Geometric Understanding of Likelihood Ratio Statistics

Abstract

Geometrie Understanding of Likelihood Ratio Statistics* Jianqing Fan, H u i - N i e n H u n g and W i n g - H u n g W o n g M a r c h 1 1 , 1998 Abstract It is well-known that twice a log-likelihood ratio statistic follows asymptotically a X -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 X -distributed. Our technical arguments are simple and can easily be understood. Jianqing Fan and Wing-Hung Wong are professors, Department of Statistics, University of California, Los Angeles, CA 90095. Hui-Nien Hung is associate professor, Institute of Statistics, National Chiao-Tong University, Taiwan. Fan's research was partially supported by NSF grant DMS-9803200 and NSA grant 96-1-0015. Wong's work was partially supported by the Mathematical Sciences Research Institute of the Chinese University of Hong Kong and by NSF grant DMS-9703918.