Asymptotic Properties of Error Density Estimator in Regression Model Under α-Mixing Assumptions

Abstract

In this article, we study the consistency of the error density estimator in nonparametric regression models when the errors form a stationary α-mixing sequences. These results improve the results of Cheng (2004) from the i.i.d. assumption to α-mixing condition and weaken the restrictions for the bandwidths a _n . Also, the rates of strong convergence for the estimator are investigated. Furthermore, we derive the asymptotic normality and the law of the iterated logarithm of the histogram-type error density estimator, which complement the conclusion in Cheng (2002) in α-mixing setting.

Keywords:

• Asymptotic normality
• Consistency
• Error density
• Law of the iterated logarithm
• α-mixing

Mathematics Subject Classification:

• Primary 62G07
• Secondary 62G08

Acknowledgments

This research was supported by the National Natural Science Foundation of China (10571136, 10871146), and also by the Grants MTM2008-03129 of the Spanish Ministry of Science and Innovation, and PGIDIT07PXIB300191PR of the Xunta de Galicia, Spain.