16CHAPTER 8.BOOTSTRAP AND JACKKNIFE ESTIMATION OF SAMPLING DISTRIBUTIONS Corollary 1 The bootstrap confidence bands {Fn+n1/2cn(a,Fn)}satisfy lim PrfFn(z)-n-1/2cn(a,Fn)<F(x)<Fn(x)+n/3cn(a,Fn)for all ER}=1-a. n-o0 The behavior of these bands,and the savings over the(conservative)asymptotic or finite-sample Kolmogorov bands has been investigated by Bickel and Krieger (1989). Bootstrapping Empirical Measures Does Theorem 4.3 carry over to Efron's bootstrap for empirical measures?The answer is "yes" as shown by Gine and Zinn(1990).For a class of functions FcL2(P),we let the envelope function Fbe defined by F()=superf().Here are the two bootstrap limit theorems of Gine and Zinn (1990): Theorem 4.4 (Gine and Zinn,1990).(Almost sure bootstrap limit theorem).Suppose that F is P-measurable.Then the following are equivalent: A.FECLT(P)and P(F2)<oo;i.e.Vn(Pn-P)=Gp where Gp is a pp-uniformly continuous P-Brownian bridge process on F. B.√m(P%-P)→Gp in oo(F)almost surely. Theorem 4.5 (Gine and Zinn,1990).(In probability bootstrap limit theorem).Suppose that F is P-measurable.Then the following are equivalent: A.FECLT(P);i.e.Vn(Pn-P)=Gp where Gp is a pp-uniformly continuous P-Brownian bridge process on F. B.√m(P*-Pn)→Gp in o(F)in probability. The proofs of these two theorem rely on "multiplier inequalities"closedly related to the "mul- tiplier central limit theorem",Poissonization inequalities,and other tools from empirical process theory.See Gine and Zinn(1990),Klaassen and Wellner (1992),and van der Vaart and Wellner (1996),chapters 3.6 and 2.9.In particular,van der Vaart and Wellner (1996),theorems 3.6.1 and 3.6.2,give a version of theorems 4.4 and 4.5 with a somewhat improved treatment of the measurability issues. The spirit of the Gine and Zinn theorems carry over to the exchangeably-weighted bootstrap methods as shown by Praestgaard and Wellner (1993).Here are the hypotheses needed on the weights: W1.The vectors W=W in Rm are exchangeable for each n. W2.Wwy≥0 for allj=1,,nand∑-1Wy=n. W3.supn lWnlla2,1<oo where lWnll2,a≡f√P(Wn1≥t)dt. W4.lim→lim sup→Supr2AtPP(Wn1≥t)=0. W5.n-1∑=1(W-1)2→pc2>0. Theorem 4.6 (Exchangeably weighted bootstrap limit theorem).Suppose that F is P-measur- able and that the random weight vectors {Wn}satisfy W1-W5.Then:16CHAPTER 8. BOOTSTRAP AND JACKKNIFE ESTIMATION OF SAMPLING DISTRIBUTIONS Corollary 1 The bootstrap confidence bands {Fn ± n −1/2 cn(α, Fn)} satisfy limn→∞ PF {Fn(x) − n −1/2 cn(α, Fn) ≤ F(x) ≤ Fn(x) + n 1/3 cn(α, Fn) for all x ∈ R} = 1 − α. The behavior of these bands, and the savings over the (conservative) asymptotic or finite-sample Kolmogorov bands has been investigated by Bickel and Krieger (1989). Bootstrapping Empirical Measures Does Theorem 4.3 carry over to Efron’s bootstrap for empirical measures? The answer is “yes” as shown by Gin´e and Zinn (1990). For a class of functions F ⊂ L2(P), we let the envelope function F be defined by F(x) ≡ supf∈F |f(x)|. Here are the two bootstrap limit theorems of Gin´e and Zinn (1990): Theorem 4.4 (Gin´e and Zinn, 1990). (Almost sure bootstrap limit theorem). Suppose that F is P−measurable. Then the following are equivalent: A. F ∈ CLT(P) and P(F 2 ) < ∞; i.e. √ n(Pn−P) ⇒ GP where GP is a ρP− uniformly continuous P− Brownian bridge process on F. B. √ n(P ∗ n − P ω n ) ⇒ GP in `∞(F) almost surely. Theorem 4.5 (Gin´e and Zinn, 1990). (In probability bootstrap limit theorem). Suppose that F is P−measurable. Then the following are equivalent: A. F ∈ CLT(P); i.e. √ n(Pn − P) ⇒ GP where GP is a ρP− uniformly continuous P− Brownian bridge process on F. B. √ n(P ∗ n − Pn) ⇒ GP in `∞(F) in probability. The proofs of these two theorem rely on “multiplier inequalities” closedly related to the “mul￾tiplier central limit theorem”, Poissonization inequalities, and other tools from empirical process theory. See Gin´e and Zinn (1990), Klaassen and Wellner (1992), and van der Vaart and Wellner (1996), chapters 3.6 and 2.9. In particular, van der Vaart and Wellner (1996), theorems 3.6.1 and 3.6.2, give a version of theorems 4.4 and 4.5 with a somewhat improved treatment of the measurability issues. The spirit of the Gin´e and Zinn theorems carry over to the exchangeably - weighted bootstrap methods as shown by Praestgaard and Wellner (1993). Here are the hypotheses needed on the weights: W1. The vectors W = Wn in R n are exchangeable for each n. W2. Wnj ≥ 0 for all j = 1, . . . , n and Pn j=1 Wnj = n. W3. supn kWn1k2,1 < ∞ where kWn1k2,1 ≡ R ∞ 0 p P(Wn1 ≥ t) dt. W4. limλ→∞ lim supn→∞ supt≥λ t 2P(Wn1 ≥ t) = 0. W5. n −1 Pn j=1(Wnj − 1)2 →p c 2 > 0. Theorem 4.6 (Exchangeably weighted bootstrap limit theorem). Suppose that F is P− measur￾able and that the random weight vectors {Wn} satisfy W1 - W5. Then: