In this post, we discuss the finite-sample Gaussian efficiency of various robust dispersion estimators. The classic standard deviation has the highest possible Gaussian efficiency of \(100\%\), but it is not robust: a single outlier can completely destroy the estimation. A typical robust alternative to the standard deviation is the Median Absolute Deviation (\(\operatorname{MAD}\)). While the \(\operatorname{MAD}\) is highly robust (the breakdown point is \(50\%\)), it is not efficient: its asymptotic Gaussian efficiency is only \(37\%\). Common alternative to the \(\operatorname{MAD}\) is the Rousseeuw-Croux \(S_n\) and \(Q_n\) scale estimators that provide higher efficiency, keeping the breakdown point of \(50\%\). In one of my recent preprints, I introduced the concept of the Quantile Absolute Deviation (\(\operatorname{QAD}\)) and its specific cases: the Standard Quantile Absolute Deviation (\(\operatorname{SQAD}\)) and the Optimal Quantile Absolute Deviation (\(\operatorname{OQAD}\)). Let us review the finite-sample and asymptotic values of the Gaussian efficiency for these estimators.

We start with reviewing the asymptotic Gaussian efficiency values:

\(\operatorname{SD}\) | \(\operatorname{MAD}\) | RC \(S_n\) | RC \(Q_n\) | \(\operatorname{SQAD}\) | \(\operatorname{OQAD}\) | |
---|---|---|---|---|---|---|

Gaussian efficiency | \(100\%\) | \(37\%\) | \(58\%\) | \(82\%\) | \(54\%\) | \(65\%\) |

Breakdown point | \(0\%\) | \(50\%\) | \(50\%\) | \(50\%\) | \(32\%\) | \(14\%\) |

As we can see, \(Q_n\) looks like the best estimator: it has the efficiency of \(82\%\) while its breakdown is \(50\%\). In the asymptotic case, \(\operatorname{SQAD}\) and \(\operatorname{OQAD}\) do not look interesting: they are less efficient and less robust.

Now, let us check the finite-sample Gaussian efficiency values:

From this picture, we can see that \(S_n\) and \(Q_n\) are not so efficient in the case of a small sample. Meanwhile, \(\operatorname{OQAD}\) is the most efficient estimator in this context for \(n \leq 20\).

When we choose an estimator, it is important to check out not only its asymptotic properties, but also finite-sample properties that are evaluated for the target sample size.

### References

- Andrey Akinshin (2022) “Quantile absolute deviation” arXiv:2208.13459
- Andrey Akinshin (2022) “Finite-sample Rousseeuw-Croux scale estimators” arXiv:2209.12268