Let X = (X1,., Xn) be a random sample from the uniform distribution on (0,0), where...

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Let X = (X1,., Xn) be a random sample from the uniform distribution on (0,0), where 0 > 0 is unknown. Let 7(0) = ba'0-(b+1) I(a.(0) be a prior density with respect to the Lebesgue measure, where b > 1 is known but a > 0 is an unknown hyperparameter. Consider the estimation of 0 under the squared error loss. (i) Show that the empirical Bayes method using the method of moments produces the empirical Bayes action 8(â), where d8(a)= 2(b-1) n bn b+n b+n-1 , max{a, X(n)}, E-1 Xi, and X(n) is the largest order statistic. i3D1 (ii) Let h(a) = a-lI0,00) (a) be an improper Lebesgue prior density for a. Obtain explicitly the generalized Bayes action. Let X = (X1,., Xn) be a random sample from the uniform distribution on (0,0), where 0 > 0 is unknown. Let 7(0) = ba'0-(b+1) I(a.(0) be a prior density with respect to the Lebesgue measure, where b > 1 is known but a > 0 is an unknown hyperparameter. Consider the estimation of 0 under the squared error loss. (i) Show that the empirical Bayes method using the method of moments produces the empirical Bayes action 8(â), where d8(a)= 2(b-1) n bn b+n b+n-1 , max{a, X(n)}, E-1 Xi, and X(n) is the largest order statistic. i3D1 (ii) Let h(a) = a-lI0,00) (a) be an improper Lebesgue prior density for a. Obtain explicitly the generalized Bayes action.

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