On the role of the shrinkage parameter in local linear smoothing |
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Authors: | Peter Hall J Stephen Marron |
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Institution: | (1) Centre for Mathematics and its Applications, Australian National University, Canberra, ACT 0200, Australia, AU;(2) Department of Statistics, University of North Carolina at Chapel Hill, CB 3260, Phillips Hall, Chapel Hill, NC 27599-3260, USA, US |
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Abstract: | Summary. It has been shown that local linear smoothing possesses a variety of very attractive properties, not least being its mean
square performance. However, such results typically refer only to asymptotic mean squared error, meaning the mean squared error of the asymptotic distribution, and in fact, the actual mean squared error
is often infinite. See Seifert and Gasser (1996). This difficulty may be overcome by shrinking the local linear estimator
towards another estimator with bounded mean square. However, that approach requires information about the size of the shrinkage
parameter. From at least a theoretical viewpoint, very little is known about the effects of shrinkage. In particular, it is
not clear how small the shrinkage parameter may be chosen without affecting first-order properties, or whether infinitely
supported kernels such as the Gaussian require shrinkage in order to achieve first-order optimal performance. In the present
paper we provide concise and definitive answers to such questions, in the context of general ridged and shrunken local linear
estimators. We produce necessary and sufficient conditions on the size of the shrinkage parameter that ensure the traditional
mean squared error formula. We show that a wide variety of infinitely-supported kernels, with tails even lighter than those
of the Gaussian kernel, do not require any shrinkage at all in order to achieve traditional first-order optimal mean square
performance.
Received: 22 May 1995 / In revised form: 23 January 1997 |
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Keywords: | : Bandwidth kernel local polynomial smoothing local regression mean squared error ridge regression nonparametric regression variance |
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