(i) Generalize Theorem9.1.1 to the weighted Bonferroni method. Hint:

Part (i) directly generalizes. To show (ii), let J = i with probability αwi and J = 0 with probability 1 − α. Let U ∼ U(0, 1) and let pˆi = αwiU if J = i; otherwise, let pˆi = (1 − αwi)U + wiα. By conditioning on K = i or K = i, show that pˆi ∼

U(0, 1). Then, the FWER is P(

s i=1

{Pˆ

i ≤ wiα}) = P(

s i=1

{J = i}) = 

i P(K = i) = α .

The first equality follows because a Type 1 error occurs iff J = 0; the second follows because the events {J = i} are disjoint.

(ii) What are the adjusted p-values for the weighted Bonferroni method?