A modified grid method for statistical separation of normal variance-mean mixtures

A modified two-stage grid method for statistical separation of normal variance-mean mixtures is described as an alternative to a pure EM (expectation-maximization) algorithm. At the first stage of this algorithm, a discrete approximation is constructed to the mixing distribution. At the second stage, the obtained discrete distribution is approximated by an absolutely continuous distribution from a predetermined family, say, by a generalized inverse Gaussian distribution. The convergence of this two-stage procedure is discussed. The monotonicity of the grid procedure used at the first stage is proved. The problem of the optimal choice of the parameters of the method is discussed in detail. First of all, the problem of the optimal choice of the grid thrown on the support of the mixing distribution is considered. Statistical estimators are proposed for the quantiles of the mixing law. The efficiency of the method is illustrated by examples of its application to the estimation of the parameters of generalized hyperbolic distributions.