= 6. Let X1, X2, ..., Xn be a random sample from a population with pdf and cdf given below, 0 > 0. e fx(x) con for 1

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6 Let X1 X2 Xn Be A Random Sample From A Population With Pdf And Cdf Given Below 0 0 E Fx X Con For 1 X 1 (105.4 KiB) Viewed 23 times

= 6. Let X1, X2, ..., Xn be a random sample from a population with pdf and cdf given below, 0 > 0. e fx(x) con for 1<x (and 0 otherwise) +1 E(X) = 0-1 Ꮎ Fx(x) = 1-- for 1<x (and 0 otherwise) Var(X) x (0-1)(0–2) Recall the sufficient statistic s={-, Inx; ~ Gam(1/0, K=n). a) (5) Derive a 100(1 – a)% equal-tail confidence interval for 0. Be as explicit as possible about the percentiles. Hint: 0 is not a location nor scale parameter. i=1 b) (3) Show that the joint pdf for f(x) has a negative monotone likelihood ratio in s=X?-, In X; . c) (4) Derive the uniformly most powerful (UMP) test of size a = 0.05 for the hypotheses Ho: Os 0. versus Ha: 0 > 0. Be as explicit as possible about your critical (rejection) region. d) (5) Determine the power function ( O') for the test in part (c). Show that this test is unbiased.