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THE LEVEL PROBABILITIES FOR A SIMPLE LOOP ORDERING

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

^{1} *Department of Statistics, University of Iowa, Iowa City, IA 52242, U.S.A.*

^{2} *Department 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 chi-squared 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.

**Source**
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