# Finite-sample Gaussian efficiency: Quantile absolute deviation vs. Rousseeuw-Croux scale estimators

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.

$$\operatorname{SD}$$$$\operatorname{MAD}$$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.

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