Let X1, X2,..., Xn be iid from a distribution with mean μ and variance Ï2, and let S2 be the usual unbiased estimator of Ï2.

Let X1, X2,..., Xn be iid from a distribution with mean μ and variance σ2, and let S2 be the usual unbiased estimator of σ2. In Example 7.3.4 we saw that, under normality, the MLE has smaller MSE than S2. In this exercise will explore variance estimates some more.
(a) Show that, for any estimator of the form aS2, where a is a constant,
MSE(aS2) = E[aS2 - σ2]2 = a2 Var(S2) + (a - 1)2 σ4.
(b) Show that

where k = E[X - μ]4 is the kurtosis.
(c) Show that, under normality, the kurtosis is 3σ4 and establish that, in this case, the estimator of the form aS2 with the minimum MSE is n-1/n+1 S2.
(d) If normality is not assumed, show that MSE(aS2) is minimized at

which is useless as it depends on a parameter.€ƒ
(e) Show that
(i) For distributions with k > 3, the optimal a will satisfy a (ii) For distributions with k

Var(S2) = 1 (k-n-34 a- n+1)12'