A Fully Automated Bandwidth Selection Method for Fitting Additive Models

By Opsomer, Jean D.; Ruppert, David | Journal of the American Statistical Association, June 1998 | Go to article overview
Save to active project

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.

The rest of this article is only available to active members of Questia

Sign up now for a free, 1-day trial and receive full access to:

  • Questia's entire collection
  • Automatic bibliography creation
  • More helpful research tools like notes, citations, and highlights
  • Ad-free environment

Already a member? Log in now.

Notes for this article

Add a new note
If you are trying to select text to create highlights or citations, remember that you must now click or tap on the first word, and then click or tap on the last word.
Loading One moment ...
Project items
Cite this article

Cited article

Citations are available only to our active members.
Sign up now to cite pages or passages in MLA, APA and Chicago citation styles.

Cited article

A Fully Automated Bandwidth Selection Method for Fitting Additive Models


Text size Smaller Larger
Search within

Search within this article

Look up

Look up a word

  • Dictionary
  • Thesaurus
Please submit a word or phrase above.
Print this page

Print this page

Why can't I print more than one page at a time?

While we understand printed pages are helpful to our users, this limitation is necessary to help protect our publishers' copyrighted material and prevent its unlawful distribution. We are sorry for any inconvenience.
Full screen

matching results for page

Cited passage

Citations are available only to our active members.
Sign up now to cite pages or passages in MLA, APA and Chicago citation styles.

Cited passage

Welcome to the new Questia Reader

The Questia Reader has been updated to provide you with an even better online reading experience.  It is now 100% Responsive, which means you can read our books and articles on any sized device you wish.  All of your favorite tools like notes, highlights, and citations are still here, but the way you select text has been updated to be easier to use, especially on touchscreen devices.  Here's how:

1. Click or tap the first word you want to select.
2. Click or tap the last word you want to select.

OK, got it!

Thanks for trying Questia!

Please continue trying out our research tools, but please note, full functionality is available only to our active members.

Your work will be lost once you leave this Web page.

For full access in an ad-free environment, sign up now for a FREE, 1-day trial.

Already a member? Log in now.

Are you sure you want to delete this highlight?