Let X ~ n(μ, Ï2), Ï2 known. For each c ¥ 0, define an interval estimator for

Let X ~ n(μ, σ2), σ2 known. For each c ‰¥ 0, define an interval estimator for μ by C(x) = [x - cσ, x + cσ] and consider the loss in (9.3.4).
a. Show that the risk function, R(μ, C), is given by
R(μ, C), b(2cσ) - P(- c ‰¤ c).
b. Using the Fundamental Theorem of Calculus, show that

and, hence, the derivative is an increasing function of c for c ‰¥ 0.
c. Show that if bσ > l/ˆš2Ï€, the derivative is positive for all c ‰¥ 0 and, hence, R(μ, C) is minimized at c = 0. That is, the best interval estimator is the point estimator C(x) = [x, x].

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