A decomposition for the likelihood ratio statistic and the bartlett correction-a Bayesian argument

Abstract

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: θ¹ = θ¹ ₀, ..., θ^j = θ^j ₀ and θˆ₀ be the unrestricted mle. Define T_j as [2n{l(θˆ_j-1)}]^1/2 sgn(θˆj_j-1 -θ^j ₀). Let T = (T₁, ..., T_p). Then under regularity conditions, the following theorem is proved: Under θ = θ₀, T is asymptotically N(n^-1/2a₀ + n^-1a, 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).