| Summary: | 碩士 === 國立交通大學 === 統計所 === 87 === It is well-known that twice a log-likelihood ratio statistic
follows asymptotically a chisquare-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 thatmore
fundamental insights can be obtained for the likelihood ratio
statistics: the result holds 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 gamma-distributed.
Even in finite sample situation, we can also use the gamma
type distributions to approximate the true distribution.Our
technical arguments are simple and can easily be understood.
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