On the Distribution of the Likelihood Ratio

1954/09/01 by Herman Chernoff · 804 citations
Computer Science · Mathematics · #Advanced Statistical Methods and Models #Bayesian Methods and Mixture Models #Combinatorics #Distribution (mathematics) #Hyperplane #Lambda #Likelihood-ratio test #Mathematical analysis #Mathematics #Matrix (chemical analysis) #Multivariate normal distribution #Multivariate statistics #Point processes and geometric inequalities #Statistics

paper · pdf · doi:10.1214/aoms/1177728725

published in The Annals of Mathematical Statistics 25(3), 573-578 (Institute of Mathematical Statistics)

openalex publication_date 1954/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/03

Abstract

A classical result due to Wilks [1] on the distribution of the likelihood ratio λ is the following. Under suitable regularity conditions, if the hypothesis that a parameter θ lies on an r-dimensional hyperplane of k-dimensional space is true, the distribution of -2 log λ is asymptotically that of χ² with k - r degrees of freedom. In many important problems it is desired to test hypotheses which are not quite of the above type. For example, one may wish to test whether θ is on one side of a hyperplane, or to test whether θ is in the positive quadrant of a two-dimensional space. The asymptotic distribution of -2 log λ is examined when the value of the parameter is a boundary point of both the set of θ corresponding to the hypothesis and the set of θ corresponding to the alternative. First the case of a single observation from a multivariate normal distribution, with mean θ and known covariance matrix, is treated. The general case is then shown to reduce to this special case where the covariance matrix is replaced by the inverse of the information matrix. In particular, if one tests whether θ is on one side or the other of a smooth (k - 1)-dimensional surface in k-dimensional space and θ lies on the surface, the asymptotic distribution of λ is that of a chance variable which is zero half the time and which behaves like χ² with one degree of freedom the other half of the time.

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