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Additional info for A Course in Mathematical Statistics and Large Sample Theory
D. t. Lebesgue measure ν on R is 1 (x − μ)2 p(x | θ) = √ exp − 2σ 2 2πσ 2 μ2 1 exp − 2 exp = √ 2σ 2πσ 2 μ 1 x − 2 x2 .
Yn ). 3)). √ Hence the conditional distribution of X given X = z is the distribution of U −1 ( n z, Y2 , . . , Yn ) which does not depend on θ. Hence X is suﬃcient for θ. 1. For statistical inference about θ (or, Pθ ), θ ∈ Θ, it is enough to know the value of a suﬃcient statistic T for θ. For, given T = t, one can simulate the random variable X0 = (X10 , X20 , . . , Xn0 ), say, whose distribution is the same as the conditional distribution, given T = t. The (unconditional, or) marginal distribution of X0 is then the same as the distribution Pθ of X.
Also, (b) d(T Proof. 2 here, noting that the function a → |a − ˆ ) − c(θ)| = E(|d(X) − c(θ) | T )| ≤ c(θ)| is convex. But, more simply, |d(T E(|d(X) − c(θ)| | T ), and taking expectation with respect to Pθ one obtains the desired result. 1(b). 3. 2 extend to the case of estimation of vector valued parametric functions c(θ) = (c1 (θ), . . , 2 ck (θ)) for additive loss functions such as L(θ, a) = 1≤i≤k (ai − ci (θ)) and L(θ, a) = 1≤i≤k |ai − ci (θ)|, respectively. 4. In general there are many suﬃcient statistics for θ.