Likelihood ratio tests and singularities

By Mathias Drton. 2007

149 citations recorded by the source corpus.

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.

Subfields: Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/46e42b9adeb9111edd6843747cbfecd3f250c4c3

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In Statistics and Probability

Citations in the corpus

Structure Learning in Graphical Modeling (2016, methodology): “An example is the so-called Verma constraint (Verma & Pearl, 1991); see also Example 3.3.14 in Drton et al. (2009).”
Discrete chain graph models (2009, background): “, normal distribution limits for maximum likelihood estimators and χ(2)-limits for likelihood ratios) is valid in smooth algebraic exponential families but may fail in non-smooth models [6].”
False discovery rate control with e‐values (2020, background): “There are many composite null hypothesis testing problems for which we may know of no direct way to construct a valid p-value even under low-dimensional asymptotics—this could happen because the model is singular or irregular and Wilks’ theorem fails to hold (Drton, 2009) and in such cases the valid”
Algebraic problems in structural equation modeling (2016, background): “At singular points, the distribution is determined by the tangent cone at the considered point [15].”
Item Response Theory—A Statistical Framework for Educational and Psychological Measurement (2021, unknown): “This asymptotic reference distribution can be derived using a more general theory for the likelihood ratio test statistic (Chernoﬀ, 1954; Drton, 2009).”
A Bayesian information criterion for singular models (2013, methodology): “We illustrate this fact in the following example.”
ESTIMATION AND INFERENCE WITH WEAK, SEMI-STRONG, AND STRONG IDENTIFICATION (2010, unknown): “…abrupt transition threshold autoregressive models are not covered by the results of the present paper; for example, see Picard (1985), Chan (1993), Bai (1997), Hansen (2000), Liu and Shao (2003), Elliott and Müller (2007, 2008), Qu and Perron (2007), and Drton (2009) for analyses of these models.”
Global identifiability of linear structural equation models (2010, background): “We briefly exemplify this issue for the models considered here and point the reader to [7] and references therein for a more detailed discussion.”