Optimal Simultaneous Confidence Bounds in Regression
Naiman, Daniel Quitt
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https://hdl.handle.net/2142/71202
Description
Title
Optimal Simultaneous Confidence Bounds in Regression
Author(s)
Naiman, Daniel Quitt
Issue Date
1982
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Abstract
The problem of finding optimal simultaneous confidence bounds for multilinear regression functions with intercept, over bounded regions, is considered. Conditions are derived which imply that the Scheffe-type bound beats the constant width bound for the case of one-sided or two-sided bounding, in the sense of having smaller average width with respect to Lebesgue measure over the region, when coverage probabilities are equated. These conditions are shown to hold for many important regression designs. A new class of bounds for which the coverage probability may be easily computed is introduced. Using this class, the Scheffe-type bound is shown to be suboptimal, for some situations. This is in contrast to a result of Bohrer (1973) which gives the optimality of Scheffe-type bounds for the non-intercept case. The notion of a simultaneous confidence bound is redefined by replacing coverage probability by expected coverage measure. Under fairly general conditions bounds can be derived which minimize a functional of the width subject to a lower bound on the expected coverage measure.
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