2
- Median absolute deviation vs. Shamos estimator (2022-02-01) 2

## Library / Papers (3)

- Alternatives to the Median Absolute Deviation (1993) by Peter J Rousseeuw et al. 3
- Geometry and Statistics (1976) by Michael Ian Shamos 1
- Investigation of finite-sample Properties of Robust Location and Scale Estimators (2020) by Chanseok Park et al. 6

### Backlinks (5)

- Effect Sizes and Asymmetry (2024-03-12) 5
- Median absolute deviation vs. Shamos estimator (2022-02-01) 2
- Customization of the nonparametric Cohen's d-consistent effect size (2021-06-08) 17 4
- Median vs. Hodges-Lehmann: compare efficiency under heavy-tailedness (2023-11-14) 2 1
- Thoughts about robustness and efficiency (2023-11-07) 2 1