Separating mixtures of probability distributions with the grid method of moments and the grid maximal likelihood method

We consider the so-called grid methods for approximate statistical separation of probability distribution mixtures based on (i) minimizing the disparity between theoretical and empirical moments and (ii) maximizing the grid likelihood function. We show that problems of type (i) can be reduced to linear programming problems. For a numerical solution of problems of type (ii) we offer the "truncated" EM algorithm and the conditional gradient algorithm. We show results of a comparative study of the suggested approaches' efficiency based on solving the decomposition problem for the volatility function of financial indices. We give examples of volatility decompositions for the CAC 40 index.