## Abstract

We develop a non-asymptotic framework for hypothesis testing in nonparametric regression where the true regression function belongs to a Sobolev space. Our statistical guarantees are exact in the sense that Type I and II errors are controlled for any finite sample size. Meanwhile, one proposed test is shown to achieve minimax rate optimality in the asymptotic sense. An important consequence of this non-asymptotic theory is a new and practically useful formula for selecting the optimal smoothing parameter in the testing statistic. Extensions of our results to general reproducing kernel Hilbert spaces and non-Gaussian error regression are also discussed.

Original language | English (US) |
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Pages (from-to) | 3709-3755 |

Number of pages | 47 |

Journal | Proceedings of Machine Learning Research |

Volume | 125 |

State | Published - 2020 |

Event | 33rd Conference on Learning Theory, COLT 2020 - Virtual, Online, Austria Duration: Jul 9 2020 → Jul 12 2020 |

## All Science Journal Classification (ASJC) codes

- Artificial Intelligence
- Software
- Control and Systems Engineering
- Statistics and Probability

## Keywords

- Kernel ridge regression
- large deviation bound
- minimax rate optimality
- non-asymptotic inference
- nonparametric testing
- smoothing spline