Parameter estimation: known vector signals in unknown Gaussian noise

摘要

This paper develops recursive, convergent estimators for the parameters of finite Gaussian mixtures with a common covariance matrix. The mean vectors (signals) of the component densities are assumed to be known. The motivation for the study stems from digital communication. The basic approach is first illustrated for the case of an independent identically distributed sequence of samples from a univariate mixture of M classes (symbols). This is accomplished through the development of a convergent stochastic approximation form of estimator for the common variance value. The asymptotic variance of the estimated variance is derived. A batch processing alternative that possesses a sufficient statistic is developed for the case of a fixed size sample set. Three generalizations are studied. The first extends from the case of the univariate data to multivariate data. The second generalization allows for the statistical dependence of successive vector signals. Finally, the case of dependent successive vector signals along with dependent successive additive noise vectors is treated. In each case, convergent estimators for all unknown parameters are developed. Many cases are illustrated with simulation experiments. Results presented are applicable to communication engineering, pattern recognition, and some special image processing problems.