Scale estimation and data-driven tuning constant selection for M-quantile regression

TL;DR

This paper evaluates existing scale estimators and proposes new data-driven methods for tuning constant selection in M-quantile regression, aiming to improve estimator efficiency and robustness.

Contribution

It introduces a new scale estimator based on the method of moments and two data-driven tuning constant selection approaches for M-quantile regression.

Findings

Data-driven tuning constants can improve estimator efficiency.

The proposed scale estimator performs well in simulations.

Methods are demonstrated on EU income data.

Abstract

M-quantile regression is a general form of quantile-like regression which usually utilises the Huber influence function and corresponding tuning constant. Estimation requires a nuisance scale parameter to ensure the M-quantile estimates are scale invariant, with several scale estimators having previously been proposed. In this paper we assess these scale estimators and evaluate their suitability, as well as proposing a new scale estimator based on the method of moments. Further, we present two approaches for estimating data-driven tuning constant selection for M-quantile regression. The tuning constants are obtained by i) minimising the estimated asymptotic variance of the regression parameters and ii) utilising an inverse M-quantile function to reduce the effect of outlying observations. We investigate whether data-driven tuning constants, as opposed to the usual fixed constant, for instance, at c=1.345, can improve the efficiency of the estimators of M-quantile regression parameters. The performance of the data-driven tuning constant is investigated in different scenarios using model-based simulations. Finally, we illustrate the proposed methods using a European Union Statistics on Income and Living Conditions data set.