# Finite-sample bias-correction factors for the median absolute deviation based on the Harrell-Davis quantile estimator and its trimmed modification

The paper is based on a series of my research notes:

- Unbiased median absolute deviation
*(2021-02-09)* - Unbiased median absolute deviation based on the Harrell-Davis quantile estimator
*(2021-02-16)* - Unbiased median absolute deviation based on the trimmed Harrell-Davis quantile estimator
*(2022-02-08)* - Unbiased median absolute deviation for n=2
*(2022-04-26)* - Preprint announcement: 'Finite-sample bias-correction factors for the median absolute deviation based on the Harrell-Davis quantile estimator and its trimmed modification'
*(2022-07-26)*

## Abstract

The median absolute deviation is a widely used robust measure of statistical dispersion. Using a scale constant, we can use it as an asymptotically consistent estimator for the standard deviation under normality. For finite samples, the scale constant should be corrected in order to obtain an unbiased estimator. The bias-correction factor depends on the sample size and the median estimator. When we use the traditional sample median, the factor values are well known, but this approach does not provide optimal statistical efficiency. In this paper, we present the bias-correction factors for the median absolute deviation based on the Harrell-Davis quantile estimator and its trimmed modification which allow us to achieve better statistical efficiency of the standard deviation estimations. The obtained estimators are especially useful for samples with a small number of elements.

## Reference

Andrey Akinshin

“Finite-sample bias-correction factors for the median absolute deviation based on the Harrell-Davis quantile estimator and its trimmed modification”(2022) arXiv:2207.12005

```
@Article{akinshin2022madfactors,
title = {Finite-sample bias-correction factors for the median absolute deviation based on the Harrell-Davis quantile estimator and its trimmed modification},
author = {Akinshin, Andrey},
year = {2022},
month = {7},
arxiv = {2207.12005},
abstract = {The median absolute deviation is a widely used robust measure of statistical dispersion. Using a scale constant, we can use it as an asymptotically consistent estimator for the standard deviation under normality. For finite samples, the scale constant should be corrected in order to obtain an unbiased estimator. The bias-correction factor depends on the sample size and the median estimator. When we use the traditional sample median, the factor values are well known, but this approach does not provide optimal statistical efficiency. In this paper, we present the bias-correction factors for the median absolute deviation based on the Harrell-Davis quantile estimator and its trimmed modification which allow us to achieve better statistical efficiency of the standard deviation estimations. The obtained estimators are especially useful for samples with a small number of elements.}
}
```