Frequentist decision-theoretic optimal design and its applications to generalised linear models

Abstract

Alphabetical optimal designs are found by minimising a scalar function of the inverse
Fisher information matrix, and represent the de-facto standard in optimal design of experiments. For example, the well-known D-optimal design is found by minimising the log determinant of the inverse Fisher information matrix. In this thesis, an alternative decision-theoretic basis for frequentist design is proposed and investigated whereby designs are found by minimising the risk function defined as the expectation of an appropriate loss function that represents the aim of an experiment (e.g, parameter estimation or model discrimination). The expectation is taken with respect to the distribution of responses that observed from the experiment given the design. Generally, the loss function compares a parameter estimator or result of a model selection procedure to the true values of the parameters or the model, respectively, given the observed data obtained from the design. The conceptual advantages of the decision-theoretic framework over alphabetical optimal designs are that it is suitable for small sample sizes and can be applied for bespoke experimental aims. This research aims to find frequentist decision-theoretic optimal designs for the experimental aim of parameter estimation and model discrimination for generalised linear models.
Finding decision-theoretic designs is accomplished by minimising the risk function over a design space X. The risk function that we minimise depends on unknown parameters. Local and pseudo-Bayesian approaches are proposed for defining an objective function in cases where the risk depends on the parameters. However, finding such designs is complicated due to the considerable computational challenges of minimising an analytically intractable risk function, leading to the need for approximation methods for the risk function. We develop novel methods to approximate the risk function, and these approximations are then used within an optimisation algorithm to find decision-theoretic optimal designs for a range of examples that have important applications in the design of experiments. Comparison between these approximations will be considered in terms of performance and computing time.

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