In this paper, using Band matrix, a method has been proposed to estimating the covariance matrix of the ARMA model and the likelihood function of the ARMA model with diagonal covariance matrix has been obtained and approximations for Kullback-Leibler and Chernoff criteria were presented. In addition, two rules for discriminating the ARMA models has been proposed. A simulation and real data sets are used to illustrate the performance of the proposed rules. Significant reduction of the calculations for large time series and low discrimination error rate are two characteristics of the proposed rules. In addition no need to normal assumption is showed in a theorem.

‎This paper is investigating the mixture autoregressive model with constant mixing weights in state space form and generalization to ARMA mixture model‎. ‎Using a sequential Monte Carlo method‎, ‎the forecasting‎, ‎filtering and smoothing distributions are approximated and parameters f the model is estimated via the EM algorithm‎. ‎The results show the dimension of parameter vector in state space representation reduces‎. ‎The results of the simulation study show that the proposed filtering algorithm has a steady state close to the real values of the state vector‎. ‎Moreover‎, ‎according to simulation results‎, ‎the mean vectors of filtering and smoothing distribution converges to state vector quickly‎.