## On combinatorial testing problems.(English)Zbl 1200.62059

Summary: We study a class of hypothesis testing problems in which, upon observing the realization of an $$n$$-dimensional Gaussian vector, one has to decide whether the vector was drawn from a standard normal distribution or, alternatively, whether there is a subset of the components belonging to a certain given class of sets whose elements have been “contaminated,” that is, have a mean different from zero. We establish some general conditions under which testing is possible and others under which testing is hopeless with a small risk. The combinatorial and geometric structure of the class of sets is shown to play a crucial role. The bounds are illustrated on various examples.

### MSC:

 62H15 Hypothesis testing in multivariate analysis 05C90 Applications of graph theory 62M99 Inference from stochastic processes 62F03 Parametric hypothesis testing 62F05 Asymptotic properties of parametric tests

### Keywords:

hypothesis testing; multiple hypotheses; Gaussian processes

LAS
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### References:

 [1] Aldous, D. J. (1990). The random walk construction of uniform spanning trees and uniform labelled trees. SIAM J. Discrete Math. 3 450-465. · Zbl 0717.05028 [2] Alon, N., Krivelevich, M. and Sudakov, B. (1999). Finding a large hidden clique in a random graph. Randoms Structures Algorithms 13 457-466. · Zbl 0959.05082 [3] Arias-Castro, E., Candès, E. J., Helgason, H. and Zeitouni, O. (2008). Searching for a trail of evidence in a maze. Ann. Statist. 36 1726-1757. · Zbl 1143.62006 [4] Arias-Castro, E., Candès, E. and Durand, A. (2009). Detection of abnormal clusters in a network. Technical report, Univ. California, San Diego. [5] Arlot, S., Blanchard, G. and Roquain, E. (2010a). Some nonasymptotic results on resampling in high dimension, I: Confidence regions. Ann. Statist. 38 51-82. · Zbl 1180.62066 [6] Arlot, S., Blanchard, G. and Roquain, E. (2010b). Some nonasymptotic results on resampling in high dimension. II. Multiple tests. Ann. Statist. 38 83-99. · Zbl 1181.62055 [7] Baraud, Y. (2002). Non-asymptotic minimax rates of testing in signal detection. Bernoulli 8 577-606. · Zbl 1007.62042 [8] Benjamini, I., Lyons, R., Peres, Y. and Schramm, O. (2001). Uniform spanning forests. Ann. Probab. 29 1-65. · Zbl 1016.60009 [9] Bhattacharyya, A. (1946). On a measure of divergence between two multinomial populations. Sankhyā 7 401-406. · Zbl 0063.00366 [10] Boucheron, S., Lugosi, G. and Massart, P. (2000). A sharp concentration inequality with applications. Random Structures Algorithms 16 277-292. · Zbl 0954.60008 [11] Broder, A. (1989). Generating random spanning trees. In 30th Annual Symposium on Foundations of Computer Science 442-447. IEEE Press, Research Triangle Park, NC. [12] Devroye, L. and Györfi, L. (1985). Nonparametric Density Estimation: The L 1 View . Wiley, New York. · Zbl 0546.62015 [13] Devroye, L., Györfi, L. and Lugosi, G. (1996). A Probabilistic Theory of Pattern Recognition . Springer, New York. · Zbl 0853.68150 [14] Donoho, D. and Jin, J. (2004). Higher criticism for detecting sparse heterogeneous mixtures. Ann. Statist. 32 962-994. · Zbl 1092.62051 [15] Dubdashi, D. and Ranjan, D. (1998). Balls and bins: A study in negative dependence. Random Structures Algorithms 13 99-124. · Zbl 0964.60503 [16] Dudley, R. M. (1978). Central limit theorems for empirical measures. Ann. Probab. 6 899-929. · Zbl 0404.60016 [17] Durot, C. and Rozenholc, Y. (2006). An adaptive test for zero mean. Math. Methods Statist. 15 26-60. [18] Feder, T. and Mihail, M. (1992). Balanced matroids. In STOC’92: Proceedings of the Twenty-Fourth Annual ACM Symposium on Theory of Computing 26-38. ACM, New York. [19] Feige, U. and Krauthgamer, R. (2000). Finding and certifying a large hidden clique in a semirandom graph. Random Structures Algorithms 16 195-208. · Zbl 0940.05050 [20] Glaz, J., Naus, J. and Wallenstein, S. (2001). Scan Statistics . Springer, New York. · Zbl 0983.62075 [21] Grimmett, G. R. and Winkler, S. N. (2004). Negative association in uniform forests and connected graphs. Random Structures Algorithms 24 444-460. · Zbl 1048.60007 [22] Haussler, D. (1995). Sphere packing numbers for subsets of the boolean n -cube with bounded Vapnik-Chervonenkis dimension. J. Combin. Theory Ser. A 69 217-232. · Zbl 0818.60005 [23] Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 13-30. JSTOR: · Zbl 0127.10602 [24] Ingster, Y. I. (1999). Minimax detection of a signal for l p n -balls. Math. Methods Statist. 7 401-428. · Zbl 1103.62312 [25] Jerrum, M., Sinclair, A. and Vigoda, E. (2004). A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries. J. ACM 51 671-697. · Zbl 1204.65044 [26] Le Cam, L. (1970). On the assumptions used to prove asymptotic normality of maximum likelihood estimates. Ann. Math. Statist. 41 802-828. · Zbl 0246.62039 [27] Ledoux, M. and Talagrand, M. (1991). Probability in Banach Spaces . Springer, New York. · Zbl 0748.60004 [28] Moon, J. W. (1970). Counting Labelled Trees. Canadian Mathematical Monographs 1 . Canadian Mathematical Congress, Montreal. · Zbl 0214.23204 [29] Propp, J. G. and Wilson, D. B. (1998). How to get a perfectly random sample from a generic Markov chain and generate a random spanning tree of a directed graph. J. Algorithms 27 170-217. · Zbl 0919.68092 [30] Romano, J. P. and Wolf, M. (2005). Exact and approximate stepdown methods for multiple hypothesis testing. J. Amer. Statist. Assoc. 100 94-108. · Zbl 1117.62416 [31] Shabalin, A. A., Weigman, V. J., Perou, C. M. and Nobel, A. B. (2009). Finding large average submatrices in high dimensional data. Ann. Appl. Statist. 3 985-1012. · Zbl 1196.62087 [32] Slepian, D. (1962). The one-sided barrier problem for Gaussian noise. Bell System Tech. J. 41 463-501. [33] Talagrand, M. (2005). The Generic Chaining . Springer, New York. · Zbl 1075.60001 [34] Tsirelson, B. S., Ibragimov, I. A. and Sudakov, V. N. (1976). Norm of Gaussian sample function. In Proceedings of the 3rd Japan-U.S.S.R. Symposium on Probability Theory. Lecture Notes in Math. 550 20-41. Springer, Berlin. · Zbl 0359.60019 [35] Vapnik, V. N. and Chervonenkis, A. Y. (1971). On the uniform convergence of relative frequences of events to their probabilities. Theory Probab. Appl. 16 264-280. · Zbl 0247.60005 [36] Vonnegut, K. (1973). Breakfast of Champions . Delacorte Press, New York.
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