Moment Matrices: Applications in Mixtures

Abstract

The use of moment matrices and their determinants are shown to elucidate the structure of mixture estimation as carried out using the method of moments. The setting is the estimation of a discrete finite support point mixing distribution. In the important class of quadratic variance exponential families it is shown for any sample there is an integer $\hat{\nu}$ depending on the data which represents the maximal number of support points that one can put in the estimated mixing distribution. From this analysis one can derive an asymptotically normal statistic for testing the true number of points in the mixing distribution. In addition, one can construct consistent nonparametric estimates of the mixing distribution for the case when the number of points is unknown or even infinite. The normal model is then examined in more detail, and in particular the case when $\sigma^2$ is unknown is given a comprehensive solution. It is shown how to estimate the parameters in a direct way for every hypothesized number of support points in the mixing distribution, and it is shown how the structure of the problem yields a decomposition of variance into model and error components very similar to the traditional analysis of variance.