8. To compare Bonferroni's test with the x2-Z-test, consider the alternative hypothesis H1 where Mi = V2.01 log n for n / of the indices i E [n] and Mi = 0 for all the other indices (assuming that n1/4 is an integer). Show that Bonferroni's test has full asymptotic power, while the x2-Z-test has asymptotic power o, i.e., it is powerless. 9. Consider another example of H1 where u = 3 for v2n indices i E [d] and Mi = 0 for all the other indices. (a) At significance level o = 0.001, show that the asymptotic power of the x'-Z-test is ex- tremely close to 1. (b) For i.i.d. Z1, . .., Zn ~ N(0, 1), it is known that maxie Zi = v2logn (1 + o(1)) where o(1) ->0 as n -> co (this is basically equivalent to the asymptotic behavior of za stated in Problem 5). Use this fact to argue that Bonferroni's test is asymptotically powerless. (It is okay to give a slightly non-rigorous argument. ) The above two problems show that Bonferroni's test is good for detecting a few strong signals, while the x2-Z-test is good for detecting many weak signals