Optimal Regression Design

Introduction

There are a variety of problems in statistics, which demand the calculation of one or more probability distributions or measures. Optimal regression design is a particular example. Other examples include parameter estimation, adaptive design and stratified sampling.

Consider the problem of selecting an experimental design to furnish information on models of the type $y \sim \pi (y\vert \underline{x},\underline{\theta },\sigma )$, where y is the response variable; $\underline{x} = {({x}_{1},{x}_{2},\ldots ,{x}_{m})}^{T}$ are design variables, $\underline{x}$ ∈ $\mathcal{X}$ ⊆ ℝ ^m, $\mathcal{X}$ is the design space; $\underline{\theta }$ = ${({\theta }_{1},{\theta }_{2},\ldots ,{\theta }_{k})}^{T}$ are unknown parameters; σ is a nuisance parameter, fixed but unknown; and π(. ) is a probability model. In most applications, $\mathcal{X}$ is taken to be compact. For each $\underline{x}$ ∈ $\mathcal{X}$, an experiment can be performed whose outcome is a...