Smoothing by Local Regression: Principles and Methods

Abstract

Local regression is an old method for smoothing data, having origins in the graduation of mortality data and the smoothing of time series in the late 19th century and the early 20th century. Still, new work in local regression continues at a rapid pace. We review the history of local regression. We discuss four of its basic components that must be chosen in using local regression in practice — the weight function, the parametric family that is fitted locally, the bandwidth, and the assumptions about the distribution of the response. A major theme of the paper is that these choices represent a modeling of the data; different data sets deserve different choices. We describe polynomial mixing, a method for enlarging polynomial parametric families. We introduce an approach to adaptive fitting,assessment of parametric localization. We describe the use of this approach to design two adaptive procedures: one automatically chooses the mixing degree of mixing polynomials at each x using cross-validation, and the other chooses the bandwidth at each x using C p . Finally, we comment on the efficacy of using asymptotics to provide guidance for methods of local regression.