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Let X1,X2,...,Xn be iid exponential random variables with unknown mean β. (b) Find the maximum likelihood...

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

(b) Find the maximum likelihood estimator of β.

(c) Determine whether the maximum likelihood estimator is unbiased for β.
(d) Find the mean squared error of the maximum likelihood estimator of β.

(e) Find the Cramer-Rao lower bound for the variances of unbiased estimators of β.
(f) What is the UMVUE (uniformly minimum variance unbiased estimator) of β? What is your reason?
(g) Determine the asymptotic distribution of the maximum likelihood estimator of β as n →∞.

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P 1 let *, , X2 -. *n belid exponential random vagiable with unknown mean UH [(*): B and roo(%) - B2 f(XB) - 1/82 VB B E xy,ara Cre) To determine the maximum likelihood estimated un biagod. Por B E (Eu (Emi) - ΣΕ(α) n Eno ß n. fp n B MALE is unbiaseEl-22log 2 3 t (De-) +1(52) ) + 2 1E() 2. 1 p3 +2nA p) 32 2n | | 32 32 n 32 (ramer-Rao Lowor bound is E (-22logi 이32 D 32 2.

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