Statistical inferences for linear regression models when the covariates have functional relationships: polynomial regression

Abstract

In many linear regression models, there are functional relationships among the covariates. The usual (partial) F tests for these models do not utilize these functional relationships, however, and so have certain undesirable properties. Improvements over these F tests are proposed in this article, and they are based on simultaneous confidence bands. For ease of illustration of the ideas, we focus on the polynomial regression model.

Keywords:

• Confidence bands
• F tests
• Linear regression
• Polynomial regression
• Simultaneous inference
• Statistical simulation

Acknowledgements

We would like to thank the British Engineering and Physics Sciences Research Council for a visiting fellowship which allowed A.J. Hayter to visit the other two authors, and the Associate Editor and the referee for useful comments.