Simultaneous Confidence Bands For Linear Regression With Covariates Restricted In Intervals And Simulation Comparisons

This article concentrates on constructing simultaneous confidence bands for a linear regression mode with covariates restricted by a rectangular region.It also provides relationships between simultaneous confidence levels and critical constants of hyperbolic bands and constant width bands,involving the one-sided bands and two-sided bands.Formulas are expressed as an integral of p-dimension for a general p,especially which are further simplified when p=1.Exact simultaneous confidence bans can be set up through the numerical quadrature method calculating critical constants for at least a small p.Then comparison between the two-sided hyperbolic and two-sided constant width bands is addressed under the minimum volume confidence set criteria.Similarly,using the average width criteria compares the onesided hyperbolic band with one-sided constant width band.It is not difficult to see that the hyperbolic band can be significantly more efficient than the constant width band when p=1,which is both applied to the one-sided and two-sided bands.And the onesided band is better than the two-sided band when p=1.It is pointed out that the simultaneous confidence band is optimal if and only if some conditions are satisfied,which are respectively the lower confidence level,the limited covariate region being the same as the design interval and the design points focusing on the two ends of the interval.