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Nonstationary Gauss-Markov Processes: Parameter Estimation and Dispersion

Tian, Peida and Kostina, Victoria (2021) Nonstationary Gauss-Markov Processes: Parameter Estimation and Dispersion. IEEE Transactions on Information Theory, 67 (4). pp. 2426-2449. ISSN 0018-9448. https://resolver.caltech.edu/CaltechAUTHORS:20210211-151615781

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Abstract

This paper provides a precise error analysis for the maximum likelihood estimate â_(ML)(uⁿ₁) of the parameter a given samples uⁿ₁ = (u₁, ... , u_n)ʹ drawn from a nonstationary Gauss-Markov process U_i = aU_(i-1) + Z)_i, i ≥ 1, where U₀ = 0, a > 1, and Zi ’s are independent Gaussian random variables with zero mean and variance σ². We show a tight nonasymptotic exponentially decaying bound on the tail probability of the estimation error. Unlike previous works, our bound is tight already for a sample size of the order of hundreds. We apply the new estimation bound to find the dispersion for lossy compression of nonstationary Gauss-Markov sources. We show that the dispersion is given by the same integral formula that we derived previously for the asymptotically stationary Gauss-Markov sources, i.e., |a|<1 . New ideas in the nonstationary case include separately bounding the maximum eigenvalue (which scales exponentially) and the other eigenvalues (which are bounded by constants that depend only on a) of the covariance matrix of the source sequence, and new techniques in the derivation of our estimation error bound.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1109/tit.2021.3050342DOIArticle
https://arxiv.org/abs/1907.00304arXivDiscussion Paper
https://resolver.caltech.edu/CaltechAUTHORS:20191004-100332992Related ItemConference Paper
ORCID:
AuthorORCID
Tian, Peida0000-0003-3665-8173
Kostina, Victoria0000-0002-2406-7440
Alternate Title:From Parameter Estimation to Dispersion of Nonstationary Gauss-Markov Processes
Additional Information:© 2021 IEEE. Manuscript received June 29, 2019; revised August 7, 2020; accepted December 22, 2020. Date of publication February 9, 2021; date of current version March 18, 2021. This work was supported in part by the National Science Foundation (NSF) under Grant CCF-1751356. This article was presented at the 2019 IEEE International Symposium on Information Theory.
Funders:
Funding AgencyGrant Number
NSFCCF-1751356
Subject Keywords:Parameter estimation, maximum likelihood estimator, unstable processes, finite blocklength analysis, lossy compression, sources with memory, rate-distortion theory, system identification, covering in stochastic processes, adaptive control
Issue or Number:4
Record Number:CaltechAUTHORS:20210211-151615781
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20210211-151615781
Official Citation:P. Tian and V. Kostina, "Nonstationary Gauss-Markov Processes: Parameter Estimation and Dispersion," in IEEE Transactions on Information Theory, vol. 67, no. 4, pp. 2426-2449, April 2021, doi: 10.1109/TIT.2021.3050342
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:108012
Collection:CaltechAUTHORS
Deposited By: George Porter
Deposited On:11 Feb 2021 23:49
Last Modified:29 Mar 2021 14:35

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