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# 1

Let X1, ..., X, be a random sample from the density f(x; 0) = 0(1 + x)-(1+/(0. .) (x) for 0 >0. (a) Estimate 6 by the method of moments assuming 0 > 1. (b) Find the maximum-likelihood estimator of 1/0. (c) Find a complete and sufficient statistic if one exists. (d) Find the Cramer-Rao lower bound for unbiased estimators of 1/0. (e) Find the UMVUE of 1/0 if such exists. Find the UMVUE of 0 if such exists.Show that 92 202 log f(X; 0)Let X1, ..., Xn denote a random sample from f(x; 0) =f.(x) = Of.(x) + (1 -0)fo(x), where 0 0