Designs enhancing Fisher information

Given a model {Y=Xβ+ϵ}$\lbrace \boldsymbol{Y}=\boldsymbol{X}\boldsymbol{\beta }+\boldsymbol{\epsilon }\rbrace$ with Fisher information matrix Ξ=X'X,$\boldsymbol{\Xi }=\boldsymbol{X}^{\prime }\boldsymbol{X},$ a principal objective is to find information enhancing transformations T$\boldsymbol{T}$ for which TΞT'⪰LΞ$\boldsymbol{T}\boldsymbol{\Xi }\boldsymbol{T}^{\prime }{\succeq }_{L}\boldsymbol{\Xi }$ under the positive definite ordering, so as to improve essentials in linear inference. This is achieved through properties of congruences together with basic orderings of linear spaces. These foundations in turn support a new class of geometric mixture models on “mixing” the original design with another to assume the role of “target,” to the following effects. Ridge, surrogate, and other solutions are often used to mitigate the effects of ill-conditioned models. Instead, in this study an ill-conditioned design matrix X$\boldsymbol{X}$ is mixed with a well-conditioned design as target, leveraging the former toward the latter as the mixing parameter evolves, thus offering an alternative approach to ill-conditioning. The methodology is demonstrated with case studies from the literature, where the geometric mixtures are compared with the ridge, surrogate, and recently found arithmetic mixture models.