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Analyticity, Convergence, and Convergence Rate
of Recursive Maximum-Likelihood Estimation
in Hidden Markov Models
by Vladislav B. Tadic
This paper considers the asymptotic properties of
the recursive maximum-likelihood estimator for hidden Markov
models. The paper is focused on the analytic properties of the
asymptotic log-likelihood and on the point-convergence and convergence
rate of the recursive maximum-likelihood estimator.
Using the principle of analytic continuation, the analyticity of the
asymptotic log-likelihood is shown for analytically parameterized
hidden Markov models. Relying on this fact and some results
from differential geometry (Lojasiewicz inequality), the almost
sure point convergence of the recursive maximum-likelihood
algorithm is demonstrated, and relatively tight bounds on the
convergence rate are derived. As opposed to the existing result
on the asymptotic behavior of maximum-likelihood estimation
in hidden Markov models, the results of this paper are obtained
without assuming that the log-likelihood function has an isolated
maximum at which the Hessian is strictly negative definite.
Keywords: Analyticity, convergence rate, hidden Markov
models, Lojasiewicz inequality, maximum-likelihood estimation,
point convergence, recursive identification.
Full text of the paper (pdf),
which has recently appeared in IEEE Transactions On Information Theory, Vol. 56, No. 12, December 2010.
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