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 have
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 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
2 CONTINUITY OF FUNCTIONALS OF F OR P 7 by the ordinary strong law of large numbers.The proof is completed by noting that the collection of bounded continuous functions on a separable metric space (S,d)is itself separable.See Dudley (1989),sections 11.2 and 11.4. Combining Varadarajan's theorem with weak continuity of T yields the following simple result. Proposition 2.1 Suppose that: (i)(,A)=(S,BBoret)where (S,d)is a separable metric space and BBoret denotes its usual Borel sigma-field. (i)T:P→R is weakly continuous at Po∈P. (iii)X1,...,Xn are i.i.d.Po. Then Tn≡T(Pn)→a.s.T(Po): Proof. By Varadarajan's theorem 2.1,Pn Po a.s.Fix wE A with Pr(A)=1 so that P%→Po.Then by weak continuity of T,Tn(P%)→T(Po).口 A difficulty in using this theorem is typically in trying to verify weak-continuity of T.Weak continuity is a rather strong hypothesis,and many interesting functions fail to have this type of continuity.The following approach is often useful. Definition 2.2 Let FC Li(P)be a collection of integrable functions.Say that PnP with respect to‖‖F if llPn-Plr=supfer|Pn(f)-P(f川一0.Furthermore,.we say that T:P→R is continuous with respect to‖·.F if Pn-PllF→0 implies that T(Pn)→T(P). Definition 2.3 IfFCL(P)is a collection of integrable functions with Pn-P0,we then say that F is a Glivenko-Cantelli class for P and write FE GC(P). Theorem 2.2 Suppose that: ()F∈GC(P)；i.e.Pn-Pl一as.0. (ii)T is continuous with respect to‖l·lF Then T(Pn)→a.s.T(P)
2. CONTINUITY OF FUNCTIONALS OF F OR P 7 by the ordinary strong law of large numbers. The proof is completed by noting that the collection of bounded continuous functions on a separable metric space (S, d) is itself separable. See Dudley (1989), sections 11.2 and 11.4. ✷ Combining Varadarajan’s theorem with weak continuity of T yields the following simple result. Proposition 2.1 Suppose that: (i) (X , A)=(S,BBorel) where (S, d) is a separable metric space and BBorel denotes its usual Borel sigma - field. (ii) T : P → R is weakly continuous at P0 ∈ P. (iii) X1, . . . , Xn are i.i.d. P0. Then Tn ≡ T(Pn) →a.s. T(P0). Proof. By Varadarajan’s theorem 2.1, Pn ⇒ P0 a.s. Fix ω ∈ A with P r(A) = 1 so that Pω n ⇒ P0. Then by weak continuity of T, Tn(Pω n) → T(P0). ✷ A difficulty in using this theorem is typically in trying to verify weak-continuity of T. Weak continuity is a rather strong hypothesis, and many interesting functions fail to have this type of continuity. The following approach is often useful. Definition 2.2 Let F ⊂ L1(P) be a collection of integrable functions. Say that Pn → P with respect to . ·. F if .Pn − P.F ≡ supf∈F |Pn(f) − P(f)| → 0. Furthermore, we say that T : P → R is continuous with respect to . · .F if .Pn − P.F → 0 implies that T(Pn) → T(P). Definition 2.3 If F ⊂ L1(P) is a collection of integrable functions with .Pn − P.∗ F → 0, we then say that F is a Glivenko-Cantelli class for P and write F ∈ GC(P). Theorem 2.2 Suppose that: (i) F ∈ GC(P); i.e. .Pn − P.∗ F →a.s. 0. (ii) T is continuous with respect to . · .F . Then T(Pn) →a.s. T(P)