  We propose an inferential approach for maximum likelihood estimation of the
hidden Markov models for continuous responses. We extend to the case of
longitudinal observations the finite mixture model of multivariate Gaussian
distributions with Missing At Random (MAR) outcomes, also accounting for
possible dropout. The resulting hidden Markov model accounts for different
types of missing pattern: (i) partially missing outcomes at a given time
occasion; (ii) completely missing outcomes at a given time occasion
(intermittent pattern); (iii) dropout before the end of the period of
observation (monotone pattern). The MAR assumption is formulated to deal with
the first two types of missingness, while to account for informative dropout we
assume an extra absorbing state. Maximum likelihood estimation of the model
parameters is based on an extended Expectation-Maximization algorithm relying
on suitable recursions. The proposal is illustrated by a Monte Carlo simulation
study and an application based on historical data on primary biliary
cholangitis.
