On the Distribution of the Likelihood Ratio (original) (raw)

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September, 1954 On the Distribution of the Likelihood Ratio

Herman Chernoff

Ann. Math. Statist. 25(3): 573-578 (September, 1954). DOI: 10.1214/aoms/1177728725

Abstract

A classical result due to Wilks [1] on the distribution of the likelihood ratio lambda\lambdalambda is the following. Under suitable regularity conditions, if the hypothesis that a parameter theta\thetatheta lies on an rrr-dimensional hyperplane of kkk-dimensional space is true, the distribution of −2loglambda-2 \log \lambda−2loglambda is asymptotically that of chi2\chi^2chi2 with k−rk - rk−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 theta\thetatheta is on one side of a hyperplane, or to test whether theta\thetatheta is in the positive quadrant of a two-dimensional space. The asymptotic distribution of −2loglambda-2 \log \lambda−2loglambda is examined when the value of the parameter is a boundary point of both the set of theta\thetatheta corresponding to the hypothesis and the set of theta\thetatheta corresponding to the alternative. First the case of a single observation from a multivariate normal distribution, with mean theta\thetatheta 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 theta\thetatheta is on one side or the other of a smooth (k−1)(k - 1)(k−1)-dimensional surface in kkk-dimensional space and theta\thetatheta lies on the surface, the asymptotic distribution of lambda\lambdalambda is that of a chance variable which is zero half the time and which behaves like chi2\chi^2chi2 with one degree of freedom the other half of the time.

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Herman Chernoff. "On the Distribution of the Likelihood Ratio." Ann. Math. Statist. 25 (3) 573 - 578, September, 1954. https://doi.org/10.1214/aoms/1177728725

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Published: September, 1954

First available in Project Euclid: 28 April 2007

Digital Object Identifier: 10.1214/aoms/1177728725