Twicing local linear kernel regression smoothers

Abstract

It is known that the local cubic smoother (LC) has a faster consistency rate than the popular local linear smoother (LL). However, LC often has a bigger mean squared error (MSE) than LL numerically for samples of finite size. By extending the idea of Stuetzle and Mittal [1979, ‘Some Comments on the Asymptotic Behavior of Robust Smoothers’, in Smoothing Techniques for Curve Estimation: Proceedings (chap. 11), eds. T. Gasser and M. Rosenbalatt, Berlin: Springer, pp. 191–195], we propose a new version of LC by ‘twicing’ the local linear smoother (TLL). Both asymptotic theory and finite sample simulations suggest that TLL has better efficiency than LL. Compared with LC, TLL has about the same asymptotic MSE (AMSE) as LC at the interior points and has a much smaller AMSE than LC at the boundary points. The TLL is also more stable than LC and has better performance than LC numerically. The application of TLL to estimate the first-order derivative of the regression function and other extensions are considered.

Keywords:

• bias reduction
• local linear kernel smoother
• mean squared error
• nonparametric regression
• ‘twicing’ kernel

Acknowledgements

We are grateful to two anonymous referees and Dr Thomas Yee for their constructive comments. This research was partially supported by the Education Department of Nature Science Research of Guizhou Province (grant no. 2010028), the Nomarch Foundation of Guizhou Province (grant no. 2010025), China, and a grant from the Risk Management Institute, National University of Singapore.