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.Comment: Published in at http://dx.doi.org/10.1214/14-AOS1259 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Can heterogeneous agent models explain the alleged mispricing of the S&P 500?

“…However, notwithstanding such limiting behavior, standard asymptotic theory is not routinely applicable in the presence of chaotic attractors because of the long-term dependence chaotic dynamics implies by its very definition. Only very recently have first results been published on the consistency of maximum likelihood estimation for certain classes of chaotic dynamics (McGoff et al, 2015a). While it is unclear whether their technical conditions for consistency would be applicable in our case, their setting is also not exactly the one we have adopted here.…”

A Large Deviation Approach to Posterior Consistency in Dynamical Systems

“…Our work can be seen as a general framework for verifying those assumptions. Additionally, the methods used in both our paper and [58] are similar to the entropy-based argument in [46], which originates from [24].…”

The Bootstrap for Dynamical Systems

“…This is carried out in a variety of fields and has wide ranging applications; for earlier surveys, see [4], [11], [35], [36]; for a recent review with numerous references, see [53]. There is also an increasing trend to study the estimation and prediction in dynamical systems theoretically, and several recent works in this vein include [29], [52], [54], [55], [58], [73]. They present the consistency and/or the rate of convergence in various point estimation or prediction settings.…”

“…They present the consistency and/or the rate of convergence in various point estimation or prediction settings. However, as far as we understand, the limiting distributions of these estimators or predictors have not been studied yet, although some, e.g., [52], [55], indicated determining limiting distributions as a future direction.…”

“…The dynamical system in (1.1) may, as in Remark 2.4 of [52], be viewed as a degenerate Markov chain with one step transition probability (1.5) p(x, A) = 1{g(x) ∈ A} whose Markov operator is, indeed, the Koopman operator of the dynamical system in (1.1). Nevertheless, each one of the Markovian settings in [5], [13], [25], [34], [56], [61], [62], [67] is significantly different from the dynamical system setting, and hence, their results do not apply to our setting.…”