Testing multivariate hypotheses with positive definite bilinear forms

In time series analysis, tests for independence, symmetry, and goodness-of-fit based on divergence measures, such as the Kullback-Leibler divergence or Hellinger distance are currently receiving much interest.We consider replacing the divergence measures in these tests by kernel-based positive definite bilinear forms. By doing so, we can use U-statistics estimators for the functional of interest (the divergence between two distributions). In this way we avoid the common practice of using plug-in estimators. In addition, our approach separates the problem of consistent estimation of the divergence measure from that of estimating the underlying joint densities consistently. The approach demonstrates why in a testing context, the bandwidth may tend to zero with the sample size slower than required for consistent density estimation. Our results are illustrated with simulations