SOME CONVERSE LIMIT THEOREMS OR EXCHANGEABLE BOOTSTRAPS Jon A. Wellner University of Washington The bootstrap Glivenko-Cantelli and bootstrap Donsker theorems of Giné and Zinn (990) contain both necessary and sufficient conditions for the asymptotic validity of Efron s nonparametric bootstrap. In the more general case of exchangeably weighted bootstraps, Praestgaard and Wellner (993) and Van der Vaart and Wellner (996) give analogues of the sufficiency half of the Theorems of Giné and Zinn (990), but did not address the corresponding necessity parts of the theorems. Here we establish a new lower bound for exchangeably weighted processes and show that the necessity half of the a.s. bootstrap Glivenko-Cantelli theorem holds for exchangeably weighted bootstraps. We also make some progress toward the conjectured necessity parts of the bootstrap Donsker theorems with exchangeable weights. AMS subject classifications: Primary: 6005, 607; secondary 60J65, 60J70. Keywords and phrases: asymptotic distributions, bootstrap, consistency, converse, empirical measures, empirical processes, exchangeable, Donsker theorem, Glivenko-Cantelli theorem, multiplier inequalities. Introduction. Giné and Zinn (990) established several beautiful limit theorems for Efron s nonparametric bootstrap of the general empirical process. One of the key tools used by Giné and Zinn (990) was the multiplier inequality used earlier in Giné and Zinn (983) with Gaussian multipliers to Gaussianize the empirical process and to relate the Gaussianized process to the symmetrized empirical process, and hence to the empirical process itself via symmetrization and de-symmetrization inequalities. Efron s nonparametric bootstrap, which involves resampling from the empirical measure P n, can be viewed as one instance of an exchangeably weighted bootstrap with the weights being the components of a random vector which has a Multinomial distribution with n cells, n trials, and vector of success probabilities (/n,..., /n). The first limit theory for exchangeably weighted bootstraps was established by Mason and Newton (992); they treated exchangeable bootstrapping of the mean and of the classical empirical process. Praestgaard and Wellner (993) extended the direct half Research supported in part by National Science oundation grant DMS-9532039 and NIAID grant 2R0 AI29968-04.
2 Jon A. Wellner of several of the theorems of Giné and Zinn (990) by use of a new type of multiplier inequality involving symmetrization with ranks rather than with Rademacher random variables. Similarly, the direct half of the bootstrap Glivenko - Cantelli theorems of Giné and Zinn (990) was established in Van der Vaart and Wellner (996); see Lemma 3.6.6, page 357. In both Praestgaard and Wellner (993) and Van der Vaart and Wellner (996), the analogue of the converse half of the theorems of Giné and Zinn (990) could not be proved for the general exchangeable bootstrap because of the lack of an appropriate analogue of the lower bound in the i.i.d. version of the multiplier inequality. In this paper we make some progress toward filling these gaps. We first establish an appropriate lower bound in the case of exchangeably weighted sums of i.i.d. random elements (Section 2). We then show how this lower bound yields the converse half of bootstrap Glivenko - Cantelli theorems (Section 3). In the case of bootstrap Donsker theorems (Section 4), we are able to establish some of the integrability needed for the converse half, but still lack an appropriate analogue of the Hoffmann-Jørgensen theorem needed to obtain the crucial uniform integrability needed to apply the L inequalities established in Section 2. or other approaches to proving bootstrap limit theorems in other cases of interest, see Csörgő and Mason (989), Einmahl and Mason (992), and Shorack (996), (997). 2 Multiplier Inequalities: a new lower bound. irst we give a statement of the multiplier inequalities for the case of i.i.d. multipliers. Let Z,..., Z n be independent and identically distributed (i.i.d.) processes indexed by a set with mean 0 (i.e. EZ i (f) = 0 for all f ), and let ξ,..., ξ n,... be i.i.d. real-valued random variables which are independent of Z,..., Z n,...; in the case of empirical processes, Z i = δ Xi P for i.i.d. random elements X i X where the basic probability space is (X, A, P ). or processes Z i we write Z i for sup f Z i (f). As in Van der Vaart and Wellner (996), we will assume throughout that X, X 2,... are defined as the coordinate projections on the first component of the product probability space (X Z, A C, P Q), and let ξ, ξ 2,... depend on the last coordinate only. Lemma 2. (Multiplier inequality for i.i.d. multipliers). Let Z,..., Z n be i.i.d stochastic processes with E Z i < independent of the Rademacher variables ɛ,..., ɛ n. Then for every i.i.d. sample ξ,..., ξ n of mean-zero
Converse Bootstrap Theorems 3 random variables independent of Z,..., Z n, and any n 0 n, 2 ξ E n () ɛ i Z i E n ξ i Z i 2(n 0 )E ξ i Z E max i n n + 2 2 ξ 2, max n 0 k n E k k ɛ i Z i. i=n 0 Here ξ 2, 0 P ( ξ > t)dt is assumed to be finite. or a proof of Lemma 2., see Giné and Zinn (983) or Van der Vaart and Wellner (996). The important thing to be observed here is that both the upper and lower bounds for the expected norm of the multiplier process n ξ iz i are provided in terms of the symmetrized process n ɛ iz i. Now we turn to the case of exchangeable multipliers. We will assume that the vector ξ n (ξ n,..., ξ nn ) is exchangeable and ξ n 2, < for each n. The upper bound half of the following inequality is a consequence of the multiplier inequality proved by Praestgaard and Wellner (993). Lemma 2.2 (Multiplier inequality for exchangeable multipliers). Let Z,..., Z n be i.i.d. stochastic processes with E Z i < independent of the Rademacher variables ɛ,..., ɛ n and of the random permutation R = (R,..., R n ) of the first n integers. Then for every exchangeable random vector ξ n = (ξ n,..., ξ nn ) independent of Z,..., Z n, and any n 0 n, 2 ξ n E n (2) ɛ i Z i E n ξ ni Z i 2(n 0 )E Z E max i n + 2 ξ n 2, max n 0 k n E k ξ ni n k i=n 0 Z Ri. Proof. The inequality on the right follows from Praestgaard and Wellner (993), Lemma 4., page 2063. It remains only to prove the inequality on the left. To do this, let Z,..., Z n be an independent copy of Z,..., Z n (canonically formed on an appropriate product probability space). Then by the triangle inequality 2E ξ ni Z i E ξ ni (Z i Z i)
4 Jon A. Wellner (3) = E ξ ni ɛ i (Z i Z i) = E ξ ni sign(ξ ni )ɛ i (Z i Z i) = E ξ ni ɛ i (Z i Z i) where the last equality holds because the vector (sign(ξ n )ɛ,..., sign(ξ nn )ɛ n ) has the same distribution as (ɛ,..., ɛ n ) and, moreover, is independent of (ξ n,..., ξ nn ). Hence by convexity of the norm and Jensen s inequality the right side of (3) is bounded below by E ξ n E ɛ i (Z i Z i) E ξ n E ɛ i Z i by convexity again and since the Z i s have mean 0; this is quite similar to the proofs of Lemma 2.3., page 08, and Lemma 2.9., page 77, Van der Vaart and Wellner (996) where the measurability details of the proof are given in detail. 3 Bootstrap Glivenko-Cantelli Theorems. Now suppose that X, X 2,... are i.i.d. P on (X, A), and let P n be the empirical measure of the first n of the X i s; P n = n δ Xi. The classical Efron nonparametric bootstrap empirical measure P n is just P n = n δ bxi where X,..., X n is a sample drawn (with replacement) from P n. Giné and Zinn (990) established the following bootstrap Glivenko-Cantelli theorem. Their notation NLDM(P ) stands for nearly linearly deviation measureable for P ; we refer to Giné and Zinn (984), page 935 for the detailed definition. Theorem 3. (Glivenko-Cantelli theorem for Efron s Bootstrap). Suppose that is NLDM(P ). Then the following are equivalent: (a) P ( ) < and P n P 0 in probability. (b) P a.s. P n P n 0 in probability.
Proof. See Giné and Zinn (990), page 860. Converse Bootstrap Theorems 5 Here is a slight reformulation of the bootstrap Glivenko-Cantelli theorem of Giné and Zinn (990) which avoids the hypothesis that is NLDM(P ). Let BL (R) be the collection of all functions h : R [0, ] such that h(x ) h(x 2 ) x x 2 for all x, x 2 R. Theorem 3.2 (Modified Glivenko-Cantelli theorem for Efron s Bootstrap). The following are equivalent: (a) P f P f < and P n P 0 in probability. (b) (P ) a.s. P n P n 0 in probability and Êh( P n P n ) Êh( P n P n ) a.s. 0 for every h BL (R). Proof. This follows as a corollary of Theorem 3.3 below. Now we turn to exchangeably weighted bootstraps. Suppose that W n = (W n,..., W nn ) satisfies the following conditions: A. W n = (W n,..., W nn ) is exchangeable for each n. A2. W ni 0 for i =,..., n and n W ni = n. A3. max i n (W ni /n) p 0. A4. lim n E W n = b > 0, lim sup n E W n 2 <. or such a vector of exchangeable weights W n, consider the exchangeably weighted bootstrap empirical measure P W n given by P W n = n W ni δ Xi. It is easily seen that the classical nonparametric bootstrap empirical measure is the special case of P W n obtained by taking W n = M n where M n = (M n,..., M nn ) Mult n (n, (/n,..., /n)). Theorem 3.3 (Glivenko-Cantelli theorem for the Exchangeable Bootstrap). Suppose that {W n } satisfies A - A4. Then the following are equivalent: (a) P f P f < and P n P 0 in probability. (b) P a.s. P W n P n 0 in probability and Êh( P W n P n ) Êh( P W n P n ) a.s. 0 for every h BL (R). Moreover, if either (a) or (b) holds it follows that (4) E W P W n P n a.s. 0.
6 Jon A. Wellner Proof. That (a) implies (b) was proved in Van der Vaart and Wellner (996), Lemma 3.6.6, page 357. It remains only to prove (b) implies (a). Suppose that we show that (b) implies P f P f < and that (4) holds. It follows that (5) E P W n P n 0. Now note that by A2 we can write P W n P n = n (W ni )(δ Xi P ) n ξ ni Z i with ξ ni W ni and Z i δ Xi P. Hence it follows from the Multiplier Lemma 2.2 and a standard symmetrization inequality (see e.g. Van der Vaart and Wellner (996), Lemma 2.3., page 08) that E P W n P n 2 E W n E n 4 E W n E n ɛ i Z i Z i = 4 E W n E P n P. In view of the non-degeneracy condition A4, this together with (5) yields E P n P 0, which implies the convergence in probability part of (a) by Markov s inequality. Now we show that (b) implies that P f P f < and that (4) holds. Suppose that (b) holds. Let X, X 2,... be an independent copy of X, X 2,..., and let ɛ,..., ɛ n be independent Rademacher random variables which are independent of the {X i }, {X i }, and {W ni, i =,..., n, n }. Set Z i δ Xi P, Z i δ X i P, and ξ ni W ni. Then we have n ξ ni Z i + n ξ ni Z i n = d n = n = d n ξ ni (Z i Z i) ξ ni ɛ i (Z i Z i) ξ ni sign(ξ ni )ɛ i (Z i Z i) ξ ni ɛ i (Z i Z i).
Converse Bootstrap Theorems 7 Since the left side of this display converges to 0 in probability a.s. respect to PZ,Z, so does the right side. That is, for every δ > 0, P ξ,ɛ ( n ) ξ ni ɛ i (Z i Z i) > δ 0 almost surely. with By independence of ξ, ɛ, and Z, Z, this yields E ξ P ɛ ( n ) ξ ni ɛ i (Z i Z i) > δ 0 almost surely, and this implies that P ɛ ( n ) ξ ni ɛ i (Z i Z i) > δ 0 in probability with respect to ξ and almost surely with respect to Z, Z. But now the summands Y i ξ ni ɛ i (Z i Z i ), i =,..., n, are, for fixed ξ ni and Z i Z i, independent and symmetric. Hence by Lévy s inequality (e.g. Proposition A..2, Van der Vaart and Wellner (996), page 43) it follows that ( P ɛ max ) n ξ ni ɛ i (Z i Z i) > δ 0 i n in probability with respect to ξ and almost surely with respect to Z, Z. But the norm is equal to n ξ ni (Z i Z i) and since this does not depend on ɛ i, it follows that (6) n max i n ξ ni (Z i Z i) p 0 in probability with respect to ξ and almost surely with respect to Z, Z. That is, P ξ ( n max (Z i Z i) (7) > ɛ) 0 i n ξ ni almost surely with respect to Z, Z. Now let R be a random permutation of the first n integers. Let b ni (/n) (Z i Z i ), and suppose that I {,..., n} satisfies max i n b ni = b ni. Then by exchangeability of the W ni s we have max ξ ni b ni = d max ξ n,r(i) b ni. i n i n
8 Jon A. Wellner Hence, conditioning on the W ni s, it follows that the probability in (7) equals E W P ( max ξ nr(i) b n,i > ɛ W ) E W P ( ξ nr(i) b ni > ɛ W ) i n = E W n [ ξnj b ni >ɛ] E W n = E W n = E W n j= j= j= j= [ ξnj > ɛ] [bni > ɛ] [ ξnj > ɛ] [bni > ɛ] [ ξnj > ɛ] (8) = P ( ξ n > ɛ) [bni > ɛ]. [bni > ɛ] Now for any non-negative random variable Y and a > 0 we have E(Y ) = E(Y [Y a] ) + E(Y [Y >a] ) a + E(Y 2 ) P (Y > a) and, choosing a = ce(y ) with c <, this yields (9) P (Y > ce(y )) ( c)2 (EY ) 2 E(Y 2 ) ; this is the Paley-Zygmund argument. Using this with Y = ξ n and cey = ɛ in combination with (7) and (8) yields (0) P ξ ( n max i n ξ ni (Z i Z i) > ɛ) ( ɛ/e ξ n ) 2 {E ξ n } 2 E ξ n 2 {n max i n (Z i Z i) > ɛ}. But by hypothesis A4 the first term on the right side of (0) converges to a positive constant. Since the left side converges to 0 almost surely by (7), it follows that n max (Z i Z i) () a.s. 0. i n But it is well known that () holds if and only if E ( (Z Z ) ) < ;
Converse Bootstrap Theorems 9 by convexity of the norm together with EZ = 0, this implies that E ( Z ) = P f(x ) P f <. This together with A4 implies that (4) holds via uniform integrability. 4 Bootstrap Donsker Theorems: Conjectures and Partial Proofs. Suppose that P n, P n, and P W n are defined as in Section 3, and define the empirical process G n, the nonparametric bootstrap empirical process Ĝn, and the exchangeably weighted empirical process ĜW n, by (2) G n = n(p n P ), Ĝ n = n( P n P n ), Ĝ W n = n( P W n P n ). In this section we will consider the processes G n, Ĝn, and ĜW n as processes indexed by functions f, where L 2 (P ) = L 2 (X, A, P ). The following bootstrap Donsker theorem results from Giné and Zinn (990) together with the measurability improvements of Van der Vaart and Wellner (996). Theorem 4. (Almost sure Donsker theorem for Efron s Bootstrap). The following are equivalent: (a) P f P f 2 < and is P Donsker; i.e. G n G P in l (). (b) sup h BL Êh(Ĝn) Eh(G P ) a.s. 0 and Êh(Ĝn) Êh(Ĝn) a.s. 0 for every h BL. Theorem 4.2 (In probability Donsker theorem for Efron s Bootstrap). The following are equivalent: (a) is P Donsker; i.e. G n G P in l (). (b) sup h BL Êh(Ĝn) Eh(G P ) p 0 and Ĝn is asymptotically measurable: Eh(Ĝn) Eh(Ĝn) 0 for every h BL. Proof. See Giné and Zinn (990), page 857; for a version of Theorem 4. under additional measurability hypotheses; and see Giné and Zinn (990), page 862 for a version of Theorem 4.2 under additional measurability hypotheses. The above statements are from Van der Vaart and Wellner (996), page 347 (where complete proofs are also given). The strong point of these theorems is they provide necessary and sufficient conditions in order for the process Ĝn to converge weakly either almost surely or in probability respectively. Now we turn to exchangeably weighted bootstraps. Our goal is to establish the analogues of the converse halves of of Theorems 4. and 4.2 in the case of the exchangeable bootstrap process ĜW n. The resulting Theorems 4.3 and 4.4 below strengthen Theorems 2. and 2.2 of Praestgaard and
0 Jon A. Wellner Wellner (993) (or see Theorem 3.6.3 of Van der Vaart and Wellner (996)) to the level of equivalence established by Giné and Zinn (990) for Efron s bootstrap. Suppose that W n = (W n,..., W nn ) satisfies the following conditions: B. W n = (W n,..., W nn ) is exchangeable for each n. B2. W ni 0 for i =,..., n and n W ni = n. B3. sup n W n, 2, <, and lim λ lim sup n sup t λ t 2 P (W n t) = 0. B4. lim n E W n = b > 0, and n n (W ni ) 2 p c 2 > 0. In view of Lemma 4.7 of Praestgaard and Wellner (993), B3 and B4 together imply that {W n } is uniformly square-integrable and hence so is n n (W ni ) 2. Therefore B3 and B4 together imply that E(W n ) 2 c 2. Theorem 4.3 (Conjectured Almost sure Donsker theorem for the Exchangeable Bootstrap). Suppose that {W n } satisfies B - B4. Then the following are equivalent: (a) P f P f 2 < and is P Donsker; i.e. G n G P in l (). (b) sup h BL Êh(ĜW n ) Eh(cG P ) a.s. 0 and Êh(ĜW n ) Êh(ĜW n ) a.s. 0 for every h BL. Theorem 4.4 (Conjectured In probability Donsker theorem for the Exchangeable Bootstrap). Suppose that {W n } satisfies B - B4. Then the following are equivalent: (a) is P Donsker; i.e. G n G P in l (). (b) sup Êh(ĜW h BL n ) Eh(cG P ) P 0 and ĜW n is asymptotically measurable. Partial Proof. That (a) implies (b) in both Theorems 4.3 and 4.4 was proved in Van der Vaart and Wellner (996), Theorem 3.6.3, page 355, under additional measurability hypotheses. It remains only to prove that (b) implies (a) in both cases. In the converse direction, we are currently able only to show that (b) of Theorem 4.3 implies the integrability condition of (a) of Theorem 4.3. Suppose that (b) holds. Then, with (Z, Z 2,...) an independent copy of (Z, Z 2,...), and ɛ, ɛ 2,... a sequence of i.i.d Rademacher random variables
independent of the W s, Z i s, and Z i s, Ĝ W n ĜW n = Converse Bootstrap Theorems n = d n = d n (W ni )(Z i Z i) W ni sign(w ni )ɛ i (Z i Z i) W ni ɛ i (Z i Z i). In view of (b) it follows that this difference converges weakly for almost all sequences Z, Z 2,..., Z, Z 2,..., to the tight Gaussian process c(g P G P ) with values in C u (), the space of uniformly continuous functions from to R. Thus the supremum c G P G P has moments of all orders, and for every ɛ > 0 there exists an x sufficiently large so that P (c G P G P x) ɛ x 2. Hence, by the Portmanteau theorem, for n N ω,ω, P W,ɛ( W ni ɛ i (Z i (ω) Z i(ω )) > x n) 2P (c G P G P x) 2ɛ x 2. But by Lévy s inequality used conditionally on the W s, the left side is E W P ɛ ( W ni ɛ i (Z i Z i) > x ) n ( 2 E W P ɛ max W ni ɛ i Z i Z i > x ) n i n = ( 2 E W P ɛ max W ni Z i Z i > x ) n. i n Thus, letting b ni n /2 (Z i Z i ) and ξ ni W ni, we have (3) P W (max i n ξ ni b ni > x) 4ɛ x 2. Now let I {,..., n} satisfy max i n b ni = b ni. Then by exchangeability of the W ni s it follows that max ξ ni b ni = d max ξ n,r(i) b ni i n i n
2 Jon A. Wellner where R is a random permutation which is independent of the W s and the Z i s, and Z i s. Thus the left side of (3) is equal to ( ) E W P R max ξ ( n,r(i) b ni > x E W P R ξn,r(i) b ni > x ) i n = E W n [ ξnj b ni >x] j= E W n [ ξnj > 2 E ξ n ] [b ni >2x/E ξ n ] (4) = E W n j= j= [ ξnj > 2 E ξ n ] [bni >2x/E ξ n ] = P ( ξ n > (/2)E ξ n ) {b ni > 2x/E ξ n } (/4)[E ξ n ] 2 [E ξ n 2 {b ni > 2x/E ξ n }. ] where we have used the inequality (9) with Y = ξ n and c = /2 in the last step. Now the first term on the right side of (4) has a positive limit inferior as n by B3-B4. Because the right side is smaller than 4ɛ/x 2 by (3), it follows that (5) lim sup n n max i n Z i Z i 4x b < almost surely where b lim n E W n = lim n E ξ n. By a standard argument, this implies that ( ) E (Z Z ) 2 <. By convexity of the norm together with EZ = 0, this implies that ( ) E Z 2 = P f(x ) P f 2 <. To complete the proof of the conjectured Theorem 4.3, it would suffice to show that (6) and (7) { ĜW n : n } is uniformly integrable. { ĜW n = n W ni (δ X ω i P ω n) : n } is P a.s. uniformly integrable.
Converse Bootstrap Theorems 3 If (6) and (7) could be proved, then the proof of the conjectured Theorems 4.3 and 4.4 are easily finished by use of the left inequality in Lemma 2.2. By general weak convergence theory, if (b) of Theorem 4.3 holds, then, for any δ n 0 and every ɛ > 0, ( ĜW ) P W n δn > ɛ a.s. 0 as n ; see e.g. Van der Vaart and Wellner (996), Theorem.5.7, page 37. This together with (7) implies that and then, in turn, E W ĜW n δn a.s. 0 as n, (8) E ĜW n δn 0 as n. Combining (8) with the left inequality in Lemma 2.2 (with replaced by δn ), ξ ni = W ni, and Z i δ Xi P, we would be able to conclude that 2 E ξ n E n ɛ i Z i δn E n ξ ni Z i δn = E ĜW n δn 0 as n, and since lim n E ξ n = b > 0 by B4, this yields (by a standard symmetrization inequality) 2 E G n δn E n ɛ i Z i δn 0 as n. It would follow (e.g. by Van der Vaart and Wellner (996), Corollary 2.3.2, page 5) that is P Donsker, so that (a) of Theorem 4.3 would hold. Unfortunately, we have not been able to establish (6) and (7). The missing link seems to be a suitable replacement in the exchangeable case for either Hoffmann-Jørgensen s inequality (see Van der Vaart and Wellner (996), Proposition A..5, page 433), or the uniform in n weak-l 2 condition for G n of Van der Vaart and Wellner (996), Lemma 2.3.9, page 3. Acknowledgements. We owe thanks to Evarist Giné, Jens Praestgaard, and Aad van der Vaart for conversations about these problems and (conjectured) theorems.
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