On the role of the ridge parameter in local linear smoothing

Research Report SRR95-023

On the role of the ridge parameter in local linear smoothing

Peter Hall, J. Stephen Marron

Abstract: 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, and in fact, the actual mean squared error is often infinite; see Seifert and Gasser (1994). This difficulty may be overcome by using a ridge parameter in the construction of the estimator, although that approach requires information about the size of the ridge. From at least a theoretical viewpoint, very little is known about the effects of ridging. In particular, it is not clear how small the ridge parameter may be chosen without affecting performance, or whether infinitely supported kernels such as the Gaussian require ridging. In the present paper we provide concise and definitive answers to such questions. We produce necessary and sufficient conditions on the size of the ridge 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 ridging at all in order to achieve traditional mean square performance.

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