An affine matrix scalarization of the Cramér–Rao inequality

We study an affine scalarization of the quadratic Cramér–Rao inequality. Instead of evaluating the usual trace product of the covariance matrix and the Fisher information matrix in a fixed Euclidean position, we minimize this product over all nonsingular linear positions. Using the positive-definite specialization of Alberti's variational characterization of matrix fidelity, while giving a self-contained proof that records all minimizers, we obtain, for positive definite matrices $S$ and $T$,

$$\underset{A\in \mathrm{G}\mathrm{L}(n)}{inf}\mathrm{t}\mathrm{r}(AS{A}^{\mathsf{T}}{)}^{1/2}\mathrm{t}\mathrm{r}(A^{-\mathsf{T}}T{A}^{-1}{)}^{1/2}=\text{tr}\left[{\left(S^{1/2}T{S}^{1/2}\right)}^{1/2}\right].$$

The matrix identity itself is classical; our contribution is its interpretation as an affine scalarization of the covariance–Fisher-information pair, including the optimal affine-position condition and the statistical equality cases. Applied to random vectors, the resulting scalar Cramér–Rao quantity is affine invariant, is no larger than the usual trace product in any fixed Euclidean position, and, when centered at the mean, attains the lower bound for every nonsingular Gaussian distribution.