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Extra resources for A Course in Mathematical Statistics and Large Sample Theory
22) Hence r(τ, d) = E(L(ϑ, d(X)) = E[E(L(ϑ, d(X)) | X] ≥ E[E(L(ϑ, d0 (X)) | X)] = E(L(ϑ, d0 (X)) = r(τ, d0 ). 1. s. 1. If the action space A is a (measurable) convex set C, containing the range of g, then under squared error loss L(θ, a) = |g(θ) − a|2 , E(g(ϑ) | X) is a Bayes estimator of g(θ). 2. Let g(θ) be a real-valued measurable function on Θ having a ﬁnite absolute ﬁrst moment under the prior τ . Let the action space A be an interval containing the range of g. , the conditional distribution of ϑ, given X) is a Bayes estimator of θ.
95. (a) Find the method-of-moments estimates of α, β. (b) Use the estimates in (a) as the initial trial solution of the likelihood equations, and apply the Newton–Raphson, or the gradient method, to compute the MLEs α, β, by iteration. Ex. 12. Consider X = (X1 , . . d. N (μ, σ 2 ) with μ known and θ = σ 2 > 0 is the unknown parameter. , 1/σ 2 has the gamma distribution G (α, β). (a) Compute the posterior distribution of σ 2 . ] (b) Find the Bayes estimator of σ 2 under squared error loss L(σ 2 , a) = (σ 2 − a)2 .
Zk ), μ = (μ1 , . . , uk ). 21) follows from the relations E(Zi −ci )2 ≡ E(Zi −μi + μi −ci )2 = E(Zi −μi )2 + (μi −ci )2 + 2(μi −ci )E(Zi −μi ) = E(Zi −μi )2 + (μi −ci )2 (i = 1, . . , k). 1. The posterior mean of ϑ is E(ϑ | X) = d0 (X), say. If d is any other decision rule (estimator), then one has, by applying the Lemma to the conditional distribution of ϑ, given X, E(L(ϑ, d(X)) | X) ≡ E(|ϑ − d(X)|2 | X) ≥ E(|ϑ − d0 (X)|2 | X) ≡ E(L(ϑ, d0 (X)) | X). 22) Hence r(τ, d) = E(L(ϑ, d(X)) = E[E(L(ϑ, d(X)) | X] ≥ E[E(L(ϑ, d0 (X)) | X)] = E(L(ϑ, d0 (X)) = r(τ, d0 ).