Data y = {y1, . . . , yn} are observed where yi iid∼ Gamma(4, 1 β ), with β > 0. (a) Derive a Method of Moments estimato

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Data y = {y1, . . . , yn} are observed where yi iid∼ Gamma(4, 1
β ), with β > 0.
(a) Derive a Method of Moments estimator βˆMM for β, based on
the first moment of f(y).
(b) Is your estimator biased? If so, what is the bias?
(c) What is the variance of your estimator?
(d) Write down the log-likelihood function `(β|y).
(e) Derive the score function ` 0 (β|y).
(f ) Derive the observed Fisher Information I(β).
(g) Derive the expected Fisher Information E[I(β)].
(h) Hence find the Maximum Likelihood Estimator βˆMLE for
β
(i) Prove that βˆMLE is the Minimum Variance Unbiased
Estimator.