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Biometrika 2004 91(4):819-834; doi:10.1093/biomet/91.4.819
© 2004 by Biometrika Trust
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A crossvalidation method for estimating conditional densities

Jianqing Fan and Tsz Ho Yim

Department of Statistics, Chinese University of Hong Kong, Shatin, Hong Kong jfan{at}sta.cuhk.edu.hk, th_yim{at}sparc20c.sta.cuhk.edu.hk

We extend the idea of crossvalidation to choose the smoothing parameters of the ‘double-kernel’ local linear regression for estimating a conditional density. Our selection rule optimises the estimated conditional density function by minimising the integrated squared error. We also discuss three other bandwidth selection rules, an ad hoc method used by Fan et al. (1996), a bootstrap method of Hall et al. (1999) for bandwidth selection in the estimation of conditional distribution functions, modified by Bashtannyk & Hyndman (2001) to cover conditional density functions, and finally a simple approach proposed by Hyndman & Yao (2002). The performance of the new approach is compared with these three methods by simulation studies, and our method performs outstandingly well. The method is illustrated by an application to estimating the transition density and the Value-at-Risk of treasury-bill data.

Key Words: Bandwidth selection; Bootstrap; Conditional density function; Crossvalidation; Diffusion process; Financial application; Transition density

Received June 2003. Revised April 2004.


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