## The Annals of Statistics

### Consistency of maximum likelihood estimation for some dynamical systems

#### 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

https://projecteuclid.org/euclid.aos/1416322034

Digital Object Identifier
doi:10.1214/14-AOS1259

Mathematical Reviews number (MathSciNet)
MR3285598

Zentralblatt MATH identifier
1319.37006

#### 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.