Optimal difference-based estimation for partially linear models
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Difference-based methods have attracted increasing attention for analyzing partially linear models in the recent literature. In this paper, we first propose to solve the optimal sequence selection problem in difference-based estimation for the linear component. To achieve the goal, a family of new sequences and a cross-validation method for selecting the adaptive sequence are proposed. We demonstrate that the existing sequences are only extreme cases in the proposed family. Secondly, we propose a new estimator for the residual variance by fitting a linear regression method to some difference-based estimators. Our proposed estimator achieves the asymptotic optimal rate of mean squared error. Simulation studies also demonstrate that our proposed estimator performs better than the existing estimator, especially when the sample size is small and the nonparametric function is rough.
KeywordsAsymptotic normality Difference-based method Difference sequence Least squares estimator Partially linear model
Yuejin Zhou’s research was supported in part by the Natural Science Foundation of Anhui Grant (No. KJ2017A087), and the National Natural Science Foundation of China Grant (No. 61472003). Yebin Cheng’s research was supported in part by the National Natural Science Foundation of China Grant (No. 11271241). Tiejun Tong’s research was supported in part by the Hong Kong Baptist University Grants FRG1/16-17/018 and FRG2/16-17/074, and the National Natural Science Foundation of China Grant (No. 11671338).