A Decomposition for the Likelihood Ratio Statistic and the Bartlett Correction--A Bayesian Argument
A Decomposition for the Likelihood Ratio Statistic and the Bartlett Correction--A Bayesian Argument
Let $l(\theta) = n^{-1} \log p(x, \theta)$ be the log likelihood of an $n$-dimensional $X$ under a $p$-dimensional $\theta$. Let $\hat{\theta}_j$ be the mle under $H_j: \theta^1 = \theta^1_0, \ldots, \theta^j = \theta^j_0$ and $\hat{\theta}_0$ be the unrestricted mle. Define $T_j$ as $\lbrack 2n\{l(\hat{\theta}_{j - 1}) - l(\hat{\theta}_j)\}\rbrack^{1/2} \operatorname{sgn}(\hat{\theta}^j_{j - …