Decomposable statistics and the hypotheses testing for groupped data

The paper deals with testing a simple hypothesis against alternatives converging to the hypothesis at rate $n^{-1/2}$ ($n$ being the sample size) in the case when the number of classes-intervals increases infinitely together with $n$.
The concept of highly sensitive criterion is introduced and general features of such criteria generated by decomposable statistics ([2]) are studied. It is shown that in the class of criteria under consideration the optimal (in a some sense) criteria are linear criteria, each of them being the most powerful test.