Let X1, X2,..., Xn be iid from a distribution with mean μ and variance Ï2, and let
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(a) Show that, for any estimator of the form aS2, where a is a constant,
MSE(aS2) = E[aS2 - Ï2]2 = a2 Var(S2) + (a - 1)2 Ï4.
(b) Show that
where k = E[X - μ]4 is the kurtosis.
(c) Show that, under normality, the kurtosis is 3Ï4 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.
(e) Show that
(i) For distributions with k > 3, the optimal a will satisfy a (ii) For distributions with k
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