Let X1,..., X, be a random sample from a Poisson distribution with mean A. (i) Find the Cramér-Rao lower bound for the variances of unbiased estimators of X (ii) Find a uniformly minimum variance unbiased estimator of A. (iii) Is the maximum likelihood estimator of A asymptotically unbiased? (iv) Find the Cramér-Rao lower bound for the variances of unbiased estimators o 7(A) = e-A. (v) It can be shown that 7 [(n- 1)/n]E. X. is an unbiased estimator of r in (iv) Find the variance Var(7) and compare it with the Cramér-Rao lower bound (o

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Let X1,..., X, be a random sample from a Poisson distribution with mean A.
(i) Find the Cramér-Rao lower bound for the variances of unbiased estimators of A.
(ii) Find a uniformly minimum variance unbiased estimator of A.
(iii) Is the maximum likelihood estimator of A asymptotically unbiased?
(iv) Find the Cramér-Rao lower bound for the variances of unbiased estimators of
T(A) = e-A.
(v) It can be shown that 7 = [(n – 1)/n]E Xi is an unbiased estimator of T in (iv).
Find the variance Var(7) and compare it with the Cramér-Rao lower bound (of
(iv)).
Transcribed Image Text:Let X1,..., X, be a random sample from a Poisson distribution with mean A. (i) Find the Cramér-Rao lower bound for the variances of unbiased estimators of A. (ii) Find a uniformly minimum variance unbiased estimator of A. (iii) Is the maximum likelihood estimator of A asymptotically unbiased? (iv) Find the Cramér-Rao lower bound for the variances of unbiased estimators of T(A) = e-A. (v) It can be shown that 7 = [(n – 1)/n]E Xi is an unbiased estimator of T in (iv). Find the variance Var(7) and compare it with the Cramér-Rao lower bound (of (iv)).
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