9. (Continued from HW11, #9) Let X1, . . . , Xn iid Beta(, 1). (a) Using 2T as a pivot, find the 100(1 )% confidence interval for . (b) Invert this confidence interval to find a rejection region of the two-sided hypotheses H0 : = 0 vs H1 : = 0. (c) Consider 0 = 1 and compare this result to the LRT you derived previously.

(HW11 #9 Let X1, . . . , Xn iid Beta(, 1). (a) Recall that a complete statistic is T = Pn i=1 log Xi . Show that under = 0, the distribution of 20T is 2 and find the appropriate degrees of freedom . (b) Find the LRT for the hypotheses H0 : = 1 vs H1 : = 1. Write the rejection region in terms of the test statistic T and give values of c in terms of chi-squared critical values. (c) Find the LRT for the hypotheses H0 : 1 vs H1 : < 1. Write the rejection region in terms of the test statistic T and give values of c in terms of chi-squared critical values.)