Semiparametric Estimation for Error-Prone Partially Linear Single-Index Models

Partially linear single-index models prove to be flexible in facilitating various types of relationships between the outcome and covariates. However, their validity is hampered by the presence of measurement error in covariates, a feature commonly encountered in applications. In this paper, we explore the use of such models to handle data subject to measurement error in both parametric and nonparametric terms. In addition, with multivariate covariates, often a few of them are informative while most of them are not. In this paper, we propose the three stage procedure to eliminate measurement error effects and select important variables for both the linear predictor term and the single-index part. To implement the proposed method efficiently, we develop a boosting algorithm that enables us to select variables and estimate the parameters without handling non-differentiable penalty functions. Theoretical results, including consistency and asymptotic normality of the estimator, are established to justify the validity of the proposed method. In addition, we examine statistical properties of the boosting algorithm, including convergence and validity of variable selection. Numerical studies, including simulation and data analysis, are conducted to assess the finite sample performance of the proposed method.