Abstract
Abstract : In certain problems, it may be expected that a regression function has a substantial overall tendency to be monotone and yet we may not be certain that all of the restrictions imposed by a simple order are satisfied. Distribution theory for liklihood ratio tests of homogeneity of a collection of normal means when the collection is 'decreasing on the average' and for testing 'decreasing on the average' as a null hypothesis, is presented. The restriction 'decreasing on the average' is less restrictive than the usual monotone restriction and allows the data to give rise to 'reversals' over short ranges of values of the parameter set. It is closely related to the 'starshaped ordering' restriction.
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