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 $T$,
$ \inf\limits_{A\in \mathrm{GL}(n)} {\rm{tr}}(ASA^{ \mathsf T})^{1/2}{\rm{tr}}(A^{- \mathsf T}TA^{-1})^{1/2} = \text{tr}\!\left[\left(S^{1/2}TS^{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.
Citation: Songjun Lv. An affine matrix scalarization of the Cramér–Rao inequality[J]. AIMS Mathematics, 2026, 11(7): 23437-23449. doi: 10.3934/math.2026945
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 $T$,
$ \inf\limits_{A\in \mathrm{GL}(n)} {\rm{tr}}(ASA^{ \mathsf T})^{1/2}{\rm{tr}}(A^{- \mathsf T}TA^{-1})^{1/2} = \text{tr}\!\left[\left(S^{1/2}TS^{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.
| [1] | H. Cramér, Mathematical methods of statistics, Princeton: Princeton University Press, 1999. |
| [2] | C. R. Rao, Information and the accuracy attainable in the estimation of statistical parameters, In: Breakthroughs in statistics, New York: Springer, 1992. https://doi.org/10.1007/978-1-4612-0919-5_16 |
| [3] | T. M. Cover, J. A. Thomas, Elements of information theory, New York: Wiley, 2005. |
| [4] |
E. Lutwak, D. Yang, G. Zhang, The Cramér–Rao inequality for star bodies, Duke Math. J., 112 (2002), 59–81. https://doi.org/10.1215/S0012-9074-02-11212-5 doi: 10.1215/S0012-9074-02-11212-5
|
| [5] |
E. Lutwak, D. Yang, G. Zhang, Moment-entropy inequalities, Ann. Probab. 32 (2004), 757–774. https://doi.org/10.1214/aop/1079021463 doi: 10.1214/aop/1079021463
|
| [6] |
E. Lutwak, D. Yang, G. Zhang, Cramér–Rao and moment-entropy inequalities for Rényi entropy and generalized Fisher information, IEEE Trans. Inform. Theory, 51 (2005), 473–478. https://doi.org/10.1109/TIT.2004.840871 doi: 10.1109/TIT.2004.840871
|
| [7] |
E. Lutwak, D. Yang, G. Zhang, Moment-entropy inequalities for a random vector, IEEE Trans. Inform. Theory, 53 (2007), 1603–1607. https://doi.org/10.1109/TIT.2007.892780 doi: 10.1109/TIT.2007.892780
|
| [8] |
E. Lutwak, S. Lv, D. Yang, G. Zhang, Extensions of Fisher information and Stam's inequality, IEEE Trans. Inform. Theory, 58 (2012), 1319–1327. https://doi.org/10.1109/TIT.2011.2177563 doi: 10.1109/TIT.2011.2177563
|
| [9] |
S. J. Lv, X. Lv, Affine Fisher information inequalities, J. Math. Anal. Appl., 371 (2010), 347–354. https://doi.org/10.1016/j.jmaa.2010.05.034 doi: 10.1016/j.jmaa.2010.05.034
|
| [10] |
A. Cianchi, E. Lutwak, D. Yang, G. Zhang, A unified approach to Cramér–Rao inequalities, IEEE Trans. Inform. Theory, 60 (2014), 643–650. https://doi.org/10.1109/TIT.2013.2284498 doi: 10.1109/TIT.2013.2284498
|
| [11] |
S. J. Lv, General Fisher information matrices of a random vector, Adv. Appl. Math., 89 (2017), 18–40. https://doi.org/10.1016/j.aam.2017.03.002 doi: 10.1016/j.aam.2017.03.002
|
| [12] |
S. Amari, T. Matsuda, Information geometry of Wasserstein statistics on shapes and affine deformations, Inf. Geom., 7 (2024), 285–309. https://doi.org/10.1007/s41884-024-00139-y doi: 10.1007/s41884-024-00139-y
|
| [13] |
D. Langharst, On moment–entropy inequalities in the space of matrices, Electron. J. Probab., 31 (2026), 1–23. https://doi.org/10.1214/26-EJP1552 doi: 10.1214/26-EJP1552
|
| [14] |
P. M. Alberti, A note on the transition probability over $C^*$-algebras, Lett. Math. Phys., 7 (1983), 25–32. https://doi.org/10.1007/BF00398708 doi: 10.1007/BF00398708
|
| [15] | J. Watrous, The theory of quantum information, Cambridge: Cambridge University Press, 2018. https://doi.org/10.1017/9781316848142 |
| [16] |
R. Bhatia, T. Jain, Y. Lim, On the Bures-Wasserstein distance between positive definite matrices, Expo. Math., 37 (2019), 165–191. https://doi.org/10.1016/j.exmath.2018.01.002 doi: 10.1016/j.exmath.2018.01.002
|
| [17] |
Y. Thanwerdas, X. Pennec, The geometry of mixed-Euclidean metrics on symmetric positive definite matrices, Differential Geom. Appl., 81 (2022), 101867. https://doi.org/10.1016/j.difgeo.2022.101867 doi: 10.1016/j.difgeo.2022.101867
|
| [18] | R. Bhatia, Matrix analysis, New York: Springer, 1997. https://doi.org/10.1007/978-1-4612-0653-8 |
Songjun Lv. An affine matrix scalarization of the Cramér–Rao inequality[J]. AIMS Mathematics, 2026, 11(7): 23437-23449. doi: 10.3934/math.2026945
DownLoad: