Curve estimation when the design density is low

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. Contexts where this problem arises include projection pursuit regression, estimation in single index models and application of the projection-slice method of Radon transform inversion. Theoretical work in these settings typically involves ignoring data toward the ends of the support of the projected design, but in practice that waste of information is not an attractive option. Motivated by these difficulties, we analyze the way in which local linear smoothing is affected by unboundedly sparse design and apply the conclusions of that study to develop empirical, adaptive bandwidth choice methods. Our results even add to knowledge in the familiar case of a design density that is bounded away from zero, where they provide adaptive bandwidth selectors that are optimal right to the ends of the design interval.