Empirical Likelihood for a Heteroscedastic Partial Linear Model

Abstract

Abstract Consider the heteroscedastic regression model , where , β is a p × 1 column vector of unknown parameter, (X i , T i , Z i ) are random design points, Y i are the response variables, g(·) is an unknown function defined on the closed interval [0, 1], {e i , ℱ i } is a sequence of martingale differences. When f is known and unknown cases, we propose the empirical log-likelihood ratio statistics for the parameter β. For each case, a nonparametric version of Wilks' theorem is derived. The results are then used to construct confidence regions of the parameter. Simulation study shows that the empirical likelihood method performs better than a normal approximation-based approach. Keywords: Confidence regionEmpirical likelihoodHeteroscedastic partial linear modelLocal polynomialMartingale difference errorMathematics Subject Classification: 62G1562E20 Acknowledgments The authors are grateful to the Editor, Associate Editor, and an anonymous referee for their valuable comments with our article. This research was supported by the National Natural Science Foundation of China (10871146), Humanities and Social Science Planning Foundation of Ministry of Education of China (10YJA910005), and Provincial Natural Science Research Project of Anhui Colleges.