.... 1. Let XX,, be a random sample from f (x; 0) = 0x-1,0 0 is unknown. n a) Find a sufficient statistic for and check for completeness. Is X, a sufficient statistic? i=1 b) Find the Cramer-Rao lower bound for the variance of unbiased estimators of 0. c) Is there a function of for which there exists an unbiased estimator whose variance coincides with the CRLB? d) Find the UMVUE of each of the following: 0, 2. Let X1, X2,..., X, be a random sample from f(x; 0) = x-(1-0)0 0. n a) Derive the minimal sufficient statistic for 0. b) Argue that the MLE of is a sufficient statistic c) Find the CRLB for the variance of unbiased estimators of 0. d) Find the UMVUE of 0. 3. Let X1, X2, X, be a random sample from (x; , ) Theorem to show that n a) IX, is sufficient for a if is known. b) c) i=1 i=1 X is sufficient for if a is known. () (a) = -xa-1e-x,x>0. Use the Neyman Factorization IXX is jointly sufficient for (a,) if both parameters are unknown. Check for completeness. i=1