LIKELIHOOD RATIO TESTS FOR A CLASS OF NON-OBLIQUE HYPOTHESES

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} ⊂ L ⊂ 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} ⊂ L ⊂ 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.