Ghosh, J. K. ; Bickel, Peter J. (1990) A decomposition for the likelihood ratio statistic and the bartlett correctiona Bayesian argument Annals of Statistics, 18 (3). pp. 10701090. ISSN 00905364

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Abstract
Let l(θ) = n^{1} log p(x, θ) be the log likelihood of an ndimensional X under a pdimensional 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_{j1})}]^{1/2} sgn(θˆj_{j1} 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}).
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