# Non asymptotic estimation lower bounds for LTI state space models with Cram\'er-Rao and van Trees

@inproceedings{Djehiche2021NonAE, title={Non asymptotic estimation lower bounds for LTI state space models with Cram\'er-Rao and van Trees}, author={Boualem Djehiche and Othmane Mazhar}, year={2021} }

We study the estimation problem for linear time-invariant (LTI) state-space 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 rate-optimal when they do. Our results extend and…

## One Citation

### Finite-Sample Analysis of Identification of Switched Linear Systems With Arbitrary or Restricted Switching

- Mathematics, Computer ScienceIEEE Control Systems Letters
- 2022

To capture the effect of the parameters of the switching strategies on the LS estimation error, finite-sample error bounds are developed in this letter and show that in the presence of unstable modes, the switching strategy should be properly designed to avoid the significant increase of the estimation error.

## References

SHOWING 1-10 OF 21 REFERENCES

### Near optimal finite time identification of arbitrary linear dynamical systems

- MathematicsICML
- 2019

This work derives finite time error bounds for estimating general linear time-invariant (LTI) systems from a single observed trajectory using the method of least squares and demonstrates that the least squares solution may be statistically inconsistent under certain conditions even when the signal-to-noise ratio is high.

### Finite impulse response models: A non-asymptotic analysis of the least squares estimator

- Mathematics
- 2019

We consider a finite impulse response system with centered independent sub-Gaussian design covariates and noise components that are not necessarily identically distributed. We derive non-asymptotic…

### Sample Complexity Lower Bounds for Linear System Identification

- Computer Science, Mathematics2019 IEEE 58th Conference on Decision and Control (CDC)
- 2019

This paper establishes problem-specific sample complexity lower bounds for linear system identification problems, and really captures the identification hardness specific to the system.

### Learning Without Mixing: Towards A Sharp Analysis of Linear System Identification

- MathematicsCOLT
- 2018

It is proved that the ordinary least-squares (OLS) estimator attains nearly minimax optimal performance for the identification of linear dynamical systems from a single observed trajectory, and generalizes the technique to provide bounds for a more general class of linear response time-series.

### Statistical estimation : asymptotic theory

- Mathematics
- 1981

when certain parameters in the problem tend to limiting values (for example, when the sample size increases indefinitely, the intensity of the noise ap proaches zero, etc.) To address the problem of…

### Upper and Lower Bounds for Stochastic Processes: Modern Methods and Classical Problems

- Mathematics
- 2014

0. Introduction.- 1. Philosophy and Overview of the Book.- 2. Gaussian Processes and the Generic Chaining.- 3. Random Fourier Series and Trigonometric Sums, I. - 4. Matching Theorems I.- 5.…

### Applications of the van Trees inequality : a Bayesian Cramr-Rao bound

- Mathematics
- 1995

We use a Bayesian version of the Cramer-Rao lower bound due to van Trees to give an elementary proof that the limiting distibution of any regular estimator cannot have a variance less than the…

### High-Dimensional Probability

- Computer Science
- 2018

A broad range of illustrations is embedded throughout, including classical and modern results for covariance estimation, clustering, networks, semidefinite programming, coding, dimension reduction, matrix completion, machine learning, compressed sensing, and sparse regression.