A New Class of Estimators Based on a General Relative Loss Function

Tao Hu,Baosheng Liang

doi:10.3390/math9101138

Abstract

Motivated by the relative loss estimator of the median, we propose a new class of estimators for linear quantile models using a general relative loss function defined by the Box–Cox transformation function. The proposed method is very flexible. It includes a traditional quantile regression and median regression under the relative loss as special cases. Compared to the traditional linear quantile estimator, the proposed estimator has smaller variance and hence is more efficient in making statistical inferences. We show that, in theory, the proposed estimator is consistent and asymptotically normal under appropriate conditions. Extensive simulation studies were conducted, demonstrating good performance of the proposed method. An application of the proposed method in a prostate cancer study is provided.

Highlights

In contrast to the mean-based regression that mainly gives an overall quantification for the central covariate effect, quantile regression can directly model a series of quantiles of the response variable to deliver a global evaluation of the covariate effect [1,2]
A traditional quantile regression is typically based on minimizing a check loss function [1], but often the relative quantile loss could be more relevant than the check loss function and might be used to gain more efficiency for inference
We propose a general class of relative loss functions via a Box–Cox transformation [14,15]

Summary

Introduction

In contrast to the mean-based regression that mainly gives an overall quantification for the central covariate effect, quantile regression can directly model a series of quantiles (from lower to higher) of the response variable to deliver a global evaluation of the covariate effect [1,2]. Khoshgoftaar et al (1992) studied the asymptotic properties of the estimators by minimizing both the squared relative loss and the absolute relative loss under nonlinear regression models [11], and made a great comparative study on them. Yi∗ = XiT β + ε∗i , and proposed to estimate the model parameters by minimizing the sum of the absolute relative error n. Their works only consider the absolute relative loss, not related to any quantile estimates of model distribution None of these studies discussed the way to apply the relative error for a general quantile regression. We apply the proposed loss function for a linear quantile regression [1] and prove that the estimates of the regression coefficients are consistent and asymptotically normal.

Model and Methods

Adaptive Criteria for Choosing γ

Simulation Study

Application

Findings

Discussion