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A PROOF OF INDEPENDENT BARTLETT CORRECTABILITY

OF NESTED LIKELIHOOD RATIO TESTS

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AKIMICHI TAKEMURA^{ 1} AND SATOSHI KURIKI^{ 2}

^{1} *Faculty of Economics, University of Tokyo,
Bunkyo-ku, Tokyo
113, Japan*

^{2} *The Institute of Statistical Mathematics, 4-6-7
Minami-Azabu,
Minato-ku, Tokyo 106, Japan*
(Received July 21, 1995; revised March 8, 1996)

**Abstract.**
It is well known that likelihood ratio
statistic is Bartlett correctable. We consider decomposition of a
likelihood ratio statistic into 1 degree of freedom components based
on sequence of nested hypotheses. We give a proof of the fact that
the component likelihood ratio statistics are distributed mutually
independently up to the order *O*(1/*n*) and each component is
independently Bartlett correctable. This was implicit in Lawley
(1956, *Biometrika*, **43**, 295-303) and proved in
Bickel and Ghosh (1990, *Ann. Statist.*, **18**,
1070-1090) using a Bayes method. We present a more direct
frequentist proof.

*Key words and phrases*:
Likelihood ratio test, Bartlett
correction, nested hypotheses, component likelihood ratio statistic.

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