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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.

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.

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.

Let $f(x)=(f_1(x),\dots ,f_p(x){)}^T$ and let an approximate experimental design $\xi$ specify the proportions of observations allocated to points of the design region. Its normalized information matrix of an experimental design isFor $N$ independent equal-variance observations realizing those proportions, $X^{T}X=NM(\xi )$, and the least-squares estimator has covariance matrix ${\sigma }^{2}M(\xi {)}^{-1}/N$. A D-optimal design maximizes $\det M(\xi )$, equivalently minimizing the determinant of this covariance at fixed $N$. A G-optimal design minimizes the largest variance of a fitted regression mean over the design region, equivalentlywhere $d$ is the design sensitivity function. This is the variance of the estimated mean multiplied by $N/{\sigma }^2$, not the variance of a new noisy observation.

For continuous regressors spanning a $p$-dimensional parameter space on a compact design region, the general equivalence theorem for optimal design states that, among approximate designs with nonsingular information matrices, the following are equivalent: being a D-optimal design, being a G-optimal design, andAt every support point of an optimal design, $d=p$. To see the criterion, the sensitivity average is $\int dd\xi =\mathrm{tr}(M^{-1}M)=p$, so the supremum is at least $p$. Adding an infinitesimal allocation at $x$ has directional derivative $d(x,\xi )-p$ for $\log \det M$. At a D-optimum each derivative is nonpositive. Conversely, if all sensitivities are at most $p$, the tangent bound for the concave function $\log \det M$ shows that no competing design increases it. A nonsingular D-optimum exists by the spanning assumption and compactness of the information-matrix set, so the smallest possible G-value is also $p$. Equality at support points follows from the average. The theorem refers to the full approximate-design class; constrained integer allocations or mandated augmentations require separate comparison.

For the linear mean ${\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2$, take $f(x)=(1,x_1,x_2{)}^T$. Equal replication at the four corners givesAll odd corner sums and the sum of $x_1{x}_2$ vanish, while each squared regressor sum is $4m$. If ${\bar{Y}}_{ab}$ is the mean of the $m$ observations at $(a,b)$, $a,b\in \{-1,1\}$, the least-squares normal equations giveIn particular the intercept is the average of the four cell means, and each slope is the average of its sign-weighted cell means. Independence and equal error variance suffice for this covariance formula; normal errors are not needed for it.

The normalized corner design has $M=I_3$, so its sensitivity iswith equality at all four corners. Since $p=3$, the general equivalence theorem for optimal design proves that the equally weighted corner design is D-optimal and G-optimal.

For the augmentation, all $m$ added observations share the regressor $f=(1,x_1,x_2{)}^T$. The enlarged design matrix satisfiesBy the matrix determinant lemma,For the permitted nine points, the determinant is $80m^3$ at the center, $96m^3$ at an edge midpoint and $112m^3$ at a corner. ThereforeAll four corners tie for the best mandated single-point augmentation. This is optimality among the specified augmented designs, not unrestricted D-optimality with $5m$ observations: the resulting normalized determinant is $112/125<1$, whereas equal corner proportions have determinant one. The restriction that the extra $m$ runs all occur at one point is what prevents recovering equal proportions.