Simultaneous Confidence Bands for Linear Regression with Covariates Constrained in Intervals

Abstract

Abstract. The focus of this article is on simultaneous confidence bands over a rectangular covariate region for a linear regression model with k>1 covariates, for which only conservative or approximate confidence bands are available in the statistical literature stretching back to Working & Hotelling (J. Amer. Statist. Assoc.24, 1929; 73–85). Formulas of simultaneous confidence levels of the hyperbolic and constant width bands are provided. These involve only a k‐dimensional integral; it is unlikely that the simultaneous confidence levels can be expressed as an integral of less than k‐dimension. These formulas allow the construction for the first time of exact hyperbolic and constant width confidence bands for at least a small k(>1) by using numerical quadrature. Comparison between the hyperbolic and constant width bands is then addressed under both the average width and minimum volume confidence set criteria. It is observed that the constant width band can be drastically less efficient than the hyperbolic band when k>1. Finally it is pointed out how the methods given in this article can be applied to more general regression models such as fixed‐effect or random‐effect generalized linear regression models.

Exact Simultaneous Confidence Bands for Quadratic and Cubic Polynomial Regression with Applications in Dose Response Study

“…The hyperbolic and constant width bands are then compared for the first time under both the average width (AW) criterion (see, e.g., Naiman ) and the minimum volume confidence set (MVCS) criterion (see, e.g., Liu & Hayter ). Note that Liu, Ah‐Kine & Zhou () consider multiple linear regression, in which the covariates are assumed to have no functional relationships among them. Hence the results of that paper are not applicable to the polynomial regression considered in this paper, in which the covariates are , and so depend on only.…”