16 CHAPTER 7.STATISTICAL FUNCTIONALS AND THE DELTA METHOD Theorem 4.4 Suppose that: (i)T is Hadamard differentiable tangentially to Cu(F,Pp)at PE P. (ii)FE CLT(P):Vn(Pn-P)Gp (where Gp takes values in Cu(F,Pp)by definition of F∈CLT(P). Then (4) √元(T(Pn)-T(P)→T(GP). Proof.Define gn PC(F)B by gn(x)=Vn(T(P+n-12x)-T(P)). Then,by(),for{△n}Coo(F)with△n-△olF一0and△o∈Cu(F,pp) gn(△n)→T(△o)=g(△o. Thus by the extended continuous mapping theorem in the Hoffmann-Jorgensen weak convergence theory (see van der Vaart and Wellner (1996),Theorem 1.11.1,page 67),gn(Gn)=g(Gp)=T(Gp), and hence(4)holds..▣ The immediate corollary for the classical Mann-Whitney form of the Wilcoxon statistic given in example 4.6 is: Corollary 1 If X1,...,Xm are i.i.d.F and independent of Yi,...,Yn which are i.i.d.G,0< Pr,G(X≤Y)<1,andλw≡m/N≡m/(m+n)→入∈(0,1)，then V{/c-∫pac}=V mn -{T(Fm.Gn)-T(F,G)} d U(F)dG-V(G)dF N(0,o(F,G) where U and V are two independent Brownian bridge processes and (F,G)=(1-)Var(G(X))+XVar(F(Y)) This is,of course,well-known,and can be proved in a variety of other ways (by treating T(Fm,Gn)as a two-sample U-statistic,or a rank statistic,or by a direct analysis),but the proof via the differentiable functional approach seems instructive and useful.(See e.g.Lehmann (1975), Statistical Methods Based on Ranks,Section 5,pages 362-371,and especially example 20,page 365.) Other interesting applications have been given by Gruibel (1988)(who studies the asymptotic theory of the length of the shorth);Pons and Turckheim (1989)(who study bivariate hazard estimators and tests of independence based thereon),and Gill and Johansen (1990)(who prove Hadamard differentiability of the "product integral").Gill,van der Laan,and Wellner (1992) give applications to several problems connected with estimation of bivariate distributions.Arcones and Gine (1990)study the delta-method in connection with M-estimation and the bootstrap. van der Vaart(1991b)shows that Hadamard differentiable functions preserve asymptotic efficiency properties of estimators.16 CHAPTER 7. STATISTICAL FUNCTIONALS AND THE DELTA METHOD Theorem 4.4 Suppose that: (i) T is Hadamard differentiable tangentially to Cu(F, ρP ) at P ∈ P. (ii) F ∈ CLT(P): √n(Pn − P) ⇒ GP (where GP takes values in Cu(F, ρP ) by definition of F ∈ CLT(P)). Then √n(T(Pn) − T(P)) ⇒ T˙ (4) (GP ). Proof. Define gn : P ⊂ +∞(F) → B by gn(x) ≡ √n(T(P + n−1/2x) − T(P)). Then, by (i), for {∆n} ⊂ +∞(F) with .∆n − ∆0.F → 0 and ∆0 ∈ Cu(F, ρP ), gn(∆n) → T˙(∆0) ≡ g(∆0). Thus by the extended continuous mapping theorem in the Hoffmann - Jorgensen weak convergence theory (see van der Vaart and Wellner (1996), Theorem 1.11.1, page 67), gn(Gn) ⇒ g(GP ) = T˙(GP ), and hence (4) holds. ✷ The immediate corollary for the classical Mann-Whitney form of the Wilcoxon statistic given in example 4.6 is: Corollary 1 If X1, . . . , Xm are i.i.d. F and independent of Y1, . . . , Yn which are i.i.d. G, 0 < PF,G(X ≤ Y ) < 1, and λN ≡ m/N ≡ m/(m + n) → λ ∈ (0, 1), then ,mn N (# FmdGn − # F dG) = ,mn N {T(Fm, Gn) − T(F, G)} →d √ 1 − λ # U(F)dG − √ λ # V(G)dF ∼ N(0, σ2 λ(F, G)) where U and V are two independent Brownian bridge processes and σ2 λ(F, G) = (1 − λ)V ar(G(X)) + λV ar(F(Y )). This is, of course, well-known, and can be proved in a variety of other ways (by treating T(Fm, Gn) as a two-sample U−statistic, or a rank statistic, or by a direct analysis), but the proof via the differentiable functional approach seems instructive and useful. (See e.g. Lehmann (1975), Statistical Methods Based on Ranks, Section 5, pages 362 - 371, and especially example 20, page 365.) Other interesting applications have been given by Gr¨ubel (1988) (who studies the asymptotic theory of the length of the shorth); Pons and Turckheim (1989) (who study bivariate hazard estimators and tests of independence based thereon), and Gill and Johansen (1990) (who prove Hadamard differentiability of the “product integral”). Gill, van der Laan, and Wellner (1992) give applications to several problems connected with estimation of bivariate distributions. Arcones and Gin´e (1990) study the delta-method in connection with M− estimation and the bootstrap. van der Vaart (1991b) shows that Hadamard differentiable functions preserve asymptotic efficiency properties of estimators