Likelihood ratio tests and singularities

Abstract

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.

Published in at http://dx.doi.org/10.1214/07-AOS571 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (4)

• Validity of the expected Euler characteristic heuristic
• Algebraic Factor Analysis: Tetrads, Pentads and Beyond
• Algebraic statistical models
• Algebraic Techniques for Gaussian Models

Cited by in corpus (34)

• A Widely Applicable Bayesian Information Criterion
• Universal Inference
• Discrete chain graph models
• Graphs for margins of Bayesian networks
• Global identifiability of linear structural equation models
• Margins of discrete Bayesian networks
• Computing maximum likelihood estimates in recursive linear models with correlated errors
• Inference for eigenvalues and eigenvectors of Gaussian symmetric matrices
• Implicit inequality constraints in a binary tree model
• Model Selection of Nested and Non-Nested Item Response Models using Vuong Tests
• A Note on Likelihood Ratio Tests for Models with Latent Variables
• Wald tests of singular hypotheses
• Item Response Theory -- A Statistical Framework for Educational and Psychological Measurement
• Identification of discrete concentration graph models with one hidden binary variable
• Marginal Likelihood Integrals for Mixtures of Independence Models
• Bayes Model Selection with Path Sampling: Factor Models and Other Examples
• Model selection and local geometry
• Singularity structures and impacts on parameter estimation in finite mixtures of distributions
• Generalised likelihood profiles for models with intractable likelihoods
• On the choice of the splitting ratio for the split likelihood ratio test
• Maximum likelihood fitting of acyclic directed mixed graphs to binary data
• Confidence in Causal Inference under Structure Uncertainty in Linear Causal Models with Equal Variances
• Algebraic Problems in Structural Equation Modeling
• Wald tests when restrictions are locally singular
• False discovery rate control with e-values
• On Testing Marginal versus Conditional Independence
• The geometry of Gaussian double Markovian distributions
• Two Robust Tools for Inference about Causal Effects with Invalid Instruments
• Smooth, identifiable supermodels of discrete DAG models with latent variables
• Likelihood ratio test for variance components in nonlinear mixed effects models
• Path Dependent Structural Equation Models
• Prior Intensified Information Criterion
• Test for homogeneity with unordered paired observations
• Colored Gaussian directed acyclic graphical models