PARAMETER ESTIMATION OF EXPONENTIAL HIDDEN MARKOV MODEL AND CONVERGENCE OF ITS PARAMETER ESTIMATOR SEQUENCE
M. Firmasyah, Berlian Setiawaty, I Gusti Putu Purnaba · International Journal of Apllied Mathematics · 2018
An exponential hidden Markov model (EHMM) is a hidden Markov model which consists of a pair of stochastic processes {X t , Y t } t∈N .{Yt } t∈N is influenced by {X t } t∈N , which is assumed to form a Markov chain.{X t } t∈N is not observed.{Y t } t∈N is an observation process and Y t given X t has exponential distribution.In this paper, we estimate the parameter of EHMM and study the convergence of the parameter estimator sequence.EHMM is characterized by a parameter φ = (A, λ) where A is a transition matrix of X t and λ is a vector of parameters of probability density function of Y t given X t .To determine the parameter estimator, a maximum likelihood method is used.Numerical approximation is used through an Expectation Maximization (EM) algorithm.Under the continuous assumption, the sequence {φ (k) } obtained by the EM algorithm, converges to φ * which is the stationary point of ln L t (φ) and the sequence {ln L t (φ (k) )} increasingly converges to ln L t (φ * ).