Asymptotic Distribution of Estimated Affinity between Multiparameter Exponential Families

Let F₁,...,F_J be the distributions of J independent multiparameter exponential families, and _J(F₁,...,F_J) denote the affinity between F₁,...,F_J. We consider the problem of estimating _J on the basis of independent random samples from these distributions. Subject to some mild regularity conditions, we derive the asymptotic distribution of the maximum likelihood estimator of _J. Applications to hypothesis testing and discriminant analysis are discussed, and an example is provided.