Estimators and D-optimal experimental designs for mixtures of binary responses

Models of mixture distributions are of great interest in many areas where several populations are mixed up. If the response is binary there is a mixture of Bernoulli distributions, which has practical applications for the classification of texts and images, biochemistry, genetics, robotics, computer science or pattern recognition. A finite mixture of probability distributions includes a set of parameters such as the proportion of the mixture and the parameters of each distribution. This simple case allows us to make some explicit computations of the estimators as well as working on the EM algorithm doing some comparisons. Some of these parameters may depend on one or more covariates through some specific model. Mixtures of exponential family distributions, except in very particular cases, are no longer within the exponential family and this means, among other things, that the expectation for computing the Information Matrix must, in most cases, be approximated. One of the main contributions of this article is its handling of this. Linear and logistic models are considered either for the proportion of one of the two populations (clusters), or for the parameters of the Bernoulli distributions. For each of these cases the analytic expression of the information matrix is calculated and optimal designs determined.