46 Section 8.3.Efficiency Suppose we want to estimate a real-valued function of 0,say g(0), g():RR.Assume that g has continuous partial derivatives. Consider estimating g(0o)by g((Xn)),where (Xn)is the MLE. Assuming that the 8 regularity conditions hold,we know that √元((Xn)-do)巳N(0,I-l(do).By the multivariate delta method,we know that √m(g((xn)-g(0o)2N(0,a(0o)'I-1(0o)a(0o) where a(0o)is the gradient of g(0)at 00
46 Section 8.3. Efficiency Suppose we want to estimate a real-valued function of θ, say g(θ), g(·) : Rk → R. Assume that g has continuous partial derivatives. Consider estimating g(θ0) by g( ˆ θ(Xn)), where ˆ θ(Xn) is the MLE. Assuming that the 8 regularity conditions hold, we know that √n( ˆ θ(Xn) − θ0) D→ N(0, I−1(θ0)). By the multivariate delta method, we know that √n(g(ˆθ(Xn)) − g(θ0)) D→ N(0, a(θ0)I−1(θ0)a(θ0)) where a(θ0) is the gradient of g(θ) at θ0
47 By the information inequality,we know that among all unbiased estimators,6(Xn),of g(0), Varoo [6(Xn)]=Eoo[(8(Xn)-g(0o))2]>a(00)'In (0o)a(0o)(1) where In(00)=nI(00). Consider an estimator,T(Xn),of g(0o)which is asymptotically normal and asymptotically unbiased,i.e., n(T(Xn)-g(0o))2 N(0,V(0o)) (2) It turns out that under some additional regularity conditions on T(Xn),we can show that V(0o)≥a(0o)'I-1(0o)a(0o) (3)
47 By the information inequality, we know that among all unbiased estimators, δ(Xn), of g(θ), V arθ0 [δ(Xn)] = Eθ0 [(δ(Xn) − g(θ0))2] ≥ a(θ0)I−1 n (θ0)a(θ0) (1) where In(θ0) = nI(θ0). Consider an estimator, T(Xn), of g(θ0) which is asymptotically normal and asymptotically unbiased, i.e., √n(T(Xn) − g(θ0)) D→ N(0, V (θ0)) (2) It turns out that under some additional regularity conditions on T(Xn), we can show that V (θ0) ≥ a(θ0)I−1(θ0)a(θ0) (3)
48 Definition:A regular estimator T(Xn)of g(0o)which satisfies (2) with V(0o)=a(0o)'I1(00)a(0o)is said to be asymptotically efficient. If g((X))is regular,then we know that it is asymptotically efficient. Remarks about Lower Bounds (1)and (3) .(1)is attained only under exceptional circumstances (i.e., usually need completeness),while (3)is obtained under quite general regularity conditions. The UMVUE tends to be unique,while asymptotically efficient estimators are not.If T(Xn)is asymptotically efficient,then so is T(Xn)+Rn;provided nRn 0. In (1),the estimator must be unbiased,whereas in(3),the estimator must be consistent and asymptotically unbiased
48 Definition: A regular estimator T(Xn) of g(θ0) which satisfies (2) with V (θ0) = a(θ0)I−1(θ0)a(θ0) is said to be asymptotically efficient. If g( ˆ θ(Xn)) is regular, then we know that it is asymptotically efficient. Remarks about Lower Bounds (1) and (3) • (1) is attained only under exceptional circumstances (i.e., usually need completeness), while (3) is obtained under quite general regularity conditions. • The UMVUE tends to be unique, while asymptotically efficient estimators are not. If T(Xn) is asymptotically efficient, then so is T(Xn) + Rn, provided √nRn P→ 0. • In (1), the estimator must be unbiased, whereas in (3), the estimator must be consistent and asymptotically unbiased
49 .V(0o)in (3)is an asymptotic variance,whereas (1)refers to the actual variance of 6(Xn). For a long time,it was believed that the regularity conditions needed to make (3)hold involved regularity conditions on the density p(x;0).This belief was exploded by Hodges
49 • V (θ0) in (3) is an asymptotic variance, whereas (1) refers to the actual variance of δ(Xn). For a long time, it was believed that the regularity conditions needed to make (3) hold involved regularity conditions on the density p(x; θ). This belief was exploded by Hodges
50 Hodges'Example of Super-Efficiency Suppose thatXn=(X1,...,Xn)where the Xi's are i.i.d. Normal(uo,1).We can show that I(uo)=1 for all uo.Consider the following estimator of uo, n|xn≥n-1/4 )|xml<n-1/4 Hodges showed that: VHECN. N(0,1)ifo≠0 The latter variance makes the asymptotic distribution of the MLE inadmissible
50 Hodges’ Example of Super-Efficiency Suppose that Xn = (X1,...,Xn) where the Xi’s are i.i.d. Normal(µ0, 1). We can show that I(µ0) = 1 for all µ0. Consider the following estimator of µ0, µ ˆ(Xn) = ⎧⎨⎩ X¯ n |X¯ n| ≥ n−1/4 0 |X¯ n| < n−1/4 Hodges showed that: √n(ˆµ(Xn) − µ0) →D ⎧⎨⎩ N(0, 1) if µ0 = 0 0 if µ0 = 0 The latter variance makes the asymptotic distribution of the MLE inadmissible