Suppose that X1, . . . , Xn form a large random sample from a Distribution for which the p.d.f. is h(x|θ) = αf(x|θ) + (1− α)g(x|θ). Here f(x|θ) is the p.d.f. of the normal distribution with unknown mean θ and variance 1, g(x|θ) is the p.d.f. of the normal Distribution with the same unknown mean θ and variance σ2, and 0 ≤ α ≤ 1. Let n and denote the sample mean and the sample median, respectively.
a. For σ2 = 100, determine the values of α for which the M.S.E. of will be smaller than the M.S.E. of n.
b. For α = 1/2, determine the values of σ2 for which the M.S.E. of will be smaller than the M.S.E. of n.