# Median absolute deviation vs. Shamos estimator

There are multiple ways to estimate statistical dispersion.
The standard deviation is the most popular one, but it’s not robust:
a single outlier could heavily corrupt the results.
Fortunately, we have robust measures of dispersions like the *median absolute deviation* and the *Shamos estimator*.
In this post, we perform numerical simulations and
compare these two estimators on different distributions and sample sizes.

## Definitions

For a sample $x = \{ x_1, x_2, \ldots, x_n \}$,
the *median absolute deviation* ($\operatorname{MAD}$) and
the Shamos Estimator are defined as follows:

where $\operatorname{median}$ is a median estimator, $C_n$ is a scale factor. In the scope of this post, we use the traditional sample median (if $n$ is odd, the median is the middle element of the sorted sample, if $n$ is even, the median is the arithmetic average of the two middle elements of the sorted sample). The $C_n$ scale factors allow using $\operatorname{MAD}$ and $\operatorname{Shamos}$ consistent estimators for the standard deviation under the normal distribution. The corresponding values of $C_n$ for both dispersion estimators could be found in park2020.

## Simulation study

Let’s perform the following experiment:

- Enumerate different distributions: the standard Normal distribution (light-tailed), the standard Gumbel distribution (light-tailed), the standard Cauchy distribution (heavy-tailed), the Frechet distribution with shape = 1 (heavy-tailed), and the Weibull distribution with shape = 0.5 (heavy-tailed.)
- Enumerate different sample sizes: 5, 10, 20.
- For each combination of the parameters, we generate $10\,000$ random samples from the given distribution of the given size, and calculate the $\operatorname{MAD}$ and $\operatorname{Shamos}$ estimations. For each group of estimations, we draw a density plot (the Sheather & Jones method, the normal kernel) and calculate some summary statistics: the mean, the median, the standard deviation (SD), the interquartile range (IQR), the $99^\textrm{th}$ percentile (P99).

Let’s start with the Normal distribution:

As we can see from the plots, $\operatorname{Shamos}$ has higher statistical efficiency than $\operatorname{MAD}$. Also, thanks to the $C_n$ scale factors from park2020, the expected value of both estimators is $1$, which makes them a robust replacement for the unbiased standard deviation.

Now let’s look at the results for the light-tailed Gumbel distribution:

$\operatorname{Shamos}$ still looks better than $\operatorname{MAD}$ because its density plot is more narrow.

Next, let’s consider the heavy-tailed Cauchy distribution (which has infinity variance):

Here we can see that $\operatorname{MAD}$ shows better robustness than $\operatorname{Shamos}$ (because it has a higher breakdown point). Similar results could be observed for heavy-tailed Frechet and Weibull distributions:

## Conclusion

Under normality, $\operatorname{Shamos}$ has better statistical efficiency than $\operatorname{MAD}$ if we consider these estimators as consistent estimators for the standard deviation. On other light-tailed distributions, $\operatorname{Shamos}$ also has a smaller dispersion than $\operatorname{MAD}$.

However, in the case of heavy-tailed distributions, $\operatorname{MAD}$ is the preferable option because it has a higher breakdown point and better resistance to outliers than $\operatorname{Shamos}$. Since we typically use robust measures of scales when we expect to have some extreme outliers, $\operatorname{MAD}$ looks like a more reasonable measure of dispersion than $\operatorname{Shamos}$.