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 $\lbrack 2n\{l(\hat{\theta}_{j - 1}) - l(\hat{\theta}_j)\}\rbrack^{1/2} \operatorname{sgn}(\hat{\theta}^j_{j - 1} - \theta^j_0).$ Let $T = (T_1, \ldots, T_p)$. Then under regularity conditions, the following theorem is proved: Under $\theta = \theta_0, T$ is asymptotically $N(n^{-1/2}a_0 + n^{-1}a, J + n^{-1}\sum) + O(n^{-3/2})$ where $J$ is the identity matrix. The result is proved by first establishing an analogous result when $\theta$ is random and then making the prior converge to a degenerate distribution. The existence of the Bartlett correction to order $n^{-3/2}$ follows from the theorem. We show that an Edgeworth expansion with error $O(n^{-2})$ for $T$ involves only polynomials of degree less than or equal to 3 and hence verify rigorously Lawley's (1956) result giving the order of the error in the Bartlett correction as $O(n^{-2})$.