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

Ghosh, J. K. ; Bickel, Peter J. (1990) A decomposition for the likelihood ratio statistic and the bartlett correction-a Bayesian argument Annals of Statistics, 18 (3). pp. 1070-1090. ISSN 0090-5364

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Official URL: http://www.jstor.org/pss/2242043

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 θˆj be the mle under Hj: θ1 = θ1 0, ..., θj = θj 0 and θˆ0 be the unrestricted mle. Define Tj as [2n{l(θˆj-1)}]1/2 sgn(θˆjj-1j 0). Let T = (T1, ..., Tp). 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).

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