THE LEVEL PROBABILITIES FOR A SIMPLE LOOP ORDERING

BAHADUR SINGH^{1} AND F. T. WRIGHT^{2}

1Department of Statistics, University of Iowa, Iowa City, IA 52242, U.S.A. 2Department of Statistics, College of Arts and Science, University of Missouri,

222 Math Sciences Bldg., Columbia, MO 65211, U.S.A.

(Received January 17, 1992; revised July 19, 1992)

Abstract. Bartholomew’s statistics for testing homogeneity of normal means with ordered alternatives have null distributions which are mixtures of chisquared or beta distributions depending on whether the variances are known or not. The mixing coefficients depend on the sample sizes and the order restriction. If a researcher knows which mean is smallest and which is largest, but does not know how the other means are ordered, then a loop ordering is appropriate. Exact expressions for the mixing coefficients for a loop ordering and arbitrary sample sizes are given for five or fewer populations and approximations are developed for more than five populations. Also, the mixing coefficients for a loop ordering with equal sample sizes are computed. These mixing coefficients also arise in testing the ordering as the null hypothesis, in testing order restrictions in exponential families and in testing order restrictions nonparametrically.

Key words and phrases: Level probabilities, likelihood ratio tests, order restricted inference, simple loop ordering.