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Let X1, X2, . . . , Xn be iid exponential random variables with unknown mean...

Let X1, X2, . . . , Xn be iid exponential random variables with unknown mean β.

(1) Find the maximum likelihood estimator of β.

(2) Determine whether the maximum likelihood estimator is unbiased for β.

(3) Find the mean squared error of the maximum likelihood estimator of β.

(4) Find the Cramer-Rao lower bound for the variances of unbiased estimators of β.

(5) What is the UMVUE (uniformly minimum variance unbiased estimator) of β? What is your reason?

(6) Determine the asymptotic distribution of the maximum likelihood estimator of β as n → ∞.

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