A chi-square type test for covariances.

In this article, we propose a test procedure based on chi-square divergence, suitable to testing hypotheses on the covariances of a measure P, such as ∫ f d P = ∫ f d P ∫ g d P, f and g belonging to given classes of functions H and K. The procedure enters in the range of minimum divergence statistics and relies on convexity and duality properties of the χ2. We use the statistic defined by Broniatowski and Leorato [Broniatowski, M. and Leorato, S., 2006, An estimation method for the Neyman chi-square divergence with application to test of hypotheses. To appear in Journal of Multivariate Analysis, 2006] suitably adapted to the covariance constraints setting. Limiting properties of the test statistic are studied, including convergence in distribution under contiguous alternatives. The method is then applied to tests of independence between two random variables.