  Online (also called "recursive" or "adaptive") estimation of fixed model
parameters in hidden Markov models is a topic of much interest in times series
modelling. In this work, we propose an online parameter estimation algorithm
that combines two key ideas. The first one, which is deeply rooted in the
Expectation-Maximization (EM) methodology consists in reparameterizing the
problem using complete-data sufficient statistics. The second ingredient
consists in exploiting a purely recursive form of smoothing in HMMs based on an
auxiliary recursion. Although the proposed online EM algorithm resembles a
classical stochastic approximation (or Robbins-Monro) algorithm, it is
sufficiently different to resist conventional analysis of convergence. We thus
provide limited results which identify the potential limiting points of the
recursion as well as the large-sample behavior of the quantities involved in
the algorithm. The performance of the proposed algorithm is numerically
evaluated through simulations in the case of a noisily observed Markov chain.
In this case, the algorithm reaches estimation results that are comparable to
that of the maximum likelihood estimator for large sample sizes.
