Constant width simultaneous confidence bands in multiple linear regression with predictor variables constrained in intervals

摘要

In the last fifty years, a great deal of research effort has been made on the construction of simultaneous confidence bands for a linear regression function. Two most frequently quoted confidence bands in the statistics literature are the Scheffe type and constant width bands over a given rectangular region of the predictor variables. For the constant width bands, a method is given by Gafarian [Gafarian, A.V., 1964, Confidence bands in straight line regression. Journal of the American Statistical Association, 59, 182-213.] for the calculation of critical constants only for the special case of one predictor variable. In this article, a method is proposed to construct constant width bands when there are any number of predictor variables. A new criterion for assessing a confidence band is also proposed; it is the probability that a confidence band excludes a false regression function and can be viewed as the power function of a. test associated, naturally, with a confidence band. Under this criterion, a numerical comparison between the Scheffe type and constant width bands is then carried out. It emerges from this comparison that the constant width bands can be better than the Scheffe type bands for certain designs.