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1. Let X1,... , Xn be IID random points from Exp(1/B). The PDF of Exp(1/B) is for x 〉 0. Let X,-1 Σー X, be the sample average. Let 3 be the parameter of interest that we want to estimate. Xi be the sample average. Let B be the parameter of (a) (1 pt) What is the bias and variance of using the sample average Xn as the estimator of 3? (b) (0.5 pt) What is the mean square error of using Xn as the estimator of β? (c) (0.5 pt) Does Xn converges to B? Why? (d) (0.5 pt) Now consider a new estimator β a × Xn, where a E R is a real number. What is the mean square error of β? (e) (0.5 pt)·To minimize the mean square error, which value of a should we ke? Does this give us an estimator that has a lower mean square error than the sample mean Xn? (Note: this new estimator is related to the Steins Shrinkage estimator.

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Stas es anola 는。 に.msElala minimum terasa 2n に』 8-P 2 2. rn) NON asnうAll the final answers are clearly mentioned in the boxes marked by red.

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