• Bernoulli
  • Volume 25, Number 4A (2019), 3041-3068.

Signal detection via Phi-divergences for general mixtures

Marc Ditzhaus

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


The family of goodness-of-fit tests based on $\Phi$-divergences is known to be optimal for detecting signals hidden in high-dimensional noise data when the heterogeneous normal mixture model is underlying. This test family includes Tukey’s popular higher criticism test and the famous Berk–Jones test. In this paper we address the open question whether the tests’ optimality is still present beyond the prime normal mixture model. On the one hand, we transfer the known optimality of the higher criticism test for different models, for example, for the heteroscedastic normal, general Gaussian and exponential-$\chi^{2}$-mixture models, to the whole test family. On the other hand, we discuss the optimality for new model classes based on exponential families including the scale exponential, the scale Fréchet and the location Gumbel models. For all these examples we apply a general machinery which might be used to show the tests’ optimality for further models/model classes in future.

Article information

Bernoulli, Volume 25, Number 4A (2019), 3041-3068.

Received: March 2018
Revised: August 2018
First available in Project Euclid: 13 September 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Berk and Jones test detection boundary $\Phi$-divergences sparse and dense signal detection Tukey’s higher criticism


Ditzhaus, Marc. Signal detection via Phi-divergences for general mixtures. Bernoulli 25 (2019), no. 4A, 3041--3068. doi:10.3150/18-BEJ1079.

Export citation


  • [1] Ali, S.M. and Silvey, S.D. (1966). A general class of coefficients of divergence of one distribution from another. J. Roy. Statist. Soc. Ser. B 28 131–142.
  • [2] Arias-Castro, E., Candès, E.J. and Plan, Y. (2011). Global testing under sparse alternatives: ANOVA, multiple comparisons and the higher criticism. Ann. Statist. 39 2533–2556.
  • [3] Arias-Castro, E. and Wang, M. (2015). The sparse Poisson means model. Electron. J. Stat. 9 2170–2201.
  • [4] Arias-Castro, E. and Wang, M. (2017). Distribution-free tests for sparse heterogeneous mixtures. TEST 26 71–94.
  • [5] Barndorff-Nielsen, O. (1978). Information and Exponential Families in Statistical Theory. Chichester: Wiley.
  • [6] Berk, R.H. and Jones, D.H. (1979). Goodness-of-fit test statistics that dominate the Kolmogorov statistics. Z. Wahrsch. Verw. Gebiete 47 47–59.
  • [7] Bingham, N.H., Goldie, C.M. and Teugels, J.L. (1987). Regular Variation. Encyclopedia of Mathematics and Its Applications 27. Cambridge: Cambridge Univ. Press.
  • [8] Cai, T.T., Jeng, X.J. and Jin, J. (2011). Optimal detection of heterogeneous and heteroscedastic mixtures. J. R. Stat. Soc. Ser. B. Stat. Methodol. 73 629–662.
  • [9] Cai, T.T. and Wu, Y. (2014). Optimal detection of sparse mixtures against a given null distribution. IEEE Trans. Inform. Theory 60 2217–2232.
  • [10] Cayon, L., Jin, J. and Treaster, A. (2004). Higher criticism statistic: Detecting and identifying non-Gaussianity in the WMAP first year data. Mon. Not. R. Astron. Soc. 362 826–832.
  • [11] Chang, L.-C. (1955). On the ratio of an empirical distribution function to the theoretical distribution function. Acta Math. Sinica 5 347–368.
  • [12] Cressie, N. and Read, T.R.C. (1984). Multinomial goodness-of-fit tests. J. Roy. Statist. Soc. Ser. B 46 440–464.
  • [13] Csiszár, I. (1967). Information-type measures of difference of probability distributions and indirect observations. Studia Sci. Math. Hungar. 2 299–318.
  • [14] Dai, H., Charnigo, R., Srivastava, T., Talebizadeh, Z. and Qing, S. (2012). Integrating P-values for genetic and genomic data analysis. J. Biom. Biostat. 3–7.
  • [15] Ditzhaus, M. (2017). The power of tests for signal detection under high-dimensional data. Ph.D. thesis, Heinrich-Heine-Univ. Duesseldorf. Available at
  • [16] Ditzhaus, M. and Janssen, A. (2017). Detectability of nonparametric signals: Higher criticism versus likelihood ratio. Available at 1709.07264v2.
  • [17] Donoho, D. and Jin, J. (2004). Higher criticism for detecting sparse heterogeneous mixtures. Ann. Statist. 32 962–994.
  • [18] Donoho, D. and Jin, J. (2009). Feature selection by higher criticism thresholding achieves the optimal phase diagram. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 367 4449–4470.
  • [19] Donoho, D. and Jin, J. (2015). Higher criticism for large-scale inference, especially for rare and weak effects. Statist. Sci. 30 1–25.
  • [20] Eicker, F. (1979). The asymptotic distribution of the suprema of the standardized empirical processes. Ann. Statist. 7 116–138.
  • [21] Feller, W. (1966). An Introduction to Probability Theory and Its Applications. Vol. II. New York: Wiley.
  • [22] Goldstein, D.B. (2009). Common genetic variation and human traits. N. Engl. J. Med. 360 1696–1698.
  • [23] Gontscharuk, V., Landwehr, S. and Finner, H. (2015). The intermediates take it all: Asymptotics of higher criticism statistics and a powerful alternative based on equal local levels. Biom. J. 57 159–180.
  • [24] Gontscharuk, V., Landwehr, S. and Finner, H. (2016). Goodness of fit tests in terms of local levels with special emphasis on higher criticism tests. Bernoulli 22 1331–1363.
  • [25] Hall, P., Pittelkow, Y. and Ghosh, M. (2008). Theoretical measures of relative performance of classifiers for high dimensional data with small sample sizes. J. R. Stat. Soc. Ser. B. Stat. Methodol. 70 159–173.
  • [26] Ingster, Y.I., Tsybakov, A.B. and Verzelen, N. (2010). Detection boundary in sparse regression. Electron. J. Stat. 4 1476–1526.
  • [27] Ingster, Yu.I. (1997). Some problems of hypothesis testing leading to infinitely divisible distributions. Math. Methods Statist. 6 47–69.
  • [28] Iyengar, S.K. and Elston, R.C. (2007). The genetic basis of complex traits: Rare variants or “common gene, common disease”? Methods Mol. Biol. 376 71–84.
  • [29] Jaeschke, D. (1979). The asymptotic distribution of the supremum of the standardized empirical distribution function on subintervals. Ann. Statist. 7 108–115.
  • [30] Jager, L. and Wellner, J. (2004). A new goodness of fit test: The reversed Berk–Jones statistic. Technical report, Dept. Statistics, Univ. Washington, DC.
  • [31] Jager, L. and Wellner, J.A. (2007). Goodness-of-fit tests via phi-divergences. Ann. Statist. 35 2018–2053.
  • [32] Jin, J. (2009). Impossibility of successful classification when useful features are rare and weak. Proc. Natl. Acad. Sci. USA 106 8859–8864.
  • [33] Jin, J., Starck, J.-L., Donoho, D.L., Aghanim, N. and Forni, O. (2005). Cosmological non-Gaussian signature detection: Comparing performance of different statistical tests. EURASIP J. Appl. Signal Process. 15 2470–2485.
  • [34] Khmaladze, E. and Shinjikashvili, E. (2001). Calculation of noncrossing probabilities for Poisson processes and its corollaries. Adv. in Appl. Probab. 33 702–716.
  • [35] Khmaladze, E.V. (1998). Goodness of fit tests for “chimeric” alternatives. Stat. Neerl. 52 90–111.
  • [36] Kulldorff, M., Heffernan, R., Hartman, J., Assunção, R. and Mostashari, F. (2005). A space–time permutation scan statistic for disease outbreak detection. PLoS Med. 2 e59.
  • [37] Mukherjee, R., Pillai, N.S. and Lin, X. (2015). Hypothesis testing for high-dimensional sparse binary regression. Ann. Statist. 43 352–381.
  • [38] Neill, D. and Lingwall, J. (2007). A nonparametric scan statistic for multivariate disease surveillance. Advances in Disease Surveillance 4 106–116.
  • [39] Saligrama, V. and Zhao, M. (2012). Local anomaly detection. JMLR W&CP 22 969–983.
  • [40] Shorack, G.R. and Wellner, J.A. (1986). Empirical Processes with Applications to Statistics. New York: Wiley.
  • [41] Strasser, H. (1985). Mathematical Theory of Statistics. De Gruyter Studies in Mathematics 7. Berlin: de Gruyter.
  • [42] Tukey, J.W. (1976). T13 N: The Higher Criticism. Coures Notes. Stat 411. Princeton: Princeton University Press.
  • [43] Tukey, J.W. (1989). Higher Criticism for individual significances in several tables or parts of tables. Internal working paper, Princeton Univ.
  • [44] Tukey, J.W. (1994). The Collected Works of John W. Tukey. Vol. VIII. London: Chapman and Hall.
  • [45] Wellner, J.A. (1978). Limit theorems for the ratio of the empirical distribution function to the true distribution function. Z. Wahrsch. Verw. Gebiete 45 73–88.