12 CHAPTER 7.STATISTICAL FUNCTIONALS AND THE DELTA METHOD A.If T exists in the Frechet sense,then it is unique,and T is Gateaux differentiable with Gateaux derivative T. B.If T is Frechet differentiable at F,then T is continuous at F. C.T(F;G-F)=∫d(G-F)=∫(y-∫ψdF)dG where the functionψis bounded and continuous.. Proof.See Huber(1981),proposition 5.1,page 37. Frechet differentiability leads to an easy proof of asymptotic normality if the metric d.is "compatible with the empirical df or empirical measure". Theorem 4.2 Suppose that T is Frechet differentiable at F with respect to d.and that (2) Vmd(Fn,F)=O(1)》 Then v元(T(Fn)-T(F)= vrdvn(Fn-F)}+op(1) 2r+o0 =1 →aN(0,E2(X) Proof.By Frechet differentiability of T at F, √元(T(Fn)-T(F)=√元 brdn+√no(d(Fn,F)】 Vn vrdFn+ d,图Vnd,.g,F d.(Fn;F) rdn+o(1)O,(1) by(2)and(1).▣ Note that if d,is the Levy metric dL or the Kolmogorov metric dk on the line,then (2)is satisfied: √ndz(Fn,F)≤Vndk(En,F)=VmFn-Fle兰IlUn(F)lo→alU(F)l川o. Unfortunately,if d.=dpr or d.=dBL,then vnd.(Fn,F)is not Op(1)in general;see Dudley (1969), Kersting (1978),and Huber-Carol (1977).Thus we are lead to consideration of other metrics such as the Kolmogorov metric and generalizations thereof for problems concerning functionals T(P)of probability distributions P.While some functionals T are Frechet differentiable with respect to the supremum or Kolmogorov metric,we can make more functionals differentiable by considering a somewhat weaker notion of differentiability as follows:12 CHAPTER 7. STATISTICAL FUNCTIONALS AND THE DELTA METHOD A. If T˙ exists in the Fr´echet sense, then it is unique, and T is Gateaux differentiable with Gateaux derivative T˙ . B. If T is Fr´echet differentiable at F, then T is continuous at F. C. T˙(F; G−F) = " ψd(G−F) = " (ψ−" ψdF)dG where the function ψ is bounded and continuous. Proof. See Huber (1981), proposition 5.1, page 37. ✷ Fr´echet differentiability leads to an easy proof of asymptotic normality if the metric d∗ is “compatible with the empirical df or empirical measure”. Theorem 4.2 Suppose that T is Fr´echet differentiable at F with respect to d∗ and that √ (2) nd∗(Fn, F) = Op(1). Then √n(T(Fn) − T(F)) = # ψF d{ √n(Fn − F)} + op(1) = 1 √n !n i=1 ψF (Xi) + op(1) →d N(0, Eψ2 F (X)). Proof. By Fr´echet differentiability of T at F, √n(T(Fn) − T(F)) = √n # ψF dFn + √no(d∗(Fn, F)) = √n # ψF dFn + o(d∗(Fn, F)) d∗(Fn, F) √nd∗(Fn, F) = √n # ψF dFn + o(1)Op(1) by (2) and (1). ✷ Note that if d∗ is the L´evy metric dL or the Kolmogorov metric dK on the line, then (2) is satisfied: √ndL(Fn, F) ≤ √ndK(Fn, F) = √n.Fn − F.∞ d = .Un(F).∞ →d .U(F).∞. Unfortunately, if d∗ = dP r or d∗ = dBL, then √nd∗(Fn, F) is not Op(1) in general; see Dudley (1969), Kersting (1978), and Huber-Carol (1977). Thus we are lead to consideration of other metrics such as the Kolmogorov metric and generalizations thereof for problems concerning functionals T(P) of probability distributions P. While some functionals T are Fr´echet differentiable with respect to the supremum or Kolmogorov metric, we can make more functionals differentiable by considering a somewhat weaker notion of differentiability as follows: