Abstract
The paper is concerned with testing nonparametric hypotheses about the underlying support G of independent and identically distributed observations. It is assumed that G belongs to a class .Cf' of compact sets with smooth upper surface called boundary fragments. It is required to distinguish the simple null hypothesis specified by a known set Go in ~ against nonparametric alternatives that G belongs to a class obtained by removing a certain neighbourhood of Go in ~. Using the asymptotic minimax approach, the problem is to determine the order of the smallest distance between the null hypothesis Ho and the alternatives for which one is able to test the null hypothesis against the alternatives with a given summarized error.
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