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August 2019 Hypothesis testing for densities and high-dimensional multinomials: Sharp local minimax rates
Sivaraman Balakrishnan, Larry Wasserman
Ann. Statist. 47(4): 1893-1927 (August 2019). DOI: 10.1214/18-AOS1729

Abstract

We consider the goodness-of-fit testing problem of distinguishing whether the data are drawn from a specified distribution, versus a composite alternative separated from the null in the total variation metric. In the discrete case, we consider goodness-of-fit testing when the null distribution has a possibly growing or unbounded number of categories. In the continuous case, we consider testing a Hölder density with exponent $0<s\leq 1$, with possibly unbounded support, in the low-smoothness regime where the Hölder parameter is not assumed to be constant. In contrast to existing results, we show that the minimax rate and critical testing radius in these settings depend strongly, and in a precise way, on the null distribution being tested and this motivates the study of the (local) minimax rate as a function of the null distribution. For multinomials, the local minimax rate has been established in recent work. We revisit and extend these results and develop two modifications to the $\chi^{2}$-test whose performance we characterize. For testing Hölder densities, we show that the usual binning tests are inadequate in the low-smoothness regime and we design a spatially adaptive partitioning scheme that forms the basis for our locally minimax optimal tests. Furthermore, we provide the first local minimax lower bounds for this problem which yield a sharp characterization of the dependence of the critical radius on the null hypothesis being tested. In the low-smoothness regime, we also provide adaptive tests that adapt to the unknown smoothness parameter. We illustrate our results with a variety of simulations that demonstrate the practical utility of our proposed tests.

Citation

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Sivaraman Balakrishnan. Larry Wasserman. "Hypothesis testing for densities and high-dimensional multinomials: Sharp local minimax rates." Ann. Statist. 47 (4) 1893 - 1927, August 2019. https://doi.org/10.1214/18-AOS1729

Information

Received: 1 June 2017; Revised: 1 May 2018; Published: August 2019
First available in Project Euclid: 21 May 2019

zbMATH: 07082274
MathSciNet: MR3953439
Digital Object Identifier: 10.1214/18-AOS1729

Subjects:
Primary: 60K35

Keywords: Local-minimax , nonparametric goodness-of-fit testing

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.47 • No. 4 • August 2019
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