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• 1. Let X1, X2, ..., Xn be a random sample of size n from the continous distribution with pdf S0(1 – x)0–1, 0 < x < 1 f(x; 0) = 10, otherwise for some parameter 0 > 0. e Consider testing Ho :0=1 versus H1 : 0 + 1. Which of the following statements are true? (Check all that apply.) The uniformly most powerful (UMP) test is the same as the generalized likelihood ratio test (GLRT). Under H., X1, X2, ..., Xn are iid from the uniform distribution over the interval (0,1). A uniformly most powerful (UMP) test does not exist. For a large sample size, the generalized likelihood ratio 1(X) has an approximate x²(1) distribution.

2. Let X1, X2,... Xn be a random sample of size n from the continous distribution with pdf S0(1 – x)8–1, 0<x< 1 f(x; 0) = 10, otherwise for some parameter 2 > 0. Consider testing Ho : 0 = 1 versus H1:0 + 1. . What is the maximum likelihood estimator needed for computing the denominator of the generalized likelihood ratio (GLR)? n O 6 = 0 O Ô = L!__61–X;) Ô 11-1(1-X;) n Ô = -n 2?_1 In(1-X;)

5. Suppose that X1, X2, ..., Xn is a random sample from the I(2,B) distribution. Suppose that n is "large". a Consider testing Ho : ß = 1 versus H1 : 3 + 1 using an approximate large sample generalized likelihood ratio test. Suppose that the generalized likelihood ratio 1(X) is observed to be lã) = 0.67. Which of the following is an approximate large sample test using a = 0.05? The rule is to reject Ho, in favor of Hy if – 2 In 1(X) > 0.0039. For the given lã), we reject Ho. The rule is to reject Ho, in favor of Hų if –2 In 1(Ă) > 5.0239. For the given (ã), we fail to reject Ho. I can not answer this question without knowing the value of n. The rule is to reject Ho, in favor of H{ if –2 In 1(X) > 3.841. For the given 1ā), we fail to reject Ho.