9,449 research outputs found
Limit Theorems in Hidden Markov Models
In this paper, under mild assumptions, we derive a law of large numbers, a
central limit theorem with an error estimate, an almost sure invariance
principle and a variant of Chernoff bound in finite-state hidden Markov models.
These limit theorems are of interest in certain ares in statistics and
information theory. Particularly, we apply the limit theorems to derive the
rate of convergence of the maximum likelihood estimator in finite-state hidden
Markov models.Comment: 35 page
Sequential Monte Carlo smoothing for general state space hidden Markov models
Computing smoothing distributions, the distributions of one or more states
conditional on past, present, and future observations is a recurring problem
when operating on general hidden Markov models. The aim of this paper is to
provide a foundation of particle-based approximation of such distributions and
to analyze, in a common unifying framework, different schemes producing such
approximations. In this setting, general convergence results, including
exponential deviation inequalities and central limit theorems, are established.
In particular, time uniform bounds on the marginal smoothing error are obtained
under appropriate mixing conditions on the transition kernel of the latent
chain. In addition, we propose an algorithm approximating the joint smoothing
distribution at a cost that grows only linearly with the number of particles.Comment: Published in at http://dx.doi.org/10.1214/10-AAP735 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org). arXiv admin note: text
overlap with arXiv:1012.4183 by other author
Efficient likelihood estimation in state space models
Motivated by studying asymptotic properties of the maximum likelihood
estimator (MLE) in stochastic volatility (SV) models, in this paper we
investigate likelihood estimation in state space models. We first prove, under
some regularity conditions, there is a consistent sequence of roots of the
likelihood equation that is asymptotically normal with the inverse of the
Fisher information as its variance. With an extra assumption that the
likelihood equation has a unique root for each , then there is a consistent
sequence of estimators of the unknown parameters. If, in addition, the supremum
of the log likelihood function is integrable, the MLE exists and is strongly
consistent. Edgeworth expansion of the approximate solution of likelihood
equation is also established. Several examples, including Markov switching
models, ARMA models, (G)ARCH models and stochastic volatility (SV) models, are
given for illustration.Comment: With the comments by Jens Ledet Jensen and reply to the comments.
Published at http://dx.doi.org/10.1214/009053606000000614;
http://dx.doi.org/10.1214/09-AOS748A; http://dx.doi.org/10.1214/09-AOS748B in
the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Statistics of sums of correlated variables described by a matrix product ansatz
We determine the asymptotic distribution of the sum of correlated variables
described by a matrix product ansatz with finite matrices, considering
variables with finite variances. In cases when the correlation length is
finite, the law of large numbers is obeyed, and the rescaled sum converges to a
Gaussian distribution. In constrast, when correlation extends over system size,
we observe either a breaking of the law of large numbers, with the onset of
giant fluctuations, or a generalization of the central limit theorem with a
family of nonstandard limit distributions. The corresponding distributions are
found as mixtures of delta functions for the generalized law of large numbers,
and as mixtures of Gaussian distributions for the generalized central limit
theorem. Connections with statistical physics models are emphasized.Comment: 6 pages, 1 figur
Convergence and Convergence Rate of Stochastic Gradient Search in the Case of Multiple and Non-Isolated Extrema
The asymptotic behavior of stochastic gradient algorithms is studied. Relying
on results from differential geometry (Lojasiewicz gradient inequality), the
single limit-point convergence of the algorithm iterates is demonstrated and
relatively tight bounds on the convergence rate are derived. In sharp contrast
to the existing asymptotic results, the new results presented here allow the
objective function to have multiple and non-isolated minima. The new results
also offer new insights into the asymptotic properties of several classes of
recursive algorithms which are routinely used in engineering, statistics,
machine learning and operations research
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