Question Three Let X = (X1, X2, . .., An) be independent and identically distributed random variables from a population with a density function 303 f(x, 0) = - where 0 > 0 is an unknown parameter. a) SUBMITTED [2 marks] Which of the following is a valid reason why T = X(1) is minimal sufficient for @? The Lehmann and Scheffe's method since L ( Y , 0 ) L(X, 0 ) IIT I(X(11,20) (0) ' which does not depend on 0 if and only if X(1) = Y(1)- The Neyman-Fisher Factorization Criterion since 3 93n L(X, 0) = II? X -(X,1,00) (0) and T' is one-dimensional. The Neyman-Fisher Factorization Criterion since 3n 93n L(X, 0 ) = and T' is one-dimensional. O Since f belongs to the one-parameter exponential family.\f() [2 marks] Find the density of T = X(1) fort > 0. Hint: You may use the density of the minimum: fx (y1) = n[1 - Fx(y1)]"-ifx(y1) O fx1) (t, 0 ) = 3n03n *3n+1 O fx) (t, 0 ) = 13n+1 O tan+1 fx1) (t, 0 ) = 3n03n O fx1 (t, 0) = tantl non O fx1) (t, 0) = #4n+1\fe) [2 marks] It can be shown that T = X(1) is also complete for 0 and 3n E(X1)) = 3n - 1 Hence, or otherwise, compute the UMVUE for 0. O 3n - 1 Gumvue = X(1) 3n O 3n dumvue = 3n - 1 X(1) O Gumvue = n - 1 X(1) n O Gumvue = X(1) O n Gumvue = n - 1 X(1)