Simultaneous confidence bands in linear regression analysis

A simultaneous confidence band provides useful information on the plausible range of anunknown regression model. For a simple linear regression model, the most frequentlyquoted bands in the statistical literature include the two-segment band, the three-segmentband and the hyperbolic band, and for a multiple linear regression model, the most com-mon bands in the statistical literature include the hyperbolic band and the constant widthband. The optimality criteria for confidence bands include the Average Width criterionconsidered by Gafarian (1964) and Naiman (1984) among others, and the Minimum AreaConfidence Set (MACS) criterion of Liu and Hayter (2007). A concise review of theconstruction of two-sided simultaneous confidence bands in simple and multiple linear re-gressions and their comparison under the two mentioned optimality criteria is provided inthe thesis. Two families of confidence bands, the inner-hyperbolic bands and the outerhyperbolicbands, which include the hyperbolic and three-segment bands as special cases,are introduced for a simple linear regression. Under the MACS criterion, the best con-fidence band within each family is found by numerical search and compared with thehyperbolic band, the best three-segment band and with each other. The inner-hyperbolicfamily of confidence bands, which include the hyperbolic and constant-width bands asspecial cases, is also constructed for a multiple linear regression model over an ellipsoidalcovariate region and the best band within the family is found by numerical search. Fora multiple linear regression model over a rectangular covariate region (i.e. the predictorvariables are constrained in intervals), no method of constructing exact simultaneous con-fidence bands has been published so far. A method to construct exact two-sided hyperbolicand constant width bands over a rectangular covariate region and compare between themis provided in this thesis when there are up to three predictor variables. A simulationmethod similar to the ones used by Liu et al. (2005a) and Liu et al. (2005b) is alsoprovided for the calculation of the average width and the minimum volume of confidenceset when there are more than three predictor variables. The methods used in this thesisare illustrated with numerical examples and the Matlab programs used are available uponrequest