Asymptotic Optimality of Fisher's Method of Combining Independent Tests

Four methods of combining independent tests of hypothesis are compared via exact Bahadur relative efficiency. The methods considered are Fisher's method, the mean of the normal transforms of the significance levels, the maximum significance level, and the minimum significance level. None of these is uniformly more powerful than the others, but, according to Bahadur efficiency, Fisher's method is the most efficient of the four. In some cases, Fisher's method is most efficient of all tests based on the data, but this is not generally true.