Multivariate Divergences with Application in Multisample Density Ratio Models

Abstract

We introduce what we will call multivariate divergences between K, $K\ge 1$, signed finite measures $(Q_1,\ldots ,Q_K)$ and a given reference probability measure P on a $\sigma$-field $(\mathcal {X},\mathcal {B})$, extending the well known divergences between two measures, a signed finite measure $Q_1$ and a given probability distribution P. We investigate the Fenchel duality theory for the introduced multivariate divergences viewed as convex functionals on well chosen topological vector spaces of signed finite measures. We obtain new dual representations of these criteria, which we will use to define new family of estimates and test statistics with multiple samples under multiple semiparametric density ratio models. This family contains the estimate and test statistic obtained through empirical likelihood. Moreover, the present approach allows obtaining the asymptotic properties of the estimates and test statistics both under the model and under misspecification. This leads to accurate approximations of the power function for any used criterion, including the empirical likelihood one, which is of its own interest. Moreover, the proposed multivariate divergences can be used, in the context of multiple samples in density ratio models, to define new criteria for model selection and multi-group classification.