IJPAM: Volume 42, No. 4 (2008)

A PRACTICAL IMPLEMENTATION FOR BANDWIDTH
SELECTION IN KERNEL DISTRIBUTION
FUNCTION ESTIMATORS

Y.L. Lio
Department of Mathematical Sciences
University of South Dakota
414, East Clark Street, Vermillion, SD 57069-2390, USA
e-mail: Yuhlong.Lio@usd.edu


Abstract.Two kernel smooth distribution estimators for a distribution function, $F(x)$ are considered. It is shown that the bandwidth $h_{0}=O(n^{-1/3})$ is asymptotically optimal in the sense

\begin{displaymath}\left[\hat{\phi}(x,n,h_{0})-F(x)\right]^{2}/\inf_{h}\left[\hat{\phi}(x,n,h)-F(x)\right]^{2}\rightarrow 1\end{displaymath}

in probability as $n \rightarrow \infty$, where $\hat{\phi}(x,n,h)$ indicates both kernel smooth estimators. To implement the optimal bandwidth selection, a Bayesian bandwidth is proposed. The computer simulation study shows that the proposed smooth distribution estimators with Bayesian bandwidth could have mean squared errors that are better than the smooth estimators with the optimal bandwidth $h_{0}$ in terms of mean squared errors.

Received: August 17, 2007

AMS Subject Classification: 62A13

Key Words and Phrases: bandwidth selection, Bayesian estimator, kernel estimation

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2008
Volume: 42
Issue: 4