Unbiased median absolute deviation based on the trimmed Harrell-Davis quantile estimator

The median absolute deviation ($\operatorname{MAD}$) is a robust measure of scale. For a sample $x = \{ x_1, x_2, \ldots, x_n \}$, it’s defined as follows:

where $\operatorname{median}$ is a median estimator, $C_n$ is a scale factor. Using the right scale factor, we can use $\operatorname{MAD}$ as a consistent estimator for the estimation of the standard deviation under the normal distribution. For huge samples, we can use the asymptotic value of $C_n$ which is

For small samples, we should use adjusted values $C_n$ which depend on the sample size. However, $C_n$ depends not only on the sample size but also on the median estimator. I have already covered how to obtain this values for the traditional median estimator and the Harrell-Davis median estimator. It’s time to get the $C_n$ values for the trimmed Harrell-Davis median estimator.

Simulation

In order to obtain the values of $C_n$, we perform the following simulation:

• Enumerate $n$ from $2$ to $100$ and some other values up to $2000$.
• For each $n$, generate $2\cdot 10^8$ samples from the normal distribution of size $n$ (the Box–Muller transform is used, see [Box1958])
• For each sample, estimate $\textrm{MAD}_n$ using $C_n = 1$ and the trimmed Harrell-Davis quantile estimator based on the highest density interval of the width $1/sqrt(n)$ (aka THD-SQRT).
• Calculate the arithmetic average of all $\textrm{MAD}_n$ values for the same $n$: the result is the value of $\hat{a}_n$
• The reverse value is the estimation of $C_n = 1/\hat{a}_n$.

Warning: on my PC, the simulation takes several days to finish.

Results for small n

Here are the results for $n \leq 100$:

Results for huge n

I have also calculated $C_n$ for some bigger sample sizes:

Following the approach from park2020, we express $C_n$ using the following equation:

With the help of linear regression, we get the following values of $\alpha$ and $beta$:

Here is a plot with both actual and predicted values of $C_n$ for different sample sizes:

References

• [Box1958]
  Box, George EP. “A note on the generation of random normal deviates.” Ann. Math. Statist. 29 (1958): 610-611.
  https://doi.org/10.1214/aoms/1177706645
• [Park2020]
  Park, Chanseok, Haewon Kim, and Min Wang. “Investigation of finite-sample properties of robust location and scale estimators.” Communications in Statistics-Simulation and Computation (2020): 1-27.
  https://doi.org/10.1080/03610918.2019.1699114