Annals of Statistics

Likelihood ratio tests and singularities

Mathias Drton

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Many statistical hypotheses can be formulated in terms of polynomial equalities and inequalities in the unknown parameters and thus correspond to semi-algebraic subsets of the parameter space. We consider large sample asymptotics for the likelihood ratio test of such hypotheses in models that satisfy standard probabilistic regularity conditions. We show that the assumptions of Chernoff’s theorem hold for semi-algebraic sets such that the asymptotics are determined by the tangent cone at the true parameter point. At boundary points or singularities, the tangent cone need not be a linear space and limiting distributions other than chi-square distributions may arise. While boundary points often lead to mixtures of chi-square distributions, singularities give rise to nonstandard limits. We demonstrate that minima of chi-square random variables are important for locally identifiable models, and in a study of the factor analysis model with one factor, we reveal connections to eigenvalues of Wishart matrices.

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Ann. Statist., Volume 37, Number 2 (2009), 979-1012.

First available in Project Euclid: 10 March 2009

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Primary: 60E05: Distributions: general theory 62H10: Distribution of statistics

Algebraic statistics factor analysis large sample asymptotics semi-algebraic set tangent cone


Drton, Mathias. Likelihood ratio tests and singularities. Ann. Statist. 37 (2009), no. 2, 979--1012. doi:10.1214/07-AOS571.

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  • [1] Anderson, T. W. and Rubin, H. (1956). Statistical inference in factor analysis. Proc. Third Berkeley Symp. Math. Statist. Probab. 1954–1955 V 111–150. Univ. California Press, Berkeley and Los Angeles.
  • [2] Basu, S., Pollack, R. and Roy, M.-F. (2003). Algorithms in Real Algebraic Geometry. Springer, Berlin.
  • [3] Benedetti, R. and Risler, J.-J. (1990). Real Algebraic and Semi-Algebraic Sets. Hermann, Paris.
  • [4] Berk, R. H. (1972). Consistency and asymptotic normality of MLE’s for exponential models. Ann. Math. Statist. 43 193–204.
  • [5] Bochnak, J., Coste, M. and Roy, M.-F. (1998). Real Algebraic Geometry. Springer, Berlin.
  • [6] Chernoff, H. (1954). On the distribution of the likelihood ratio. Ann. Math. Statist. 25 573–578.
  • [7] Cox, D., Little, J. and O’Shea, D. (1997). Ideals, Varieties, and Algorithms, 2nd ed. Springer, New York.
  • [8] Drton, M. (2006). Algebraic techniques for Gaussian models. In Prague Stochastics (M. Hušková and M. Janžura, eds.) 81–90. Matfyzpress, Charles Univ. Prague.
  • [9] Drton, M. and Sullivant, S. (2007). Algebraic statistical models. Statist. Sinica 17 1273–1297.
  • [10] Drton M., Sturmfels B. and Sullivant, S. (2007). Algebraic factor analysis: Tetrads, pentads and beyond. Probab. Theory Related Fields 138 463–493.
  • [11] Fernando, J. F. and Gamboa, J. M. (2003). Polynomial images of Rn. J. Pure Appl. Algebra 179 241–254.
  • [12] Ferrarotti, M., Fortuna, E. and Wilson, L. (2000). Real algebraic varieties with prescribed tangent cones. Pacific J. Math. 194 315–323.
  • [13] Geiger, D., Heckerman, D., King H. and Meek, C. (2001). Stratified exponential families: Graphical models and model selection. Ann. Statist. 29 505–529.
  • [14] Geyer, C. J. (1994). On the asymptotics of constrained M-estimation. Ann. Statist. 22 1993–2010.
  • [15] Greuel, G.-M., Pfister, G. and Schönemann, H. (2005). Singular 3.0. A computer algebra system for polynomial computations Centre for Computer Algebra, Univ. Kaiserslautern. Available at
  • [16] Hanumara, R. C. and Thompson, W. A. (1968). Percentage points of the extreme roots of a Wishart matrix. Biometrika 55 505–512.
  • [17] Hayashi K., Bentler, P. M. and Yuan, K.-H. (2007). On the likelihood ratio test for the number of factors in exploratory factor analysis. Struct. Equ. Model. 14 505–526.
  • [18] Kass, R. E. and Vos, P. W. (1997). Geometrical Foundations of Asymptotic Inference. Wiley, New York.
  • [19] Lin, Y. and Lindsay, B. G. (1997). Projections on cones, chi-bar squared distributions, and Weyl’s formula. Statist. Probab. Lett. 32 367–376.
  • [20] Miller, J. J. (1977). Asymptotic properties of maximum likelihood estimates in the mixed model of the analysis of variance. Ann. Statist. 5 746–762.
  • [21] O’Shea, D. B. and Wilson, L. C. (2004). Limits of tangent spaces to real surfaces. Amer. J. Math. 126 951–980.
  • [22] Pachter, L. and Sturmfels, B. (2005). Algebraic Statistics for Computational Biology. Cambridge Univ. Press, Cambridge.
  • [23] Rockafellar, R. T. and Wets, R. J.-B. (1998). Variational Analysis. Springer, Berlin.
  • [24] Self, S. G. and Liang, K.-Y. (1987). Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions. J. Amer. Statist. Assoc. 82 605–610.
  • [25] Shapiro, A. (1985). Asymptotic distribution of test statistics in the analysis of moment structures under inequality constraints. Biometrika 72 133–144.
  • [26] Shapiro, A. (1988). Towards a unified theory of inequality constrained testing in multivariate analysis. Internat. Statist. Rev. 56 49–62.
  • [27] Shapiro, A. (2000). On the asymptotics of constrained local M-estimators. Ann. Statist. 28 948–960.
  • [28] Shapiro, A. (2007). Statistical inference of moment structures. In Handbook of Latent Variable and Related Models (S.-Y. Lee, ed.) 229–260. Elsevier, Amsterdam.
  • [29] Silvapulle, M. J. and Sen, P. K. (2005). Constrained Statistical Inference. Wiley, Hoboken, NJ.
  • [30] Spirtes P., Glymour, C. and Scheines, R. (2000). Causation, Prediction, and Search, 2nd ed. MIT Press, Cambridge, MA.
  • [31] Takemura, A. and Kuriki, S. (1997). Weights of χ̅2 distribution for smooth or piecewise smooth cone alternatives. Ann. Statist. 25 2368–2387.
  • [32] Takemura, A. and Kuriki, S. (2002). On the equivalence of the tube and Euler characteristic methods for the distribution of the maximum of Gaussian fields over piecewise smooth domains. Ann. Appl. Probab. 12 768–796.
  • [33] Taylor, J., Takemura, A. and Adler, R. J. (2005). Validity of the expected Euler characteristic heuristic. Ann. Probab. 33 1362–1396.
  • [34] van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge Univ. Press, Cambridge.