The Annals of Statistics

Consistency of maximum likelihood estimation for some dynamical systems

Kevin McGoff, Sayan Mukherjee, Andrew Nobel, and Natesh Pillai

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Abstract

We consider the asymptotic consistency of maximum likelihood parameter estimation for dynamical systems observed with noise. Under suitable conditions on the dynamical systems and the observations, we show that maximum likelihood parameter estimation is consistent. Our proof involves ideas from both information theory and dynamical systems. Furthermore, we show how some well-studied properties of dynamical systems imply the general statistical properties related to maximum likelihood estimation. Finally, we exhibit classical families of dynamical systems for which maximum likelihood estimation is consistent. Examples include shifts of finite type with Gibbs measures and Axiom A attractors with SRB measures.

Article information

Source
Ann. Statist., Volume 43, Number 1 (2015), 1-29.

Dates
First available in Project Euclid: 18 November 2014

Permanent link to this document
https://projecteuclid.org/euclid.aos/1416322034

Digital Object Identifier
doi:10.1214/14-AOS1259

Mathematical Reviews number (MathSciNet)
MR3285598

Zentralblatt MATH identifier
1319.37006

Subjects
Primary: 37A50: Relations with probability theory and stochastic processes [See also 60Fxx and 60G10] 37A25: Ergodicity, mixing, rates of mixing 62B10: Information-theoretic topics [See also 94A17] 62F12: Asymptotic properties of estimators 62M09: Non-Markovian processes: estimation
Secondary: 37D20: Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) 60F10: Large deviations 62M05: Markov processes: estimation 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 94A17: Measures of information, entropy

Keywords
Dynamical systems hidden Markov models maximum likelihood estimation strong consistency

Citation

McGoff, Kevin; Mukherjee, Sayan; Nobel, Andrew; Pillai, Natesh. Consistency of maximum likelihood estimation for some dynamical systems. Ann. Statist. 43 (2015), no. 1, 1--29. doi:10.1214/14-AOS1259. https://projecteuclid.org/euclid.aos/1416322034


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Supplemental materials

  • Supplementary material: Supplement to “Consistency of maximum likelihood estimation for some dynamical systems”. We provide three technical appendices. In Appendix A, we present proofs of Propositions 4.1, 4.2 and 4.3. In Appendix B, we discuss shifts of finite type and Gibbs measures and prove Theorem 5.1. Finally, Appendix C contains definitions for Axiom A systems, as well as a proof of Theorem 5.7.