Gleser and Healy (1976) give a detailed treatment of the estimation problem in the n(, a2) family,

Question:

Gleser and Healy (1976) give a detailed treatment of the estimation problem in the n(θ, aθ2) family, where o is a known constant (of which Exercise 7.50 is a special case). We explore a small part of their results here. Again let X1... ,Xn be iid n(θ,θ2), θ > 0, and let and cS be as in Exercise 7.50. Define the class of estimators
T = {T:T = ai+ a2(cS)} ,
where we do not assume that a1 + 02 = 1.
(a) Find the estimator T ∈ T that minimizes Eθ(0 - T)2; call it T*.
(b) Show that the MSE of 7* is smaller them the MSE of the estimator derived in Exercise 7.50(b).
(c) Show that the MSE of T*+ = max{0,T*} is smaller than the MSE of T*.
(d) Would θ be classified as a location parameter or a scale parameter? Explain.