On the Optimality of Gaussian Kernel Based Nonparametric Tests against Smooth Alternatives

Understand

Nonparametric tests via kernel embedding of distributions have witnessed a great deal of practical successes in recent years.

• However, statistical properties of these tests are largely unknown beyond consistency against a fixed alternative.
• To fill in this void, we study here 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 Gaussian kernel with an appropriately chosen scaling parameter are minimax optimal against smooth alternatives in all three settings.