Geometric understanding of likelihood ratio statistics

Jianqing Fan*, Wing Hung Wong, Hui-Nien Hung

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

15 Scopus citations


It is well known that twice a log-likelihood ratio statistic follows asymptotically a chi-square distribution. The result is usually understood and proved via Taylor's expansions of likelihood functions and by assuming asymptotic normality of maximum likelihood estimators (MLEs). We obtain more general results by using a different approach: The Wilks type of results hold as long as likelihood contour sets are fan-shaped. The classical Wilks theorem corresponds to the situations in which the likelihood contour sets are ellipsoidal. This provides a 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 chi-square distributed. Our technical arguments are simple and easily understood.

Original languageEnglish
Pages (from-to)836-841
Number of pages6
JournalJournal of the American Statistical Association
Issue number451
StatePublished - 1 Sep 2000


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