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, a global maximum likelihood approach is not appropriate. Hence, an alternative approach is taken whereby a finite local maximum of the likelihood surface is sought. This approach, which is termed the quasi-maximum 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.