Simultaneous testing of hypotheses and alternatives

Abstract: To identify statistically significant conclusions, it is proposed to simultaneously test hypotheses and alternatives. It is shown that, under the condition of free combination of hypotheses and alternatives, the closure method leads to single-step procedures for the simultaneous testing of hypotheses and alternatives. Using an example it is shown the result is lost when hypotheses and alternatives do not satisfy the condition of free combination of hypotheses and alternatives. It is shown that the average number of insignificant conclusions is an uncontrolled part of the risk function of a single-step procedure with an appropriate choice of the loss function.