A Decomposition for the Likelihood Ratio Statistic and the Bartlett Correction--A Bayesian Argument

Let l(θ) = n -1 log p(x, θ) be the log likelihood of an n-dimensional X under a p-dimensional θ. Let θˆ j be the mle under H j : θ 1 = θ 1 0 , ..., θ j = θ j 0 and θˆ 0 be the unrestricted mle. Define T j as [2n{l(θˆ j-1 )}] 1/2 sgn(θˆj j-1 -θ j 0 ). Let T = (T 1 , ..., T p ). Then under regularity conditions, the following theorem is proved: Under θ = θ 0 , T is asymptotically N(n -1/2 a 0 + n -1 a, J + n -1 ∑) + O(n -3/2 ) where J is the identity matrix. The result is proved by first establishing an analogous result when θ 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 ).