Asymptotic properties of particle filter-based maximum likelihood estimators for state space models

We study the asymptotic performance of approximate maximum likelihood estimators for state space models obtained via sequential Monte Carlo methods. The state space of the latent Markov chain and the parameter space are assumed to be compact. The approximate estimates are computed by, firstly, running possibly dependent particle filters on a fixed grid in the parameter space, yielding a pointwise approximation of the log-likelihood function. Secondly, extensions of this approximation to the whole parameter space are formed by means of piecewise constant functions or B-spline interpolation, and approximate maximum likelihood estimates are obtained through maximization of the resulting functions. In this setting we formulate criteria for how to increase the number of particles and the resolution of the grid in order to produce estimates that are consistent and asymptotically normal.

[1]  Jun S. Liu,et al.  Blind Deconvolution via Sequential Imputations , 1995 .

[2]  Dan Crisan,et al.  Particle Filters - A Theoretical Perspective , 2001, Sequential Monte Carlo Methods in Practice.

[3]  N. Gordon,et al.  Novel approach to nonlinear/non-Gaussian Bayesian state estimation , 1993 .

[4]  A. V. D. Vaart,et al.  Asymptotic Statistics: U -Statistics , 1998 .

[5]  G. Churchill.  Stochastic models for heterogeneous DNA sequences. , 1989, Bulletin of mathematical biology.

[6]  Biing-Hwang Juang,et al.  Hidden Markov Models for Speech Recognition , 1991 .

[7]  Nando de Freitas,et al.  An Introduction to Sequential Monte Carlo Methods , 2001, Sequential Monte Carlo Methods in Practice.

[8]  R. Douc,et al.  Asymptotic properties of the maximum likelihood estimator in autoregressive models with Markov regime , 2004, math/0503681.

[9]  Brian D. Ripley,et al.  Stochastic Simulation , 2005 .

[10]  D. Cox,et al.  Complex stochastic systems , 2000 .

[11]  Eric Moulines,et al.  On the Convergence of the Monte Carlo Maximum Likelihood Method for Latent Variable Models , 2002 .

[12]  Laurent Mevel,et al.  Exponential Forgetting and Geometric Ergodicity in Hidden Markov Models , 2000, Math. Control. Signals Syst..

[13]  V. V. Petrov.  Limit Theorems of Probability Theory: Sequences of Independent Random Variables , 1995 .

[14]  A. Wald.  Note on the Consistency of the Maximum Likelihood Estimate , 1949 .

[15]  Richard L. Tweedie,et al.  Markov Chains and Stochastic Stability , 1993, Communications and Control Engineering Series.

[16]  Pierre Del Moral,et al.  Feynman-Kac formulae , 2004 .

[17]  T. Lindvall.  Lectures on the Coupling Method , 1992 .

[18]  Hans Kiinsch,et al.  State Space and Hidden Markov Models , 2000 .

[19]  Simon J. Godsill,et al.  On sequential Monte Carlo sampling methods for Bayesian filtering , 2000, Stat. Comput..

[20]  P. Moral,et al.  Branching and interacting particle systems. Approximations of Feynman-Kac formulae with applications to non-linear filtering , 2000 .

[21]  Nando de Freitas,et al.  Sequential Monte Carlo Methods in Practice , 2001, Statistics for Engineering and Information Science.

[22]  P. Moral.  Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications , 2004 .

[23]  L. Schumaker,et al.  Local Spline Approximation Methods , 1975 .

[24]  T. Louis.  Finding the Observed Information Matrix When Using the EM Algorithm , 1982 .

[25]  N. Shephard,et al.  Markov chain Monte Carlo methods for stochastic volatility models , 2002 .

[26]  R. Douc,et al.  Asymptotics of the maximum likelihood estimator for general hidden Markov models , 2001 .

[27]  O. Cappé,et al.  Sequential Monte Carlo smoothing with application to parameter estimation in nonlinear state space models , 2006, math/0609514.

[28]  Arnaud Doucet,et al.  A survey of convergence results on particle filtering methods for practitioners , 2002, IEEE Trans. Signal Process..

[29]  B. Leroux.  Maximum-likelihood estimation for hidden Markov models , 1992 .

[30]  P. Moral,et al.  On the stability of interacting processes with applications to filtering and genetic algorithms , 2001 .

[31]  P. Bickel,et al.  Asymptotic normality of the maximum-likelihood estimator for general hidden Markov models , 1998 .