 General equivalence theorem for optimal design

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.