(i) If p is uniform on (0, 1), show that 2 log(p) has the Chisquared distribution with...

(i) If ˆp is uniform on (0, 1), show that −2 log(ˆp) has the Chisquared distribution with 2 degrees of freedom.

(ii) Suppose ˆp1,..., pˆs are i.i.d. uniform on (0, 1). Let F = −2 log(ˆp1 ··· pˆs). Argue that F has the Chi-squared distribution with 2s degrees of freedom. What can you say about F if the ˆpi are independent and satisfy P{pˆi ≤ u} ≤ u for all 0 ≤ u ≤ 1? [Fisher (1934a) proposed F as a means of combining p-values from independent experiments.]

Section 3.4