Asymptotic normality of parametric part in semiparametric regression in the presence of measurement error

In this study, we investigate a partially linear regression model where the covariate entering the nonparametric component is measured with error. A key challenge in such models is that the measurement error distribution is unknown, and this setting is further complicated by the limited availability of data; specifically, only one repeated observation of the mismeasured regressor is accessible. To address this, we propose an estimation procedure that leverages the Fourier transform, a powerful analytical tool that transforms complex convolution equations into tractable algebraic forms. By modifying an existing approach rooted in Fourier analysis, we construct a novel estimator that accommodates the unknown error distribution and efficiently utilizes the repeated measurement. We establish the asymptotic normality of the proposed estimator, demonstrating its theoretical validity and robustness. To assess its practical performance, we conduct a series of Monte Carlo simulations under various scenarios. These simulations provide strong evidence of the estimator's effectiveness in finite samples, particularly in terms of bias reduction and variance control. The methodology offers a flexible and computationally feasible framework for dealing with measurement error in semiparametric models without requiring knowledge of the error distribution. This contributes to the growing literature on measurement error models by extending nonparametric estimation techniques to more realistic and constrained data settings.