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Open Access Publications from the University of California

Geometric Understanding of Likelihood Ratio Statistics

  • Author(s): Fan, Jianqing
  • Hung, Hui-Nien
  • Wong, Wing-Hung
  • et al.

It is well-known that twice a log-likelihood ratio statistic follows asymptotically a X2-distribution. The result is usually understood and proved via Taylor's expansions of likelihood functions and by assuming asymptotic normality of maximum likelihood estimators. We contend that more fundamental insights can be obtained for likelihood ratio statistics: the Wilks type of results hold as long as likelihood contour sets are of fan-shape. The classical Wilks theorem corresponds to the situations where the likelihood contour sets are ellipsoid. This provides an insightful geometric understanding and a useful extension of the likelihood ratio theory. As a result, even if the MLEs are not asymptotically normal, the likelihood ratio statistics can still be asymptotically X2 distributed. Our technical arguments are simple and can easily be understood.

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