A note on response mean confidence band for linear regression models

Abstract

From fitting (training) a multiple linear regression model with p basis function predictors (e.g., polynomial, trigonometric) we study a type of confidence band covering an entire set of the response means, where a constant (C) is utilized to scale individual confidence interval of each response mean under consideration. We prove that the coverage rate profiles from two special one-dimensional (p = 1) models are identical. When the testing domain equals the training domain, this “basis-free” property applies to the multi-dimensional scenarios. We further study the prediction scenario where the testing domain differs from the training domain. Although the overall coverage rate profile comparison becomes more complicated among different predictor types (e.g., polynomial, trigonometric, independent normal components), the coverage rate range profiles are quite similar among different predictor types. We finally briefly discuss hypothesis testing using this type of confidence band.

Keywords:

• Basis function
• Confidence band
• Linear regression
• Predictor
• Response mean
• Testing domain
• Training domain

AMS CLASSIFICATION:

• 62J05