Universal Hypothesis Testing with Kernels: Asymptotically Optimal Tests for Goodness of Fit
share
We characterize the asymptotic performance of nonparametric goodness of fit testing, otherwise known as the universal hypothesis testing that dates back to Hoeffding (1965). The exponential decay rate of the type-II error probability is used as the asymptotic performance metric, hence an optimal test achieves the maximum decay rate subject to a constant level constraint on the type-I error probability. We show that two classes of Maximum Mean Discrepancy (MMD) based tests attain this optimality on R^d, while a Kernel Stein Discrepancy (KSD) based test achieves a weaker one under this criterion. In the finite sample regime, these tests have similar statistical performance in our experiments, while the KSD based test is more computationally efficient. Key to our approach are Sanov's theorem from large deviation theory and recent results on the weak convergence properties of the MMD and KSD.
READ FULL TEXT
