3. [13 marks] Let X = (X1, ... , Xn) be independent and identically distributed random variables from a population with

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3 13 Marks Let X X1 Xn Be Independent And Identically Distributed Random Variables From A Population With 1 (95.49 KiB) Viewed 24 times

3. [13 marks] Let X = (X1, ... , Xn) be independent and identically distributed random variables from a population with density M o 6+1 f(x,0,0) 0 x > 0, 0 > 0, 0 > 0, where 0 and are unknown parameters. a) [2 marks] Apply the Factorization Criterion to show that a two-dimensional sufficient statistic for (0,0) is n T = (T1, T2) II X;, X(1) i=1 b) [2 marks] Show that the sufficient statistic T in part (a) is also minimal sufficient statistic for (0,0). c) [2 marks] Show that the density of the statistic T = X(1) is given by no 10 nø+1 fx, (x) = for x > 0 otherwise zero. Hint: The density for the minimum is given by fxu (yı) = n[1 – Fx(yı)]"-1fx(yı). = d) [2 marks] For the remainder of this question assume that the parameter o is known. Find the MLE of 0 and provide appropriate justification. e) [2 marks] Show that the MLE is a biased estimator for 0. f) [3 marks] Show that T = X(1) is complete and hence find the UMVUE of 0.