Approximate designs and linear-model optimality criteria

Let $\mathcal{X}$ be a finite nonempty candidate set, let $p\ge 1$, and let $f:\mathcal{X}\to {\mathbb{R}}^p$ be the declared regression-vector map. Work under the stated linear mean model $Y=f(x{)}^{\mathsf{T}}\beta +ϵ$, with mean-zero independent run errors of common variance ${\sigma }^2>0$. These are modeling assumptions, not claims about an experiment's empirical adequacy.

An approximate design is a weight function $\xi :\mathcal{X}\to [0,1]$ with ${\sum }_{x\in \mathcal{X}}{\xi }_x=1$. Its support is $\mathrm{supp}(\xi )=\{x:{\xi }_x>0\}$ and its information matrix is $M(\xi )={\sum }_{x\in \mathcal{X}}{\xi }_{x}f(x)f(x{)}^{\mathsf{T}}\mathrm{.}$ This matrix is symmetric and positive semidefinite. A design is nonsingular when $M(\xi )$ is positive definite. For a singular design, the page's estimability convention is that a specified coefficient contrast $c^{\mathsf{T}}\beta$ is estimable exactly when $c\in \mathrm{span}\{f(x):x\in \mathrm{supp}(\xi )\}$; the inverse-matrix criteria below are defined only for nonsingular designs.

For nonsingular designs on this fixed $f$ and $\mathcal{X}$, D-optimality maximizes $\log \det M(\xi )$, A-optimality minimizes $\mathrm{tr}(M(\xi {)}^{-1})$, E-optimality maximizes ${\lambda }_{\min }(M(\xi ))$, and $c$-optimality for a fixed $c\in {\mathbb{R}}^p$ minimizes $c^{\mathsf{T}}M(\xi {)}^{-1}c$; $c=0$ gives a trivial tie. The smallest eigenvalue is real because $M(\xi )$ is real symmetric (Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis). These are different model- and candidate-dependent objectives; none is a universal design ranking.

An exact $n$-run design, for an integer $n\ge 1$, has weights ${\xi }_x=n_x\mathrm{/}n$ with nonnegative integer counts $n_x$ summing to $n$; optimizing over those integer allocations is a separate finite problem.

The equivalence criterion for D-optimality is proved in Equivalence criterion for D-optimal approximate designs.