For kernel-based estimators, smoothness conditions ensure that the asymptotic rate at which the bias goes to zero is determined by the kernel order. In a finite sample, the leading term in the expansion of the bias may provide a poor approximation. We explore the relation between smoothness and bias and provide estimators for the degree of the smoothness and the bias. We demonstrate the existence of a linear combination of estimators whose trace of the asymptotic mean-squared error is reduced relative to the individual estimator at the optimal bandwidth. We examine the finite-sample performance of a combined estimator that minimizes the trace of the MSE of a linear combination of individual kernel estimators for a multimodal density. The combined estimator provides a robust alternative to individual estimators that protects against uncertainty about the degree of smoothness.
The authors thank the participants of the Advances in Econometrics Conference in honor of Aman Ullah and the participants of the Canadian Economics Association meeting, in particular Heng Chen. We also thank the anonymous referees who provided insightful comments and useful criticism. This work was supported by the Social Sciences and Humanities Research Council of Canada (SSHRC) and by the Fonds québecois de la recherche sur la société et la culture (FQRSC).
Kotlyarova, Y., Schafgans, M.M.A. and Zinde-Walsh, V. (2016), "Smoothness: Bias and Efficiency of Nonparametric Kernel Estimators", Essays in Honor of Aman Ullah (Advances in Econometrics, Vol. 36), Emerald Group Publishing Limited, pp. 561-589. https://doi.org/10.1108/S0731-905320160000036025Download as .RIS
Emerald Group Publishing Limited
Copyright © 2016 Emerald Group Publishing Limited