Goodness-of-Fit Testing for Hölder-Continuous Densities: Sharp Local Minimax Rates

09/09/2021
by   Julien Chhor, et al.
0

We consider the goodness-of fit testing problem for Hölder smooth densities over ℝ^d: given n iid observations with unknown density p and given a known density p_0, we investigate how large ρ should be to distinguish, with high probability, the case p=p_0 from the composite alternative of all Hölder-smooth densities p such that p-p_0_t ≥ρ where t ∈ [1,2]. The densities are assumed to be defined over ℝ^d and to have Hölder smoothness parameter α>0. In the present work, we solve the case α≤ 1 and handle the case α>1 using an additional technical restriction on the densities. We identify matching upper and lower bounds on the local minimax rates of testing, given explicitly in terms of p_0. We propose novel test statistics which we believe could be of independent interest. We also establish the first definition of an explicit cutoff u_B allowing us to split ℝ^d into a bulk part (defined as the subset of ℝ^d where p_0 takes only values greater than or equal to u_B) and a tail part (defined as the complementary of the bulk), each part involving fundamentally different contributions to the local minimax rates of testing.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
06/30/2017

Hypothesis Testing For Densities and High-Dimensional Multinomials: Sharp Local Minimax Rates

We consider the goodness-of-fit testing problem of distinguishing whethe...
research
12/26/2020

Sharp Local Minimax Rates for Goodness-of-Fit Testing in Large Random Graphs, multivariate Poisson families and multinomials

We consider the identity testing problem in large inhomogeneous random g...
research
07/06/2021

Goodness-of-fit testing for Hölder continuous densities under local differential privacy

We address the problem of goodness-of-fit testing for Hölder continuous ...
research
12/03/2017

The local geometry of testing in ellipses: Tight control via localized Kolomogorov widths

We study the local geometry of testing a mean vector within a high-dimen...
research
12/03/2017

The local geometry of testing in ellipses: Tight control via localized Kolmogorov widths

We study the local geometry of testing a mean vector within a high-dimen...
research
11/06/2019

Minimax Nonparametric Two-sample Test

We consider the problem of comparing probability densities between two g...
research
10/19/2022

A kernel Stein test of goodness of fit for sequential models

We propose a goodness-of-fit measure for probability densities modeling ...

Please sign up or login with your details

Forgot password? Click here to reset