Estimating the conditional single-index error distribution with a partial linear mean regression

In this paper, we present a method for estimating the conditional distribution function of the model error. Given the covariates, the conditional mean function is modeled as a partial linear model, and the conditional distribution function of model error is modeled as a single-index model. To estimate the single-index parameter, we propose a semi-parametric global weighted least-squares estimator coupled with an indicator function of the residuals. We derive a residual-based kernel estimator to estimate the unknown conditional distribution function. Asymptotic distributions of the proposed estimators are derived, and the residual-based kernel process constructed by the estimator of the conditional distribution function is shown to converge to a Gaussian process. Simulation studies are conducted and a real dataset is analyzed to demonstrate the performance of the proposed estimators.