# Investigation of finite-sample properties of robust location and scale estimators

## TL;DR

This paper studies the finite-sample properties of robust location and scale estimators, providing closed-form breakdown points, unbiasing factors, and efficiencies through simulations up to size 100, and offers predictive formulas for larger samples.

## Contribution

It fills a gap by analyzing finite-sample behavior of robust estimators, offering explicit formulas and simulation results for practical sample sizes.

## Key findings

- Finite-sample breakdown points are derived in closed form.
- Unbiasing factors significantly improve estimator performance.
- Predictive formulas for unbiasing factors for samples over 100 are provided.

## Abstract

When the experimental data set is contaminated, we usually employ robust alternatives to common location and scale estimators such as the sample median and Hodges-Lehmann estimators for location and the sample median absolute deviation and Shamos estimators for scale. It is well known that these estimators have high positive asymptotic breakdown points and are Fisher-consistent as the sample size tends to infinity. To the best of our knowledge, the finite-sample properties of these estimators, depending on the sample size, have not well been studied in the literature. In this paper, we fill this gap by providing their closed-form finite-sample breakdown points and calculating the unbiasing factors and relative efficiencies of the robust estimators through the extensive Monte Carlo simulations up to the sample size 100. The numerical study shows that the unbiasing factor improves the finite-sample performance significantly. In addition, we provide the predicted values for the unbiasing factors obtained by using the least squares method which can be used for the case of sample size more than 100.