Abstract
A test for a hypothesized parameter is generalized by replacing the indicator function of the test critical region with a function (‘weight of evidence for the alternative’) having values in [0,1] and estimating the value 1 when the alternative is true and 0 otherwise. It is a ‘guarded’ weight of evidence if a bound is placed on the Type I risk. The focus of this paper is on a guarded weight of evidence which is a function of the likelihood ratio of the sign statistic for a two‐sided alternative to a point hypothesis regarding the centre of a symmetric distribution. Inversion of a family of such guarded weights of evidence yields an ‘acceptability profile’ for the median which is more informative than the traditional confidence interval for the median. The main results, with the exception of the comparison of the Type II risks with an envelope risk, are based entirely on permutation arguments.
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