On the Distribution of the Likelihood Ratio

Chernoff, Herman

doi:10.1214/aoms/1177728725

Cited by 713 publications

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“…When I 11 = I * 11 , the asymptotic distribution reduces to a p s , 0·5 and 0·5 − p s mixture of χ 2 2 , χ 2 1 and χ 2 0 , respectively, which agrees with the results in Case 7 of Self & Liang (1987) and Example 3 in Chernoff (1954).…”

Section: Theorem 2 If the Regularity Conditions Insupporting

confidence: 84%

“…For simplicity of notation, let θ 0 = 0. Under the same regularity conditions as Chernoff (1954), for any θ in a neighbourhood N of θ = 0, we have…”

Section: Acknowledgementmentioning

confidence: 97%

“…Here we give the consistency result when the true value of θ lies on the boundary. Chernoff (1954) hold, the intersection of ϕ and the closure of neighbourhoods centred about the true value of the parameter of interest, θ 0 , constitute closed subsets of R p , and θ 0 is a limit point of ϕ, then with probability tending to 1 as n → ∞, there exists a sequence of points in ϕ,θ ϕ , at which local maxima of log L * (θ) occur, and that converges to θ 0 in probability.…”

Section: ·2 the Parameter Of Interest Lies On The Boundarymentioning

confidence: 99%

“…Chernoff (1954) are satisfied, θ 0 is a limiting point of both 0 and 1 , and the sets 0 and 1 are approximated by nonnull and disjoint cones C 0 and C 1 , respectively, then, when γ = γ 0 , the asymptotic distribution of the pseudolikelihood ratio statistic, T , is the same as the distribution of the likelihood ratio test of θ ∈ C 0 against θ ∈ C 1 based on one observation from a population with multivariate normal distribution with mean θ and unknown covariance matrix I −1 11 I * 11 I −1 11 while the covariance matrix is misspecified as I −1 11 in the likelihood ratio test. A sketch of the proof is given in Appendix A.…”

Section: Lemma 1 If the Regularity Conditions Inmentioning

confidence: 99%

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On the asymptotic behaviour of the pseudolikelihood ratio test statistic with boundary problems

Chen

Liang

2010

Biometrika

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SUMMARYThis paper considers the asymptotic distribution of the likelihood ratio statistic T for testing a subset of parameter of interest θ, θ = (γ, η), H 0 : γ = γ 0 , based on the pseudolikelihood L(θ,φ), whereφ is a consistent estimator of φ, the nuisance parameter. We show that the asymptotic distribution of T under H 0 is a weighted sum of independent chi-squared variables. Some sufficient conditions are provided for the limiting distribution to be a chi-squared variable. When the true value of the parameter of interest, θ 0 , or the true value of the nuisance parameter, φ 0 , lies on the boundary of parameter space, the problem is shown to be asymptotically equivalent to the problem of testing the restricted mean of a multivariate normal distribution based on one observation from a multivariate normal distribution with misspecified covariance matrix, or from a mixture of multivariate normal distributions. A variety of examples are provided for which the limiting distributions of T may be mixtures of chi-squared variables. We conducted simulation studies to examine the performance of the likelihood ratio test statistics in variance component models and teratological experiments.

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Section: Theorem 2 If the Regularity Conditions Insupporting

confidence: 84%

“…For simplicity of notation, let θ 0 = 0. Under the same regularity conditions as Chernoff (1954), for any θ in a neighbourhood N of θ = 0, we have…”

Section: Acknowledgementmentioning

confidence: 97%

Section: ·2 the Parameter Of Interest Lies On The Boundarymentioning

confidence: 99%

Section: Lemma 1 If the Regularity Conditions Inmentioning

confidence: 99%

See 2 more Smart Citations

On the asymptotic behaviour of the pseudolikelihood ratio test statistic with boundary problems

Chen

Liang

2010

Biometrika

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show abstract

“…Here I(θ 0 ) 1/2 can be any matrix square root of I(θ 0 ). This theorem has its origins in work by Chernoff [20]. A textbook proof can be found in [95,Theorem 16.7].…”

Section: Likelihood Ratio Testsmentioning

confidence: 99%