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Biometrika 1995 82(3):575-587; doi:10.1093/biomet/82.3.575
© 1995 by Biometrika Trust
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On partial local smoothing rules for curve estimation

PETER HALL1, J. STEPHEN MARRON2 and D. M. TITTERINGTON3

1 Centre for Mathematics and its Applications, Australian National University ACT 0200, Canberra, Australia
2 Department of Statistics, University of North Carolina Chapel Hill, North Carolina 27599-3260, USA.
3 Department of Statistics, University of Glasgow, University Gardens Glasgow G12 8QW, UK.

Received for publication 1 August 1994. Revision received 1 April 1995.
  Abstract

We compare the performances of local and global rules for smoothing parameter choice, in terms of asymptotic mean squared errors of the resulting estimators. In some instances there is surprisingly little to choose between local and global approaches; our analysis identifies contexts where the differences are small or large. This work motivates development of smoothing rules that form a ‘half-way house’ between local and global smoothing. There, interpolation provides a basis for partial local smoothing. A key result shows that interpolation on even a coarse grid can produce a very good approximation to full local smoothing. Our theoretical and numerical results lead us to suggest linear interpolation of a bandwidth obtained by integral approximations on discrete intervals.

Key Words: Adaptive method • Bandwidth • Global smoothing • Kernel method • Local linear smoothing • Local smoothing • Smoothing parameter


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