Optimal and efficient experimental design for nonparametric regression with application to functional data

Abstract

Functional data is ubiquitous in modern science, technology and medicine. An example, which motivates the work in this thesis, is an experiment in tribology to investigate wear in automotive transmission.

The research in this thesis provides methods for the design of experiments when the response is assumed to be a realisation of a smooth function. In the course of the research, two areas were investigated: designs for local linear smoothers and designs for discriminating between two functional linear models.

Designs that are optimal for minimising the prediction variance of a smooth function were found across an interval using two kernel smoothing methods: local linear regression and Gasser and Muller estimation. The values of the locality parameter and run size were shown to affect the optimal design. Optimal designs for best prediction using local linear regression were applied to the tribology experiment. A compound optimality criterion is proposed which is a weighted average of the integrated prediction variance and the inverse of the trace of the smoothing matrix using the Gasser and Muller estimator. The complexity of the model to be fitted was shown to influence the selection of optimal design points. The robustness of these optimal designs to misspecification of the kernel function for the compound criterion was also critically assessed.

A criterion and method for finding T-optimal designs was developed for discriminating between two competing functional linear models. It was proved that the choice of optimal design is independent of the parameter values when discriminating between two nested functional linear models that differ by only one term. The performance of T-optimal designs was evaluated in simulation studies which calculated the power of the test for assessing the fit of one model using data generated from the competing model.

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Identifiers

ORCID for David C. Woods: orcid.org/0000-0001-7648-429X

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