Optimal experimental design

Optimal experimental design chooses observation locations and allocations to optimize a specified criterion for a specified statistical model. Criteria can control the determinant of an estimator's covariance matrix or its worst prediction variance. Optimality depends on the model and the permitted design class, including whether allocations are integer or approximate.

For continuous regressors spanning a $p$-dimensional space on a compact design region, and approximate designs with nonsingular information matrix of an experimental design, the following are equivalent: D-optimal design, G-optimal design, and ${\sup }_{x}d(x,\xi )=p$. A D-optimal design has nonpositive directional derivatives $d(x,\xi )-p$. Conversely, if every such derivative is nonpositive, concavity of $\log \det$ bounds every competing information matrix by the tangent at $M(\xi )$, proving D-optimality. The sensitivity average is $p$, and a D-optimal design exists by compactness of the information-matrix set and the spanning assumption; hence the optimal G-value is $p$. Sensitivity equals $p$ at every support point of an optimal design. The equivalence concerns approximate allocations; integer restrictions can change exact-design optima.

A G-optimal design minimizes the maximum design sensitivity function over the full design region. Thus it minimizes the worst variance of a fitted regression mean, excluding the separate noise variance of a new observation. The sensitivity average is $p$, so its supremum is at least $p$.

A D-optimal design maximizes the determinant of the information matrix of an experimental design, equivalently minimizing the determinant of the covariance matrix of a fixed-size least-squares estimator. It minimizes the volume of a fixed-level confidence ellipsoid for the model coefficients under the usual Gaussian model.

An approximate design is a probability measure on the design region, with weight $w_i$ specifying a proportion of observations at $x_i$. For $N$ independent observations and common error variance ${\sigma }^2$, a linear regression with vector $f(x)$ has normalized information matrix of an experimental design $M(\xi )=\int f(x)f(x{)}^{T}d\xi (x)$; its least-squares estimator has covariance ${\sigma }^{2}M(\xi {)}^{-1}/N$ when the allocations realize these proportions. Exact designs restrict $N{w}_i$ to integers.

The normalized information matrix for a homoscedastic linear regression model is the average outer product of its regressor vector. It is positive-definite precisely when the supported regressors span the parameter space. The full Fisher information matrix under normal errors is $NM(\xi )/{\sigma }^2$.

The sensitivity function is the normalized variance of a fitted regression mean. Its average under the design is the number $p$ of model coefficients, since $\int dd\xi =\mathrm{tr}(M^{-1}M)=p$. The directional derivative of $\log \det M$ towards a point mass at $x$ is $d(x,\xi )-p$.