Asymptotic normality of variance estimator in a heteroscedastic model with dependent errors

Abstract

Consider the heteroscedastic regression model Y _ni =g(x _ni )+σ _ni ε _ni (1≤i≤n), where , the design points (x _ni , u _ni ) are known and nonrandom, g(·) and f(·) are unknown functions defined on [0, 1], and the random errors {ε _ni , 1≤i≤n} are assumed to have the same distribution as {ξ _i , 1≤i≤n}, which is a stationary and α-mixing time series with Eξ _i =0. Under appropriate conditions, we study the asymptotic normality of an estimator of the function f(·). At the same time, we derive a Berry–Esseen-type bound for the estimator. As a corollary, by making a certain choice of the weights, the Berry–Esseen-type bound of the estimator can attain O(n ^−1/12(log n)^−1/3). Finite sample behaviour of this estimator is investigated too.

Keywords:

• heteroscedastic regression model
• asymptotic normality
• Berry–Esseen-type bound
• α-mixing

AMS 2000 Subject Classifications :

• 62G08
• 62G20

Acknowledgements

The authors are grateful to the editor, associate editor, and anonymous referees for their helpful comments. This research was supported by the National Natural Science Foundation of China (10871146), and also by the Grants MTM2008-03129 of the Spanish Ministry of Science and Innovation, the Xunta de Galicia under the INBIOMED project (DXPCTSUG, Ref. 2009/063), and Project 10PXIB300068PR of the Xunta de Galicia, Spain.