= General equivalence theorem for optimal design
= Kiefer-Wolfowitz equivalence theorem
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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.
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