The Annals of Statistics

The geometry of hypothesis testing over convex cones: Generalized likelihood ratio tests and minimax radii

Yuting Wei, Martin J. Wainwright, and Adityanand Guntuboyina

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Abstract

We consider a compound testing problem within the Gaussian sequence model in which the null and alternative are specified by a pair of closed, convex cones. Such cone testing problem arises in various applications, including detection of treatment effects, trend detection in econometrics, signal detection in radar processing and shape-constrained inference in nonparametric statistics. We provide a sharp characterization of the GLRT testing radius up to a universal multiplicative constant in terms of the geometric structure of the underlying convex cones. When applied to concrete examples, this result reveals some interesting phenomena that do not arise in the analogous problems of estimation under convex constraints. In particular, in contrast to estimation error, the testing error no longer depends purely on the problem complexity via a volume-based measure (such as metric entropy or Gaussian complexity); other geometric properties of the cones also play an important role. In order to address the issue of optimality, we prove information-theoretic lower bounds for the minimax testing radius again in terms of geometric quantities. Our general theorems are illustrated by examples including the cases of monotone and orthant cones, and involve some results of independent interest.

Article information

Source
Ann. Statist., Volume 47, Number 2 (2019), 994-1024.

Dates
Received: April 2017
Revised: March 2018
First available in Project Euclid: 11 January 2019

Permanent link to this document
https://projecteuclid.org/euclid.aos/1547197246

Digital Object Identifier
doi:10.1214/18-AOS1701

Mathematical Reviews number (MathSciNet)
MR3909958

Zentralblatt MATH identifier
07033159

Subjects
Primary: 62F03: Hypothesis testing
Secondary: 52A05: Convex sets without dimension restrictions

Keywords
Hypothesis testing closed convex cone likelihood ratio test minimax rate Gaussian complexity

Citation

Wei, Yuting; Wainwright, Martin J.; Guntuboyina, Adityanand. The geometry of hypothesis testing over convex cones: Generalized likelihood ratio tests and minimax radii. Ann. Statist. 47 (2019), no. 2, 994--1024. doi:10.1214/18-AOS1701. https://projecteuclid.org/euclid.aos/1547197246


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References

  • [1] Amelunxen, D., Lotz, M., McCoy, M. B. and Tropp, J. A. (2014). Living on the edge: Phase transitions in convex programs with random data. Inf. Inference 3 224–294.
  • [2] Baraud, Y. (2002). Non-asymptotic minimax rates of testing in signal detection. Bernoulli 8 577–606.
  • [3] Barlow, R. E., Bartholomew, D. J., Bremner, J. M. and Brunk, H. D. (1972). Statistical Inference Under Order Restrictions. The Theory and Application of Isotonic Regression. Wiley, New York.
  • [4] Bartholomew, D. J. (1959). A test of homogeneity for ordered alternatives. Biometrika 46 36–48.
  • [5] Bartholomew, D. J. (1959). A test of homogeneity for ordered alternatives. II. Biometrika 46 328–335.
  • [6] Bartlett, P. L., Bousquet, O. and Mendelson, S. (2005). Local Rademacher complexities. Ann. Statist. 33 1497–1537.
  • [7] Batu, T., Kumar, R. and Rubinfeld, R. (2004). Sublinear algorithms for testing monotone and unimodal distributions. In Proceedings of the 36th Annual ACM Symposium on Theory of Computing 381–390. ACM, New York.
  • [8] Besson, O. (2006). Adaptive detection of a signal whose signature belongs to a cone. In Proceedings SAM Conference.
  • [9] Bühlmann, P. (2013). Statistical significance in high-dimensional linear models. Bernoulli 19 1212–1242.
  • [10] Chatterjee, S. (2014). A new perspective on least squares under convex constraint. Ann. Statist. 42 2340–2381.
  • [11] Chen, D. and Plemmons, R. J. (2010). Nonnegativity constraints in numerical analysis. In The Birth of Numerical Analysis 109–139. World Sci. Publ., Hackensack, NJ.
  • [12] Chetverikov, D. (2012). Testing regression monotonicity in econometric models Technical report, UCLA. Available at arXiv:1212.6757.
  • [13] Dykstra, R. L. and Robertson, T. (1983). On testing monotone tendencies. J. Amer. Statist. Assoc. 78 342–350.
  • [14] Edwards, A. W. F. (1984). Likelihood. Cambridge Univ. Press, Cambridge. Reprint of the 1972 original.
  • [15] Ermakov, M. S. (1991). Minimax detection of a signal in Gaussian white noise. Theory Probab. Appl. 35 667–679.
  • [16] Fan, J. and Jiang, J. (2007). Nonparametric inference with generalized likelihood ratio tests. TEST 16 409–444.
  • [17] Fan, J., Zhang, C. and Zhang, J. (2001). Generalized likelihood ratio statistics and Wilks phenomenon. Ann. Statist. 29 153–193.
  • [18] Greco, M., Gini, F. and Farina, A. (2008). Radar detection and classification of jamming signals belonging to a cone class. IEEE Trans. Signal Process. 56 1984–1993.
  • [19] Hu, X. and Wright, F. T. (1994). Likelihood ratio tests for a class of nonoblique hypotheses. Ann. Inst. Statist. Math. 46 137–145.
  • [20] Ingster, Yu. I. (1987). A minimax test of nonparametric hypotheses on the density of a distribution in $L_{p}$ metrics. Theory Probab. Appl. 31 333–337.
  • [21] Ingster, Yu. I. and Suslina, I. A. (2003). Nonparametric Goodness-of-Fit Testing Under Gaussian Models. Lecture Notes in Statistics 169. Springer, New York.
  • [22] Koltchinskii, V. (2001). Rademacher penalties and structural risk minimization. IEEE Trans. Inform. Theory 47 1902–1914.
  • [23] Kudô, A. (1963). A multivariate analogue of the one-sided test. Biometrika 50 403–418.
  • [24] Lehmann, E. L. (2012). On likelihood ratio tests. In Selected Works of E. L. Lehmann (J. Rojo, ed.) 209–216. Springer, New York.
  • [25] Lehmann, E. L. and Romano, J. P. (2005). Testing Statistical Hypotheses, 3rd ed. Springer, New York.
  • [26] Lepski, O. V. and Spokoiny, V. G. (1999). Minimax nonparametric hypothesis testing: The case of an inhomogeneous alternative. Bernoulli 5 333–358.
  • [27] Lepski, O. V. and Tsybakov, A. B. (2000). Asymptotically exact nonparametric hypothesis testing in sup-norm and at a fixed point. Probab. Theory Related Fields 117 17–48.
  • [28] McCoy, M. B. and Tropp, J. A. (2014). From Steiner formulas for cones to concentration of intrinsic volumes. Discrete Comput. Geom. 51 926–963.
  • [29] Meinshausen, N. (2013). Sign-constrained least squares estimation for high-dimensional regression. Electron. J. Stat. 7 1607–1631.
  • [30] Menéndez, J. A., Rueda, C. and Salvador, B. (1992). Dominance of likelihood ratio tests under cone constraints. Ann. Statist. 20 2087–2099.
  • [31] Menéndez, J. A., Rueda, C. and Salvador, B. (1992). Testing nonoblique hypotheses. Comm. Statist. Theory Methods 21 471–484.
  • [32] Menéndez, J. A. and Salvador, B. (1991). Anomalies of the likelihood ratio tests for testing restricted hypotheses. Ann. Statist. 19 889–898.
  • [33] Meyer, M. C. (2003). A test for linear versus convex regression function using shape-restricted regression. Biometrika 90 223–232.
  • [34] Moreau, J.-J. (1962). Décomposition orthogonale d’un espace hilbertien selon deux cônes mutuellement polaires. C. R. Acad. Sci. Paris 255 238–240.
  • [35] Perlman, M. D. and Wu, L. (1999). The emperor’s new tests. Statist. Sci. 14 355–381.
  • [36] Pisier, G. (1986). Probabilistic methods in the geometry of Banach spaces. In Probability and Analysis (Varenna, 1985). Lecture Notes in Math. 1206 167–241. Springer, Berlin.
  • [37] Raubertas, R. F., Lee, C.-I. C. and Nordheim, E. V. (1986). Hypothesis tests for normal means constrained by linear inequalities. Comm. Statist. Theory Methods 15 2809–2833.
  • [38] Robertson, T. (1978). Testing for and against an order restriction on multinomial parameters. J. Amer. Statist. Assoc. 73 197–202.
  • [39] Robertson, T. (1986). On testing symmetry and unimodality. In Advances in Order Restricted Statistical Inference (Iowa City, Iowa, 1985). Lecture Notes in Statistics 37 231–248. Springer, Berlin.
  • [40] Robertson, T. and Wegman, E. J. (1978). Likelihood ratio tests for order restrictions in exponential families. Ann. Statist. 6 485–505.
  • [41] Robertson, T. and Wright, F. T. (1982). On measuring the conformity of a parameter set to a trend, with applications. Ann. Statist. 10 1234–1245.
  • [42] Robertson, T., Wright, F. T. and Dykstra, R. L. (1988). Order Restricted Statistical Inference. Wiley, Chichester.
  • [43] Scharf, L. L. (1991). Statistical Signal Processing: Detection, Estimation and Time Series Analysis. Addison-Wesley, Reading, MA.
  • [44] Sen, B. and Meyer, M. (2017). Testing against a linear regression model using ideas from shape-restricted estimation. J. R. Stat. Soc. Ser. B. Stat. Methodol. 79 423–448.
  • [45] Shapiro, A. (1988). Towards a unified theory of inequality constrained testing in multivariate analysis. Int. Stat. Rev. 56 49–62.
  • [46] Slawski, M. and Hein, M. (2013). Non-negative least squares for high-dimensional linear models: Consistency and sparse recovery without regularization. Electron. J. Stat. 7 3004–3056.
  • [47] Spokoiny, V. G. (1998). Adaptive and spatially adaptive testing of a nonparametric hypothesis. Math. Methods Statist. 7 245–273.
  • [48] van de Geer, S. A. (2000). Applications of Empirical Process Theory. Cambridge Series in Statistical and Probabilistic Mathematics 6. Cambridge Univ. Press, Cambridge.
  • [49] van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes: With Applications to Statistics. Springer, New York.
  • [50] Warrack, G. and Robertson, T. (1984). A likelihood ratio test regarding two nested but oblique order-restricted hypotheses. J. Amer. Statist. Assoc. 79 881–886.
  • [51] Wei, Y. and Wainwright, M. J. (2016). Sharp minimax bounds for testing discrete monotone distributions. In IEEE International Symposium on Information Theory (ISIT) 2684–2688. IEEE, Los Alamitos, CA.
  • [52] Wei, Y., Wainwright, M. J and Guntuboyina, A. (2019). Supplement to “The geometry of hypothesis testing over convex cones: Generalized likelihood ratio tests and minimax radii.” DOI:10.1214/18-AOS1701SUPP.
  • [53] Zarantonello, E. H. (1971). Projections on convex sets in Hilbert space and spectral theory. In Contributions to Nonlinear Functional Analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1971) 237–424. Academic Press, New York.

Supplemental materials

  • Supplement to “The geometry of hypothesis testing over convex cones: Generalized likelihood ratio tests and minimax radii”. The supplementary material includes the explanation for the GLRT suboptimality and the proofs of more technical aspects.