## The Annals of Statistics

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

#### 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
Revised: May 2018
First available in Project Euclid: 21 May 2019

https://projecteuclid.org/euclid.aos/1558425634

Digital Object Identifier
doi:10.1214/18-AOS1729

Mathematical Reviews number (MathSciNet)
MR3953439

#### 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

#### References

• [1] Addario-Berry, L., Broutin, N., Devroye, L. and Lugosi, G. (2010). On combinatorial testing problems. Ann. Statist. 38 3063–3092.
• [2] Arias-Castro, E., Candès, E. J. and Durand, A. (2011). Detection of an anomalous cluster in a network. Ann. Statist. 39 278–304.
• [3] Arias-Castro, E., Pelletier, B. and Saligrama, V. (2018). Remember the curse of dimensionality: The case of goodness-of-fit testing in arbitrary dimension. J. Nonparametr. Stat. 30 448–471.
• [4] Balakrishnan, S. and Wasserman, L. (2019). Supplement to “Hypothesis testing for densities and high-dimensional multinomials: Sharp local minimax rates.” DOI:10.1214/18-AOS1729SUPP.
• [5] Balakrishnan, S. and Wasserman, L. (2018). Hypothesis testing for high-dimensional multinomials: A selective review. Ann. Appl. Stat. To appear.
• [6] Barron, A. R. (1989). Uniformly powerful goodness of fit tests. Ann. Statist. 17 107–124.
• [7] Batu, T., Fischer, E., Fortnow, L., Kumar, R., Rubinfeld, R. and White, P. (2001). Testing random variables for independence and identity. In 42nd IEEE Symposium on Foundations of Computer Science (Las Vegas, NV, 2001) 442–451. IEEE Computer Soc., Los Alamitos, CA.
• [8] Berthet, Q. and Rigollet, P. (2013). Optimal detection of sparse principal components in high dimension. Ann. Statist. 41 1780–1815.
• [9] Cai, T. T. and Low, M. G. (2015). A framework for estimation of convex functions. Statist. Sinica 25 423–456.
• [10] Carpentier, A. (2015). Testing the regularity of a smooth signal. Bernoulli 21 465–488.
• [11] Casella, G. and Berger, R. L. (2002). Statistical Inference. Duxbury, Pacific Grove, CA.
• [12] Chatterjee, S. (2014). A new perspective on least squares under convex constraint. Ann. Statist. 42 2340–2381.
• [13] Cramér, H. (1928). On the composition of elementary errors. Scand. Actuar. J. 1928 13–74.
• [14] Devroye, L. and Györfi, L. (1985). Nonparametric Density Estimation: The $L_{1}$ View. Wiley, New York.
• [15] Diaconis, P. and Mosteller, F. (2006). Methods for studying coincidences. In Selected Papers of Frederick Mosteller (S. E. Fienberg and D. C. Hoaglin, eds.) 605–622. Springer, New York.
• [16] Diakonikolas, I. and Kane, D. M. (2016). A new approach for testing properties of discrete distributions. In 57th Annual IEEE Symposium on Foundations of Computer Science—FOCS 2016 685–694. IEEE Computer Soc., Los Alamitos, CA.
• [17] Fienberg, S. E. (1979). The use of chi-squared statistics for categorical data problems. J. Roy. Statist. Soc. Ser. B 41 54–64.
• [18] Giné, E. and Nickl, R. (2016). Mathematical Foundations of Infinite-Dimensional Statistical Models. Cambridge Series in Statistical and Probabilistic Mathematics 40. Cambridge Univ. Press, New York.
• [19] Goldreich, O. and Ron, D. (2011). On testing expansion in bounded-degree graphs. In Studies in Complexity and Cryptography. Lecture Notes in Computer Science 6650 68–75. Springer, Heidelberg.
• [20] Ingster, Y. I. (1990). Minimax detection of a signal in $\ell_{p}$-metrics. J. Math. Sci. 68 503–515.
• [21] Ingster, Y. I. (1997). Adaptive chi-square tests. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 244 150–166, 333.
• [22] Ingster, Y. I. and Suslina, I. A. (2003). Nonparametric Goodness-of-Fit Testing Under Gaussian Models. Lecture Notes in Statistics 169. Springer, New York.
• [23] Ingster, Y. I., Tsybakov, A. B. and Verzelen, N. (2010). Detection boundary in sparse regression. Electron. J. Stat. 4 1476–1526.
• [24] LeCam, L. (1973). Convergence of estimates under dimensionality restrictions. Ann. Statist. 1 38–53.
• [25] Lehmann, E. L. and Casella, G. (1998). Theory of Point Estimation, 2nd ed. Springer, New York.
• [26] Lehmann, E. L. and Romano, J. P. (2005). Testing Statistical Hypotheses, 3rd ed. Springer, New York.
• [27] Marriott, P., Sabolova, R., Van Bever, G. and Critchley, F. (2015). Geometry of goodness-of-fit testing in high dimensional low sample size modelling. In Geometric Science of Information. Lecture Notes in Computer Science 9389 569–576. Springer, Cham.
• [28] Morris, C. (1975). Central limit theorems for multinomial sums. Ann. Statist. 3 165–188.
• [29] Neyman, J. and Pearson, E. S. (1933). On the problem of the most efficient tests of statistical hypotheses. Philos. Trans. Roy. Soc. Lond. Ser. A 231 289–337.
• [30] Nickl, R. and van de Geer, S. (2013). Confidence sets in sparse regression. Ann. Statist. 41 2852–2876.
• [31] Paninski, L. (2008). A coincidence-based test for uniformity given very sparsely sampled discrete data. IEEE Trans. Inform. Theory 54 4750–4755.
• [32] Pearson, K. (1900). On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling. Philos. Mag. Ser. 5 50 157–175.
• [33] Read, T. R. C. and Cressie, N. A. C. (1988). Goodness-of-Fit Statistics for Discrete Multivariate Data. Springer, New York.
• [34] Ron, D. (2008). Property testing: A learning theory perspective. Found. Trends Mach. Learn. 1 307–402.
• [35] Smirnoff, N. (1939). On the estimation of the discrepancy between empirical curves of distribution for two independent samples. Moscow Univ. Math. Bull. 2 3–14.
• [36] Snedecor, G. W. and Cochran, W. G. (1980). Statistical Methods, 7th ed. Iowa State Univ. Press, Ames, IA.
• [37] Valiant, G. and Valiant, P. (2014). An automatic inequality prover and instance optimal identity testing. In 55th Annual IEEE Symposium on Foundations of Computer Science—FOCS 2014 51–60. IEEE Computer Soc., Los Alamitos, CA.
• [38] von Mises, R. (1951). Wahrscheinlichkeit, Statistik und Wahrheit. Springer, Vienna.
• [39] Wilks, S. S. (1938). The large-sample distribution of the likelihood ratio for testing composite hypotheses. Ann. Math. Stat. 9 60–62.

#### 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.