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LIKELIHOOD RATIO TESTS FOR A CLASS OF

NON-OBLIQUE HYPOTHESES

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XIAOMI HU AND F. T. WRIGHT

*Department of Statistics, University of Missouri-Columbia,
Columbia, MO 65211, U.S.A.*
(Received October 21, 1992; revised May 6, 1993)

**Abstract.**
In this paper, we study the likelihood ratio tests
for situations in which the null and alternative hypotheses are
determined by two polyhedral cones, *C*_{1} and *C*_{2}, which are nested so
that *C*_{1} \subset *L* \subset *C*_{2} and *L* is a linear space. The two cones
are proved to be non-oblique. Members which satisfy this nesting
condition are easily identified and include the cases in which *C*_{1} = *L* or
*L* = *C*_{2}. When testing two non-oblique hypotheses with variances unknown,
the least favorable point within the null hypothesis has not been
determined in general. However, for the situation considered here, the
zero vector is shown to be least favorable within the null hypothesis.
Two sets of hypotheses are said to be equivalent if they lead to the same
likelihood ratio test. For two non-oblique polyhedral cones, *C*_{1} and
*C*_{2}, four sets of equivalent hypotheses are identified. If *C*_{1} \subset
*L* \subset *C*_{2}, then the two cones in each of these four sets of
hypotheses are similarly nested with a linear space in between.

*Key words and phrases*:
Inferences subject to inequality
constraints, iterated projection property, non-oblique cones, order
restricted inferences.

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