STATS 300 A : Theory of Statistics Fall 2015 Lecture 10 — October 22
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چکیده
X1, . . . , Xn iid ∼ N (θ, σ), with σ known. Our goal is to estimate θ under squared-error loss. For our first guess, pick the natural estimator X. Note that it has constant risk σ 2 n , which suggests minimaxity because we know that Bayes estimators with constant risk are also minimax estimators. However, X is not Bayes for any prior, because under squared-error loss unbiased estimators are Bayes estimators only in the degenerate situations of zero risk (TPE Theorem 4.2.3), and X is unbiased. Thus, we cannot conclude by our previous results (e.g., TPE Corollary 5.1.5) that X is minimax. We might try to consider the wider class of estimators δa,μ0 (X) = aX + (1− a)μ0 for a ∈ (0, 1) and μ0 ∈ R, because many of the Bayes estimators we’ve encountered are convex combinations of a prior and a data mean. Note however that the worst case risk for these estimators is infinite:
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