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

Optimal simultaneous confidence bands in simple linear regression

A simultaneous confidence band provides useful information on the plausible range of an unknown regression model. For simple linear regression models, the most frequently quoted bands in the statistical literature include the hyperbolic band and the three-segment bands. One interesting question is whether one can construct confidence bands better than the hyperbolic and three-segment bands. The optimality criteria for confidence bands include the average width criterion considered by Gafarian (1964) and Naiman (1984) among others, and the minimum area confidence set (MACS) criterion of Liu and Hayter (2007). In this paper, two families of exact 1?? confidence bands, the inner-hyperbolic bands and the outer-hyperbolic bands, which include the hyperbolic and three-segment bands as special cases, are introduced in simple linear regression. Under the MACS criterion, the best confidence band within each family is found by numerical search and compared with the hyperbolic band, the best three-segment band and with each other. The methodologies are illustrated with a numerical example and the Matlab programs used are available upon request

Simultaneous confidence bands for linear regression with covariates constrained in intervals

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