On the role of the shrinkage parameter in local linear smoothing

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 in®nite. See Seifert and Gasser (1996). This diculty 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 e€ects of shrinkage. In particular, it is not clear how small the shrinkage parameter may be chosen without a€ecting ®rst-order properties, or whether in®nitely supported kernels such as the Gaussian require shrinkage in order to achieve ®rst-order optimal performance. In the present paper we provide concise and de®nitive answers to such questions, in the context of general ridged and shrunken local linear estimators. We produce necessary and sucient conditions on the size of the shrinkage parameter that ensure the traditional mean squared error formula. We show that a wide variety of in®nitelysupported kernels, with tails even lighter than those of the Gaussian kernel, do not require any shrinkage at all in order to achieve traditional ®rst-order optimal mean square performance.