Simultaneous confidence bands in linear regression analysis

Abstract

A simultaneous confidence band provides useful information on the plausible range of an
unknown regression model. For a simple linear regression model, the most frequently
quoted bands in the statistical literature include the two-segment band, the three-segment
band 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 width
band. 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). A concise review of the
construction of two-sided simultaneous confidence bands in simple and multiple linear re-
gressions and their comparison under the two mentioned optimality criteria is provided in
the thesis. Two families of confidence bands, the inner-hyperbolic bands and the outerhyperbolic
bands, 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 the
hyperbolic band, the best three-segment band and with each other. The inner-hyperbolic
family of confidence bands, which include the hyperbolic and constant-width bands as
special cases, is also constructed for a multiple linear regression model over an ellipsoidal
covariate region and the best band within the family is found by numerical search. For
a multiple linear regression model over a rectangular covariate region (i.e. the predictor
variables 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 hyperbolic
and constant width bands over a rectangular covariate region and compare between them
is provided in this thesis when there are up to three predictor variables. A simulation
method similar to the ones used by Liu et al. (2005a) and Liu et al. (2005b) is also
provided for the calculation of the average width and the minimum volume of confidence
set when there are more than three predictor variables. The methods used in this thesis
are illustrated with numerical examples and the Matlab programs used are available upon
request.

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Identifiers

ORCID for W. Liu: orcid.org/0000-0002-4719-0345

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