A PROOF OF INDEPENDENT BARTLETT CORRECTABILITY OF NESTED LIKELIHOOD RATIO TESTS

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

1Faculty of Economics, University of Tokyo, Bunkyo-ku, Tokyo 113, Japan

2The 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.