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 23:1 log pi in Fisher's combination test is difcult to analyze. Instead, we analyze a qualitatively similar test statistic |[Y|| : 121 Y3. Continuing with the Gaussian sequence model Y; ~ N(u,;, l), we new test H0 : psi = 0 for all i E [71] against H1 : m- a 0 for at least one i E [n]. 6. We may do the Xz-test with test statistic ||Y||. However, to further simplify the analysis, let us consider the standardized statistic : ||Y||3 n_ Zn : V 2n Use the central limit theorem to do a Gaussian approximation of Zn under H0 and H1 respec tively. (It is useful to know ELLA] : 3 for z N Na}, 1))