Consider the loss function for a positive parameter L(0, a) = (0-a)2/0. This measures the squared...

 

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Consider the loss function for a positive parameter L(0, a) = (0-a)2/0. This measures the squared error relative to the magnitude of the parameter, so that if 0 is large we penalize mistakes slightly less. (a) Show that the formula (in terms of posterior expectations of 0) for the Bayes estimator under this loss function is 8(x) = 1/E{0-1 | x). (b) If X1, ..., Xn~ iid Poisson (0) with prior 8~ Gamma(a, B), find the Bayes estimator under this loss function. (c) Show that this estimator can be written as a weighted average of a data piece, X, and a prior piece, [E{0 -¹1-¹. (d) Compare your result in (b) to the usual posterior mean estimator. Consider the loss function for a positive parameter L(0, a) = (0-a)2/0. This measures the squared error relative to the magnitude of the parameter, so that if 0 is large we penalize mistakes slightly less. (a) Show that the formula (in terms of posterior expectations of 0) for the Bayes estimator under this loss function is 8(x) = 1/E{0-1 | x). (b) If X1, ..., Xn~ iid Poisson (0) with prior 8~ Gamma(a, B), find the Bayes estimator under this loss function. (c) Show that this estimator can be written as a weighted average of a data piece, X, and a prior piece, [E{0 -¹1-¹. (d) Compare your result in (b) to the usual posterior mean estimator.

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