Parameter estimation for autoregressive Gaussian-mixture processes: the EMAX algorithm

The problem of estimating parameters of discrete-time non-Gaussian autoregressive (AR) processes is addressed. The subclass of such processes considered is restricted to those whose driving noise samples are statistically independent and identically distributed according to a Gaussian-mixture probability density function (pdf). Because the likelihood function for this problem is typically unbounded in the vicinity of undesirable, degenerate parameter estimates, the maximum likelihood approach is not fruitful. Hence, an alternative approach is taken whereby a finite local maximum of the likelihood surface is sought. This approach, which is termed the quasimaximum likelihood (QML) approach, is used to obtain estimates of the AR parameters as well as the means, variances, and weighting coefficients that define the Gaussian-mixture pdf. A technique for generating solutions to the QML problem is derived using a generalized version of the expectation-maximization principle. This technique, which is referred to as the EMAX algorithm, is applied in four illustrative examples; its performance is compared directly with that of previously proposed algorithms based on the same data model and that of conventional least-squares techniques.