The above discussion shows that Bonferroni's test is powerful when there is one strong signal (or few strong signals) in the alternative. In the setting where there are many weak signals in the alternative, Fisher's combination test works better than Bonferroni's test. However, the test statistic Tn = 2 ELI log pt- in Fisher's combination test is difcult to analyze. Instead, we analyze a qualitatively similar test statistic ||Y|| 2 21:1 K2. Continuing with the Gaussian sequence model Y} ~ Arm-5,1), we now test H0 : n1- : 0 for all i E [n] against H; : pt,- 5% D for at least one i E [n]. 8. To compare Bonferroni's test with the X2-Z-test, consider the alternative hypothesis H1 where puz- : m for 711/ 4 of the indices 2' E [n] and pe- : 0 for all the other indices (assuming that Til/4 is an integer). Show that Bonferroni's test has full asymptotic power, while the X2-Z-test has asymptotic power 0!, i.e., it is powerless