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**Extra info for A Course in Mathematical Statistics and Large Sample Theory**

**Sample text**

Thus d0 and d1 are both Bayes estimators. But d1 is clearly inadmissible, since for all 0 < θ < 1, R(θ, d0 ) < R(θ, d1 ), while R(θ, d0 ) = R(θ, d1 ) for θ = 0, 1. To rule out situations like this, one may require a property such as P1 below. Property P1 There exists a subset Θ of Θ such that (1) τ (Θ) > 0 and (2) if Pθ0 (A) = 0 for some θ0 ∈ Θ (and for some event A ⊂ X ), then Pθ (A) = 0 for all θ ∈ Θ. 7 does not satisfy P1 . 4. In a decision problem with loss c(θ − a)2 (c > 0 independent of θ), suppose a prior τ has the property P1 .

Indeed, one can deﬁne this space canonically as Ω = X × Θ with the product sigma-ﬁeld, and with the probability measure P speciﬁed by P (C × D) = Pθ (C)dτ (θ) for all C ∈ B(X ) and D ∈ B(Θ). 13) then may be expressed as r(τ, d0 ) = EL(ϑ, d(X)). , with respect to the prior τ ). 13) says r(τ, d) = EL(ϑ, d(X)) = E[E(L(ϑ, d(X)) | ϑ)]. 16) One may reverse the order of integration, by ﬁrst taking the conditional expectation, given X, and then integrating this conditional expectation over the (marginal) distribution of X: r(τ, d) = EL(ϑ, d(X)) = E[E(L(ϑ, d(X)) | X)].

57) Ex. 5. Let the observation space be X = {0, 1, . . }n ≡ Zn+ , n n e−θ f (x | θ) = i=1 θ xi θ 1 xi ≡ e−nθ n xi ! i=1 xi ! 58) where θ ∈ Θ = (0, ∞), A = [0, ∞). 4). 36 3 Introduction to General Methods of Estimation n Ex. 6. Let Θ = [0, 1] = A , X = {0, 1}n, Pθ ({x}) = θr (1 − θ)n−r (r = 1 xi ), the beta prior L(θ, a) = c(θ − a)2 (c > 0). 5, the Bayes rule for √ Be (α, β) is (r + α)/(n + α + β) ≡ d0 (X). 59) is minimax, as well as admissible. Ex. 7. Admissibility of the Sample Mean from a Poisson Distribution.