A Fully Automated Bandwidth Selection Method for Fitting Additive Models
Opsomer, Jean D., Ruppert, David, Journal of the American Statistical Association
Additive models (Hastie and Tibshirani 1990) are a popular multivariate nonparametric fitting technique. The additive model assumes that the conditional expectation function of the dependent variable Y can be written as the sum of smooth terms in the covariates [X.sub.1], . . ., [X.sub.D],
E(Y[where]X = ([x.sub.1], . . ., [x.sub.D])) = [m.sub.1]([x.sub.1]) + . . . + [m.sub.D]([x.sub.D]). (1)
The additive model's appeal is that the fitted models are free of restrictive parametric assumptions, as with any other nonparametric method, but unlike most of them, the effects of individual covariates on the dependent variable can still be easily interpreted, regardless of the number of covariates D. The availability of easy-to-use model estimation software in S-PLUS (Chambers and Hastie 1992) has further contributed to its widespread use. As an example of additive modeling, consider the following question: What is the relationship between housing value and various sociodemographic variables such as tax rate and student/teacher ratio? A dataset was collected by Harrison and Rubinfeld (1978) to answer that question, and they fitted a parametric model to the data to develop a marginal willingness-to-pay model for housing. Clearly, the quality of their economic model depends on the validity of their selected parametric model. Fitting an additive model allows researchers to perform exploratory data analysis and avoid the problems associated with selecting an inappropriate model. But the nonparametric fit can still be misleading if the bandwidth parameters are not selected with care. Researchers fitting additive models can, of course, select bandwidths by "trial and error," but this is a tedious and somewhat arbitrary process, especially for models with many covariates. This article aims to provide a fast, data-driven method for fitting additive models, which would include a theoretically valid, objective bandwidth selection mechanism. The application of this method to the development of a housing value model is explored in Section 5.
Most automated bandwidth selection methods proposed for the additive model rely on cross-validation or one of its approximations (Hastie and Tibshirani 1990). Despite its intuitive appeal and simplicity, this approach suffers from two drawbacks, which are illustrated in simulation experiments in a later section. The first concerns the properties of the bandwidth estimators. In the closely related regression smoothing context, cross-validation estimators have been shown to be limited to a [O.sub.p]([n.sup.-1/10]) relative rate of convergence and to display large sample-to-sample variability (Hardle, Hall, and Marron 1988). Perhaps even more important from a practical standpoint, the second drawback is that these bandwidth selectors are very computation intensive; for a model with D covariates, the search for the "optimal" bandwidth has to take place by numerical approximation over [Mathematical Expression Omitted]. Although methods are available to make this search more efficient (e.g., Gu and Wahba 1988), it still necessitates the calculation of numerous additive model fits.
In this article we develop plug-in bandwidth estimators for the additive model that address both of these drawbacks. Plug-in bandwidth estimators are well known in kernel smoothing, kernel regression, and local polynomial regression, and several authors have developed estimators with good theoretical and practical properties. (For an overview of the literature on this subject, see Wand and Jones 1995.) In a recent article (Opsomer and Ruppert 1997), we explored the asymptotic bias and variance properties of the bivariate additive model fitted by the backfitting algorithm of Buja, Hastie, and Tibshirani (1989). That article provided the theoretical framework that we apply in developing a plug-in bandwidth selection method for additive models.
The article is organized as follows. …