Finite Sample Performance of Linear Least Squares Estimators under Sub-Gaussian Martingale Difference Noise

Michael Krikheli, Amir Leshem

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

5 Scopus citations

Abstract

Linear Least Squares is a very well known technique for parameter estimation, which is used even when sub-optimal, because of its very low computational requirements and the fact that exact knowledge of the noise statistics is not required. Surprisingly, bounding the probability of large errors with finitely many samples has been left open, especially when dealing with correlated noise with unknown covariance. In this paper we analyze the finite sample performance of the linear least squares estimator under sub-Gaussian martingale difference noise. In order to analyze this important question we used concentration of measure bounds. When applying these bounds we obtained tight bounds on the tail of the estimator's distribution. We show the fast exponential convergence of the number of samples required to ensure a given accuracy with high probability. We provide probability tail bounds on the estimation error's norm. Our analysis method is simple and uses simple L-{\infty} type bounds on the estimation error. The tightness of the bounds is tested through simulation. The proposed bounds make it possible to predict the number of samples required for least squares estimation even when least squares is sub-optimal and used for computational simplicity. The finite sample analysis of least squares models with this general noise model is novel.

Original languageAmerican English
Title of host publication2018 IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2018 - Proceedings
PublisherInstitute of Electrical and Electronics Engineers Inc.
Pages4444-4448
Number of pages5
ISBN (Print)9781538646588
DOIs
StatePublished - 10 Sep 2018
Externally publishedYes
Event2018 IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2018 - Calgary, Canada
Duration: 15 Apr 201820 Apr 2018

Publication series

NameICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings
Volume2018-April
ISSN (Print)1520-6149

Conference

Conference2018 IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2018
Country/TerritoryCanada
CityCalgary
Period15/04/1820/04/18

Bibliographical note

Publisher Copyright:
© 2018 IEEE.

Keywords

  • Concentration bounds
  • Confidence bounds
  • Estimation
  • Finite sample
  • Large deviations
  • Linear least squares
  • Martingale difference sequence
  • Non-Gaussian

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