Abstract
Abstract In this paper we propose a new nonparametric kernel-based estimator for a density function f which achieves bias reduction relative to the classical Rosenblatt–Parzen estimator. Contrary to some existing estimators that provide for bias reduction, our estimator has a full asymptotic characterisation including uniform consistency and asymptotic normality. In addition, we show that bias reduction can be achieved without the disadvantage of potential negativity of the estimated density – a deficiency that results from using higher order kernels. Our results are based on imposing global Lipschitz conditions on f and defining a novel corresponding kernel. A Monte Carlo study is provided to illustrate the estimator's finite sample performance. Keywords: bias reductionkernel density estimationLipschitz conditions AMS 2000 Classification codes : 62G0762G20 Acknowledgements We thank two anonymous referees for their comments. The remaining errors are the authors' responsibility. Notes We have several examples and graphical illustrations for which M k >0 with the Cauchy seed, but we have been unable to establish this fact analytically. Results for samples of size n=600 are not reported but are available upon request from the authors. As expected from asymptotic theory, when n=600 bias and MSE for all estimators across all densities are reduced. See Fuller (1976) Fuller, W. 1976. Introduction to Statistical Time Series, New York: Wiley. [Google Scholar], Dickey and Fuller (1979) Dickey, D. A. and Fuller, W. 1979. Distribution of the Estimators for Autoregressive Time Series with a Unit Root. Journal of the American Statistical Association, 74: 427–431. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar] and Pagan and Ullah (1999) Pagan, A. and Ullah, A. 1999. Nonparametric Econometrics, Cambridge: Cambridge University Press. [Crossref] , [Google Scholar].