Notes / Shamos Estimator

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.

References (3)