On local linear smoothing when the design density is low

Research Report SRR95-027

On local linear smoothing when the design density is low

Peter Hall, J. Stephen Marron, N.H. Neumann and D.M. Titterington

Abstract: In problems where a high-dimensional design is projected into a lower number of dimensions, the density of the new design is typically not bounded away from zero over its support, even if the original one was. This difficulty motivates us to analyse the performance of nonparametric regression estimators when the design density is arbitrarily low. A careful analysis of properties of the regression estimator leads to locally adaptive bandwidth selectors that do not require prior knowledge of the way in which the design density behaves at the ends of the design interval. These rules are translated into empirical methods, which are shown to produce first-order optimal bandwidths. Our results and conclusions apply also to more familiar cases where the design density is bounded away from zero. Even there they add significantly to existing knowledge, for example by providing adaptive bandwidth selectors that are optimal right to the ends of the design interval.

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