Kernel embedding methods have witnessed a great deal of practical success in the area of nonparametric hypothesis testing in recent years. But ever since its first proposal, there exists an inevitable problem that researchers in this area have been trying to answer--what kernel should be selected, because the performance of the associated nonparametric tests can vary dramatically with different kernels. While the way of kernel selection is usually ad hoc, we wonder if there exists a principled way of kernel selection so as to ensure that the associated nonparametric tests have good performance. As consistency results against fixed alternatives do not tell the full story about the power of the associated tests, we study their statistical performance within the minimax framework. First, focusing on the case of goodness-of-fit tests, our analyses show that a vanilla version of the kernel embedding based test could be suboptimal, and suggest a simple remedy by moderating the kernel. We prove that the moderated approach provides optimal tests for a wide range of deviations from the null and can also be made adaptive over a large collection of interpolation spaces.

Then, we study the asymptotic properties of goodness-of-fit, homogeneity and independence tests using Gaussian kernels, arguably the most popular and successful among such tests. Our results provide theoretical justifications for this common practice by showing that tests using a Gaussian kernel with an appropriately chosen scaling parameter are minimax optimal against smooth alternatives in all three settings. In addition, our analysis also pinpoints the importance of choosing a diverging scaling parameter when using Gaussian kernels and suggests a data-driven choice of the scaling parameter that yields tests optimal, up to an iterated logarithmic factor, over a wide range of smooth alternatives. Numerical experiments are presented to further demonstrate the practical merits of our methodology.