SIMEX estimation for single-index model with covariate measurement error

TL;DR

This paper introduces a SIMEX estimation method for single-index models with measurement error in covariates, which does not require distributional assumptions and is validated through simulations and real data analysis.

Contribution

The paper develops a novel SIMEX algorithm for single-index models with measurement error, avoiding distributional assumptions and providing asymptotic properties.

Findings

Estimator is asymptotically normal under regularity conditions.

Method effectively reduces bias in the presence of measurement error.

Simulation studies confirm the estimator's good performance.

Abstract

In this paper, we consider the single-index measurement error model with mismeasured covariates in the nonparametric part. To solve the problem, we develop a simulation-extrapolation (SIMEX) algorithm based on the local linear smoother and the estimating equation. For the proposed SIMEX estimation, it is not needed to assume the distribution of the unobserved covariate. We transform the boundary of a unit ball in $\mathbb{R}^p$ to the interior of a unit ball in $\mathbb{R}^{p-1}$ by using the constraint $\|\beta\|=1$. The proposed SIMEX estimator of the index parameter is shown to be asymptotically normal under some regularity conditions. We also derive the asymptotic bias and variance of the estimator of the unknown link function. Finally, the performance of the proposed method is examined by simulation studies and is illustrated by a real data example.