Non asymptotic estimation lower bounds for LTI state space models with CramérRao and van Trees
Abstract
We study the estimation problem for linear timeinvariant (LTI) statespace models with Gaussian excitation of an unknown covariance. We provide non asymptotic lower bounds for the expected estimation error and the mean square estimation risk of the least square estimator, and the minimax mean square estimation risk. These bounds are sharp with explicit constants when the matrix of the dynamics has no eigenvalues on the unit circle and are rateoptimal when they do. Our results extend and improve existing lower bounds to lower bounds in expectation of the mean square estimation risk and to systems with a general noise covariance. Instrumental to our derivation are new concentration results for rescaled sample covariances and deviation results for the corresponding multiplication processes of the covariates, a differential geometric construction of a prior on the unit operator ball of small Fisher information, and an extension of the CramérRao and van Treesinequalities to matrixvalued estimators.
 Publication:

arXiv eprints
 Pub Date:
 September 2021
 arXiv:
 arXiv:2109.08582
 Bibcode:
 2021arXiv210908582D
 Keywords:

 Mathematics  Statistics Theory;
 Mathematics  Probability;
 Statistics  Machine Learning;
 62J05;
 62M10;
 60B20;
 60E15;
 62C20;
 62B11
 EPrint:
 41 pages