4. DIFFERENTIABILITY OF FUNCTIONALS T OF F OR P 13 Definition 4.4 A functional T:FR is Hadamard differentiable at F with respect to the Kolmogorov distance dK =llloo (or compactly differentiable with respect to dk)if there exists T(F;)continuous and linear satisfying IT(F)-T(F)-T(F;F-F)I=o(1) 判 for all{F}satisfyingt-(F-F)-△lo→0 for some function△. The motivation for this definition is simply that we can write V(T(E)-T(F))=T(F+n-IPnP(E-F)-T(F) n-1/2 where vn(Fn-F)4U(F)U(F).Hence we can easily deduce the following theorem. Theorem 4.3 Suppose that T:F-R is Hadamard-differentiable at F with respect to llloo Then V元(T(En)-T(F)→aN(0,E(T2(F:1-o,(X)-F)) Moreover, Vn(T(Fn)-T(F))-T(F;Vn(Fn-F))=op(1). Proof.This is easily proved using a Skorokhod construction of the empirical process,or by the extended continuous mapping theorem.Gill(1989)used the Skorokhod approach;Wellner (1989) pointed out the extended continuous mapping proof. One way of treating all the kinds of differentiability we have discussed so far is as follows.Define T(F)-T(F)-T(F;F-F)=Rem(F+th); Here h =t-(F-F).Let S be a collection of subsets of the metric space (F,d.).Then T is S-differentiable at F with derivative T if for all SE S Rem(F+th→0ast→0 aniformly inhS. Now different choices of S yield different degrees of "goodness"of the linear approximation of T by T at F.The three most common choices are just those we have discussed: A.When S=fall singletons of (F,d.),T is called Gateaux or directionally differentiable. B.When S=fall compact subsets of (F,d.)},T is called Hadamard or compactly differentiable. C.When S=fall bounded subsets of F,d,)},T is called Frechet (or boundedly)differentiable. Here is a simple example of a function T defined on pairs of probability distributions (or,in this case,distribution functions)which is compactly differentiable with respect to the familiar supremum (or uniform or Kolomogorov)norm,but which is not Frechet differentiable with respect to this norm.4. DIFFERENTIABILITY OF FUNCTIONALS T OF F OR P 13 Definition 4.4 A functional T : F → R is Hadamard differentiable at F with respect to the Kolmogorov distance dK = . ·. ∞ (or compactly differentiable with respect to dK) if there exists T˙(F; ·) continuous and linear satisfying |T(Ft) − T(F) − T˙(F; Ft − F)| |t| = o(1) for all {Ft} satisfying .t −1(Ft − F) − ∆.∞ → 0 for some function ∆. The motivation for this definition is simply that we can write √n(T(Fn) − T(F)) = T(F + n−1/2n1/2(Fn − F)) − T(F) n−1/2 where √n(Fn − F) d = Un(F) ⇒ U(F). Hence we can easily deduce the following theorem. Theorem 4.3 Suppose that T : F → R is Hadamard - differentiable at F with respect to . · .∞. Then √n(T(Fn) − T(F)) →d N(0, E(T˙ 2(F; 1(−∞,·](X) − F))). Moreover, √n(T(Fn) − T(F)) − T˙(F; √n(Fn − F)) = op(1). Proof. This is easily proved using a Skorokhod construction of the empirical process, or by the extended continuous mapping theorem. Gill (1989) used the Skorokhod approach; Wellner (1989) pointed out the extended continuous mapping proof. ✷ One way of treating all the kinds of differentiability we have discussed so far is as follows. Define T(Ft) − T(F) − T˙(F; Ft − F) ≡ Rem(F + th); Here h = t −1(Ft − F). Let S be a collection of subsets of the metric space (F, d∗). Then T is S−differentiable at F with derivative T˙ if for all S ∈ S Rem(F + th) t → 0 as t → 0 uniformly in h ∈ S. Now different choices of S yield different degrees of “goodness” of the linear approximation of T by T˙ at F. The three most common choices are just those we have discussed: A. When S = {all singletons of (F, d∗)}, T is called Gateaux or directionally differentiable. B. When S = {all compact subsets of (F, d∗)}, T is called Hadamard or compactly differentiable. C. When S = {all bounded subsets of F, d∗)}, T is called Fr´echet (or boundedly) differentiable. Here is a simple example of a function T defined on pairs of probability distributions (or, in this case, distribution functions) which is compactly differentiable with respect to the familiar supremum (or uniform or Kolomogorov) norm, but which is not Fr´echet differentiable with respect to this norm