Consider the two-parameter normal density function Show that the unbiased estimator does not attain the information bound for its variance. Let X1,...,Xn be n i.i.d. random variables having...

Consider the two-parameter normal density function

Show that the unbiased estimator

does not attain the information bound for its variance.

Let X₁,...,X_n
be n i.i.d. random variables having normal distribution with location parameter µand scale parameter σ > 0. Express E(X) and Var(X) in terms of θ = (µ, σ) and, hence, use the method of moment to estimate θ. Also, work out the case when X has a Laplace/logistic distribution with location and scale parameters θ₀
and θ₁, respectively. Comment on their optimality properties.