Simultaneous confidence bands in linear regression analysis

A simultaneous confidence band provides useful information on the plausible range of an<br/>unknown regression model. For a simple linear regression model, the most frequently<br/>quoted bands in the statistical literature include the two-segment band, the three-segment<br/>band and the hyperbolic band, and for a multiple linear regression model, the most com-<br/>mon bands in the statistical literature include the hyperbolic band and the constant width<br/>band. The optimality criteria for confidence bands include the Average Width criterion<br/>considered by Gafarian (1964) and Naiman (1984) among others, and the Minimum Area<br/>Confidence Set (MACS) criterion of Liu and Hayter (2007). A concise review of the<br/>construction of two-sided simultaneous confidence bands in simple and multiple linear re-<br/>gressions and their comparison under the two mentioned optimality criteria is provided in<br/>the thesis. Two families of confidence bands, the inner-hyperbolic bands and the outerhyperbolic<br/>bands, which include the hyperbolic and three-segment bands as special cases,<br/>are introduced for a simple linear regression. Under the MACS criterion, the best con-<br/>fidence band within each family is found by numerical search and compared with the<br/>hyperbolic band, the best three-segment band and with each other. The inner-hyperbolic<br/>family of confidence bands, which include the hyperbolic and constant-width bands as<br/>special cases, is also constructed for a multiple linear regression model over an ellipsoidal<br/>covariate region and the best band within the family is found by numerical search. For<br/>a multiple linear regression model over a rectangular covariate region (i.e. the predictor<br/>variables are constrained in intervals), no method of constructing exact simultaneous con-<br/>fidence bands has been published so far. A method to construct exact two-sided hyperbolic<br/>and constant width bands over a rectangular covariate region and compare between them<br/>is provided in this thesis when there are up to three predictor variables. A simulation<br/>method similar to the ones used by Liu et al. (2005a) and Liu et al. (2005b) is also<br/>provided for the calculation of the average width and the minimum volume of confidence<br/>set when there are more than three predictor variables. The methods used in this thesis<br/>are illustrated with numerical examples and the Matlab programs used are available upon<br/>request.

Name dc:type.qualificationname

Ph.D.

Level dc:type.qualificationlevel

doctoral

Grantor dc:publisher.institution

University of Southampton

Year dc:date.issued

2010

Author dc:creator

• Ah-Kine, Pascal Soon Shien

Advisor dc:contributor.advisor

• Liu, W.