Semi-parametric inference in a bivariate (multivariate) mixture model

We consider estimation in a bivariate mixture model in which the component distributions can be decomposed into identical distributions. Previous approaches to estimation involve parametrizing the distributions. In this paper, we use a semi-parametric approach. The method is based on the exponential tilt model of Anderson (1979), where the log ratio of probability (density) functions from the bivariate components is linear in the observations. The proposed model does not require training samples, i.e., data with confirmed component membership. We show that in bivariate mixture models, parameters are identifiable. This is in contrast to previous works, where parameters are identifiable if and only if each univariate marginal model is identifiable (Teicher (1967)).