The Annals of Statistics

Hypothesis testing for densities and high-dimensional multinomials: Sharp local minimax rates

Sivaraman Balakrishnan and Larry Wasserman

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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.

Article information

Source
Ann. Statist., Volume 47, Number 4 (2019), 1893-1927.

Dates
Received: June 2017
Revised: May 2018
First available in Project Euclid: 21 May 2019

Permanent link to this document
https://projecteuclid.org/euclid.aos/1558425634

Digital Object Identifier
doi:10.1214/18-AOS1729

Mathematical Reviews number (MathSciNet)
MR3953439

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Local-minimax nonparametric goodness-of-fit testing

Citation

Balakrishnan, Sivaraman; Wasserman, Larry. Hypothesis testing for densities and high-dimensional multinomials: Sharp local minimax rates. Ann. Statist. 47 (2019), no. 4, 1893--1927. doi:10.1214/18-AOS1729. https://projecteuclid.org/euclid.aos/1558425634


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Supplemental materials

  • Supplement to “Hypothesis testing for densities and high-dimensional multinomials: Sharp local minimax rates.”. The Supplementary Material contains detailed technical proofs. It also includes a brief study of limiting distributions of the test statistics we study. Finally, the Supplementary Material includes the design and analysis of tests that are adaptive to various parameters.