觀察對數概似比函數在不平滑模型下之行為

Abstract

兩倍的對數概似比函數的極限分配為卡方分配是我們所熟知 的定理。但此定理的證明須利用到概似函數的泰勒展開式，且需 要假設最大概似統計量的極限分配為常態分配。 在我們的論文中，只需對數概似比函數的輪廓的結構為扇形 。若Wilks定理成立，則此處的扇形結構為一橢圓。即使概似函 數不甚平滑或最大概似統計量的極限分配不為常態，仍可證得對 數概似比函數的極限分配為加碼分配。同時在小樣本的情況下， 對數概似比函數的分配仍很接近加碼分配。

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.