## Abstract

We characterize the asymptotic performance of nonparametric goodness of fit testing. The exponential decay rate of the type-II error probability is used as the asymptotic performance metric, and a test is optimal if it achieves the maximum 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 the quadratic-time Kernel Stein Discrepancy (KSD) based tests achieve the maximum exponential decay rate under a relaxed level constraint. Under the same performance metric, we proceed to show that the quadratic-time MMD based two-sample tests are also optimal for general two-sample problems, provided that kernels are bounded continuous and characteristic. Key to our approach are Sanov's theorem from large deviation theory and the weak metrizable properties of the MMD and KSD.

Original language | English (US) |
---|---|

State | Published - 2020 |

Event | 22nd International Conference on Artificial Intelligence and Statistics, AISTATS 2019 - Naha, Japan Duration: Apr 16 2019 → Apr 18 2019 |

### Conference

Conference | 22nd International Conference on Artificial Intelligence and Statistics, AISTATS 2019 |
---|---|

Country | Japan |

City | Naha |

Period | 4/16/19 → 4/18/19 |

## ASJC Scopus subject areas

- Artificial Intelligence
- Statistics and Probability