New estimation for heteroscedastic single-index measurement error models

Abstract

In this paper, we propose a novel estimation for heteroscedastic single-index models with covariate measurement errors, in which both the conditional mean and conditional variance functions of the response given the covariates have a single-index structure. When covariates are directly observable, we show that the index parameter vector can be estimated consistently by the estimation obtained by fitting a misspecified linear quantile regression model under some mild regularity conditions. It is well known that naively treating mismeasured covariates as error-free usually leads to inconsistent estimators. To account for measurement errors in covariates, we establish a new estimation procedure based on corrected quantile loss function, and obtain the asymptotic consistency and normality of the resulting estimators. Finally, the finite sample performance of the proposed estimation method is illustrated by simulation studies and an empirical analysis of a real dataset.

Keywords:

• Corrected quantile loss function
• heteroscedasticity
• linearity condition
• measurement error
• single-index model

2020 Mathematics Subject Classification:

• 62G05

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Ding's research was partially supported by the National Natural Science Foundation of China [grant number 11901286]. Zhang's research was partially supported by the National Natural Science Foundation of China [grant number 11971171], 111 Project [grant number B14019], and Open Research Fund of Key Laboratory of Advanced Theory and Application in Statistics and Data Science-MOE, ECNU.