Some problems relating to simultaneous confidence bands

Abstract

The thesis comprises two distinct areas of research involving the use of simultaneous confidence bands, specifically relating to the Scheffe type simultaneous confidence band of Scheffe (1953), and the constant width simultaneous confidence band of Gafarian (1964).

The first relates to establishing the optimal design of experiments for simultaneous confidence bands, considered over a finite range of a covariate. Methodology is developed for this area that focuses on establishing continuous optimal designs under two optimality criteria specific to simultaneous confidence bands, the Average Width and Minimum Area Confidence Set criteria. We develop a method of numeric investigation which allows the search area to be constrained over intervals. From this, we conclude that the optimal continuous designs for 95 percent simultaneous confidence bands, considered over the range [-1, 1] are D-optimal for specific values of N. Also investigated is the application of traditional analytic methods used to obtain optimal continuous designs.

Secondly, simultaneous confidence bands can be used to construct simultaneous confidence sets for the Effective Doses (ED) of a the logistic regression model (cf. Walter (1983)), by inverting the bounds of a Scheffe type simultaneous confidence band. We introduce an improvement to this method, guaranteed to exhibit closer to nominal simultaneous coverage by constructing specialised simultaneous confidence sets for a specific number (k) of ED's. Two sided sets are fully defined for k - 2 for a multiple covariate model, and k - 3 or more for a one covariate model. This new methodology is then applied to construct simultaneous confidence sets for two additional situations: (i) when the ED can be assumed to lie over a finite interval, (ii) when one sided simultaneous confidence sets are sought. These methods may be applied to any generalised linear model, and the improvements over the original methods are illustrated with examples.

All numeric methods provided in the thesis, including illustrations, were carried out by custom programs made in R, which are available by request.

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Identifiers

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

ORCID for Stefanie Biedermann: orcid.org/0000-0001-8900-8268

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