A dual approach for matrix-derivatives

Summary

A dual approach for matrix by matrix derivatives is proposed, based on the work ofBalestra and the idea ofDwyer/MacPhail. The two derivative concepts are calledB-type derivative for the form ∂B/∂A = (∂b _kl /∂A), because it was studied fully inBalestra, andA-type derivative for the form ∂B/ /∂A = (∂B/∂a _ij ). Both derivative concepts are linked by permutation matrices, which also reveil the duality aspect more clearly, and can be transformed to each other very easily. The derivatives are applied to least squares estimates and posterior means in the general regression model and the “seemingly unrelated regression” (SUR)-system, introduced byZellner. The derivatives with respect to the covariance matrix is related to the local sensitivity concept ofLeamer [1978], while the derivatives with respect to the data-matrix, also called local resistance, is linked with the robustness concept ofTukey. The newly definedB-derivative enables an easier interpretation of the results.