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 andC 2, which are nested so thatC 1 ⊂L ⊂C 2 andL 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 whichC 1=L orL=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 andC 2, four sets of equivalent hypotheses are identified. IfC 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.
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This research was supported in part by the National Institutes for Health under Grant 1 R01 GM42584-01A1.
Part of this work is taken from this author's dissertation submitted in partial fulfillment for the Ph.D. Degree at the University of Missouri-Columbia.
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Hu, X., Wright, F.T. Likelihood ratio tests for a class of non-oblique hypotheses. Ann Inst Stat Math 46, 137–145 (1994). https://doi.org/10.1007/BF00773599
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DOI: https://doi.org/10.1007/BF00773599