Asymptotically Optimal Methods of Combining Tests

A criterion for the asymptotic optimality of combined independent tests is given. Methods other than Fisher's are shown to satisfy the criterion. Certain problems call for a weighted combination of tests. It is shown that many weighted procedures are optimal. In particular, a weighted procedure based on gamma transforms is optimal. This weighted procedure has the additional advantage that critical values are easily determined. Good's weighting procedure is shown not to be asymptotically optimal.