Tests and optimal guarded weights of evidence

Optimal guarded weights of evidence for simple alternatives to a hypothesis are shown to belong to two distinct classes, depending on whether or not they include test critical functions as limiting cases for distant alternatives. In particular, it is shown that for shift families with monotone likelihood ratio, as the alternative moves infinitely far from the hypothesis, the minimum risk level-α weight of evidence for the alternative approaches a level-α Neyman-Pearson test critical function if and only if the score function is unbounded. The results are extended to a class of one-parameter families with monotone likelihood ratio, in particular exponential families. Examples include scale and shape parameters.