6 CHAPTER 7.STATISTICAL FUNCTIONALS AND THE DELTA METHOD 2 Continuity of Functionals of F or P One of the basic properties of a functional T is continuity (or lack thereof).The sense in which we will want our functionals T to be continuous is in the sense of weak convergence. Definition 2.1 A.T:F-R is weakly continuous at Fo if Fn Fo implies T(Fn)T(Fo). T:F-R is weakly lower-semicontinuous at Fo if Fn=Fo implies lim infnT(Fn)>T(Fo). B.T:P→R is weakly continuous at Po∈P if Pn→Po implies T(Pn)一T(Po) Example 2.1 T(F)=fxdF(x)is discontinuous at every Fo:if Fn=(1-n-1)Fo+n-16a,then Fn→Fo since,for boundedψ bdEn=(1-n-l)pd+n-1b(an）→bdo, But T(Fn)=(1-n)T(Fo)+nanoo if we choose an so that n-anoo. Example 2.2 T(F)=(1-2a)-1f-F-1(u)du with 0<a<1/2 is continuous at every Fo: Fn=Fo implies that Fn(t)Fo(t)a.e.Lebesgue.Hence B）=a-2aah 1-a -(1-2a)-1 F(u)du=T(Fo) by the dominated convergence theorem. Example 2.3 T(F)=F-1(1/2)is continuous at every Fo such that Fo is continuous at 1/2. Example 2.4 (A lower-semicontinuous functional T).Let T(F)=Varr(X)=(=-ErX)2dF(a)=EF(X-X) where X,X'F are independent;recall example 1.3.If Fnd F,then liminfnT(Fn)>T(F); this follows from Skorokhod and Fatou. Here is the basic fact about empirical measures that makes weak continuity of a functional T useful: Theorem 2.1 (Varadarajan).If X1,...,Xn are i.i.d.P on a separable metric space (S,d),then Pr(Pn→P)=1. Proof. For each fixed bounded and continuous function we have6 CHAPTER 7. STATISTICAL FUNCTIONALS AND THE DELTA METHOD 2 Continuity of Functionals of F or P One of the basic properties of a functional T is continuity (or lack thereof). The sense in which we will want our functionals T to be continuous is in the sense of weak convergence. Definition 2.1 A. T : F → R is weakly continuous at F0 if Fn ⇒ F0 implies T(Fn) → T(F0). T : F → R is weakly lower-semicontinuous at F0 if Fn ⇒ F0 implies lim infn→∞ T(Fn) ≥ T(F0). B. T : P → R is weakly continuous at P0 ∈ P if Pn ⇒ P0 implies T(Pn) → T(P0). Example 2.1 T(F) = " xdF(x) is discontinuous at every F0: if Fn = (1 − n−1)F0 + n−1δan , then Fn ⇒ F0 since, for bounded ψ # ψdFn = (1 − n−1) # ψdF0 + n−1ψ(an) → # ψdF0. But T(Fn) = (1 − n−1)T(F0) + n−1an → ∞ if we choose an so that n−1an → ∞. Example 2.2 T(F) = (1 − 2α)−1 " 1−α α F −1(u)du with 0 < α< 1/2 is continuous at every F0: Fn ⇒ F0 implies that F −1 n (t) → F −1 0 (t) a.e. Lebesgue. Hence T(Fn) = (1 − 2α) −1 # 1−α α F −1 n (u)du → (1 − 2α) −1 # 1−α α F −1 0 (u)du = T(F0) by the dominated convergence theorem. Example 2.3 T(F) = F −1(1/2) is continuous at every F0 such that F −1 0 is continuous at 1/2. Example 2.4 (A lower-semicontinuous functional T). Let T(F) = V arF (X) = # (x − EF X) 2dF(x) = 1 2 EF (X − X% ) 2 where X, X% ∼ F are independent; recall example 1.3. If Fn →d F, then lim infn→∞ T(Fn) ≥ T(F); this follows from Skorokhod and Fatou. Here is the basic fact about empirical measures that makes weak continuity of a functional T useful: Theorem 2.1 (Varadarajan). If X1, . . . , Xn are i.i.d. P on a separable metric space (S, d), then P r(Pn ⇒ P) = 1. Proof. For each fixed bounded and continuous function ψ we have Pnψ ≡ # ψdPn = 1 n !n i=1 ψ(Xi) →a.s. Pψ ≡ # ψdP