7. Let Xâ‚, Xâ‚‚, ..., Xâ‚™ be a random sample from a normal distribution with mean μ and standard deviation σ.
a) Find the maximum likelihood estimators of μ and σ.
b) Show that the estimator of μ is unbiased but the estimator of σ is biased.
c) Compute the bias of the estimator of σ and comment on it. Hence, write a function of this estimator that is unbiased, find its variance, and show that it is consistent.
d) Find the Cramer-Rao lower bound variance of σ and hence find the absolute efficiency of the unbiased estimator you proposed in (c).