Computational aspects of nonparametric smoothing with illustrations from the sm library

Smoothing techniques such as density estimation and nonparametric regression are widely used in applied work and the basic estimation procedures can be implemented relatively easily in standard statistical computing environments. However, computationally e2cient procedures quickly become necessary with large datasets, many evaluation points or more than one covariate. Further computational issues arise in the use of smoothing techniques for inferential, rather than simply descriptive, purposes. These issues are addressed in two ways by (i) deriving e2cient matrix formulations of nonparametric smoothing methods and (ii) by describing further simple modi6cations to these for the use of ‘binned’ data when sample sizes are large. The implications for other graphical and inferential aspects of the estimators are also discussed. These issues are dealt with in an algorithmic manner, to allow implementation in any programming environment, but particularly those which are geared towards vector and matrix representations of data. Speci6c examples of S-Plus code from the sm library of Bowman and Azzalini (Applied Smoothing Techniques for Data Analysis: the Kernel Approach With S-Plus Illustrations, Oxford University Press, Oxford, 1997) are given in an appendix as illustrations. c © 2002 Elsevier Science B.V. All rights reserved.