A new approximate likelihood estimator for ARMA-filtered hidden Markov models

Hidden Markov models (HMM’s) are successfully applied in various fields of time series analysis. Colored noise, e.g., due to filtering, violates basic assumptions of the model. Although it is well known how to consider autoregressive (AR) filtering, there is no algorithm to take into account moving-average (MA) filtering in parameter estimation exactly. We present an approximate likelihood estimator for MA-filtered HMM that is generalized to deal with an autoregressive moving-average (ARMA) filtered HMM. The approximation order of the likelihood calculation can be chosen. Therefore, we obtain a sequence of estimators for the HMM parameters as well as for the filter coefficients. The recursion equations for an efficient algorithm are derived from exact expressions for the forward iterations. By simulations, we show that the derived estimators are unbiased in filter situations where standard HMM’s are not able to recover the true dynamics. Special implementation strategies together with small approximations yield further acceleration of the algorithm.