A PROOF OF INDEPENDENT BARTLETT CORRECTABILITY OF NESTED LIKELIHOOD RATIO TESTS

It is well known that likelihood ratio statistic is Bartlett correctab le. We consider decomposition of a likelihood ratio statistic into 1 d egree of freedom components based on sequence of nested hypotheses. We give a proof of the fact that the component likelihood ratio statisti cs are distributed mutually independently up to the order O(1/n) and e ach component is independently Bartlett correctable. This was implicit in Lawley (1956, Biometrika, 43, 295-303) and proved in Bickel and Gh osh (1990, Ann. Statist., 18, 1070-1090) using a Bayes method. We pres ent a more direct frequentist proof.