Abstract: Estimating the residual variance is an important question in nonparametric regression. Among the existing estimators, the optimal difference-based variance estimation proposed in Hall, Kay, and Titterington(1990) is widely used in practice. Their method is restricted to the situation when the errors are independent and identically distributed. In this paper, we propose the optimal difference-based variance estimation in heteroscedastic nonparametric regression under settings of uncorrelated errors and correlated errors. The proposed estimators are shown to be asymptotically unbiased and their mean squared errors are derived. Simulation studies indicate that the proposed estimators perform better than existing competitors in finite sample settings. In addition, data examples are analyzed to demonstrate the practical usefulness of the proposed method. The proposed method has many applications and we apply it to the nonparametric regression model with repeated measurements and the semiparametric partially linear model.

Key words and phrases: Difference-based estimator, heteroscedasticity, homoscedasticity, nonparametric regression, repeated measurements, variance estimation.