INDEPENDENCE OF LIKELIHOOD RATIO CRITERIA FOR HOMOGENEITY OF SEVERAL POPULATIONS

TAKESI HAYAKAWA

Faculty of Economics, Hitotsubashi University, Kunitachi, Tokyo 186-8601, Japan

(Received October 17, 2000; revised November 8, 2001)

Abstract. Let Π_{i} be an i-th population with a probability density function f(· | θ_{i}) with one dimensional unknown parameter θ_{i}, i = 1, 2, …, k. Let n_{i} sample be drawn from each Π_{i}. The likelihood ratio criteria λ_{j|(j−1)} for testing hypothesis that the first j parameters are equal against alternative hypothesis that the first (j −1) parameters are equal and the j-th parameter is different with the previous ones are defined, j = 2, 3, …, k. The paper shows the asymptotic independence of λ_{j|(j−1)}’s up to the order 1/n under a hypothesis of equality of k parameters, where n is a number of total samples.

Key words and phrases: Likelihood ratio criterion, asymptotic expansion, homogeneity of parameters, asymptotic independence.