Bahadur representations for bootstrap quantiles 1

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1 Bahadur representations for bootstrap quantiles 1 Yijun Zuo Department of Statistics and Probability, Michigan State University East Lansing, MI 48824, USA zuo@msu.edu 1 Research partially supported by NSF grants DMS and DMS

2 Summary A bootstrap sample may contain more than one replica of original data points. To extend the classical Bahadur type representations for the sample quantiles in the independent identical distributed case to bootstrap sample quantiles therefore is not a trivial task. This manuscript fulfils the task and establishes the asymptotic theory of bootstrap sample quantiles. AMS 1991 Classification: Primary 62G20; Secondary 62G09, 62G05. Key words and phrase: bootstrap, Bahadur representation, sample quantile, asymptotics. Short running title: Bahadur representations for bootstrap quantiles 1. Introduction Let F be a given probability distribution function. The p th quantile of F, often denoted by F 1 p or simply ξ p for the fixed F, is defined as F 1 p := inf {x, F x p}, for any p 0, 1. 1 It is well-known that the quantiles ξ p, p 0, 1, characterize the distribution F. Statistical inference about ξ p is therefore an important problem in practice. Let F n be the empirical distribution assigning mass 1/n to each of i.i.d. X i, i = 1,, n, from F. Then ˆξ pn := Fn 1 p, the sample p th quantile, is a natural estimator of the population counterpart ξ p. Indeed, when the derivative of F at ξ p, F ξ p, exists and is positive, ˆξ pn is consistent for ξ p and [see Smirnov 1952 and Serfling 1980] n ˆξpn ξ p d N 0, p1 p/f ξ p 2. 2 The sample quantiles thus can be employed to infer the population quantiles for large sample. For example, an asymptotic 1 2α confidence interval for ξ p is: I Qn = p1 p 1/2, p1 p 1/2 ] [ˆξpn z 1 α F ξ p 2 ˆξpn + z 1 α n F ξ p 2, 3 n where z r is the rth quantile of the standard normal distribution function Φx. Unfortunately, F ξ p the density of F at ξ p has to be estimated first before this inference procedure becomes practically useful and relevant. 1

3 Bahadur s 1966 representation of the sample quantile offers a most ingenious way to fulfill the above task. For F twice differentiable at ξ p with F ξ p = fξ p > 0, ˆξ pn ξ p = p F n ξ p /fξ p + R n, 4 where R n = On 3/4 log n almost surely a.s. as n. This and other results in Bahadur 1966 imply that for any integers 1 k 1n < k 2n n such that k 1n n = p z 1 αp1 p 1/2 n 1/2 +on 1/2, k 2n n = p+ z 1 αp1 p 1/2 n 1/2 +on 1/2 5 as n, the following is an asymptotic 1 2α confidence interval for ξ p : I Wn = [X k1n, X k2n ], 6 where X 1 X n are order statistics; see Serfling Woodruff 1952 first proposed this procedure empirically. The interval thus is also called Woodruff interval in the literature. Nevertheless, it is Bahadur s representation that provides a theoretical justification of this asymptotic confidence interval procedure. Bahadur s work gave tremendous impetus to further research on the topic of quantile representations under various conditions. Keifer 1967, 1970a, b provided refinements of Bahadur s results. Under the weaker assumption that F is first order differentiable at ξ p with fξ p > 0, Ghosh 1971 obtained a weaker version of 4: ˆξ pn ξ p = p F n ξ p /fξ p + o p n 1/2, 7 which suffices for a lot of but some important statistical applications. This Ghosh type representations of sample quantiles for complex survey data were discussed in Francisco and Fuller 1991 and Shao and Wu Extensions of Bahadur s result to dependent random variables and processes were pursued in Sen 1968, 1972, Hesse 1990, Ho and Hsing 1996, and Wu An excellent review of sample quantile representations and inference procedures is given in Serfling Bootstrap quantiles can provide excellent approximations to sample quantiles in the inference of population quantiles; see, e.g., Bickel and Freedman 1981 and Singh 1981 in general and Shao and Chen 1998 for the special survey situation. Ghost type representations of the bootstrap quantiles turn out to be vital for the inference procedures in Shao and Chen

4 The bootstrap idea leads to not only a general mechanism of generating quantiles but also a general variance and distribution estimation method. The latter offers, in turn, a bootstrap based confidence interval procedure for quantiles Section 2. Establishing Bahadur type representations of the bootstrap quantiles under a general setting, which, to the best of our knowledge, has not been fulfilled and documented in the literature, is the major objective of this paper. A Bahadur type representation not only provides consistency and asymptotic normality results about sample quantiles but also reveals the rate of convergence of sample quantiles to population counterparts. Furthermore, in the case for bootstrap quantiles, results here recover some important results in the literature. For a smooth function T of the sample mean, Hall 1991 studied the Bahadur representations of the quantiles of the bootstrap versions T 1,, T B of T X n. His result, however, covers neither the non-smooth T case nor the bootstrap quantiles. It is not the purpose of this manuscript and it is impossible for us to review all the important references on the massive topic of bootstrapping related methods. For more complete reviews/surveies, please see the specific review/survey articles on the topic, e.g., DiCiccio and Efron 1996 and the references cited therein. The rest of the paper is organized as follows. Section 2 introduces the bootstrap confidence intervals for quantiles. Section 3 deals with the large sample properties and the Bahadur representations of bootstrap sample quantiles. The asymptotic normality of the bootstrap quantiles follows as a by-product of the representations. The paper ends with some brief concluding remarks. 2. Bootstrap sample quantiles and confidence intervals Let X 1,, X n be a random sample from the empirical distribution F n. It is often called a bootstrap sample. Denote by F n the empirical distribution based on this sample. The p th quantile based on this sample, ˆξ pn, is called the bootstrap sample p th quantile. In addition to Woodruff s confidence interval for ξ p in 6 we now consider a bootstrap type confidence interval. Let the estimator ˆθ n = T F n of a functional θ = T F satisfy d n ˆθ n θ N0, σf 2 in our case T G = G 1 p. Note that σ F is usually unknown. Let H n be the c.d.f. of n ˆθ n θ. Then for any finite sample [ˆθn n 1/2 H 1 n 1 α, ˆθn n 1/2 H 1 n α ] 8 is a confidence interval for θ with an approximate level 1 2α. Since H n is still 3

5 unknown in general, we thus consider its bootstrap version H defined as H x = P n ˆθ n ˆθ n x := P n ˆθ n ˆθ n x X 1,, X n with ˆθ n = T Fn. For many T, H 1 t Hn 1 t 0 a.s. as n for t 0, 1; see, e.g., Bickel and Freedam 1981 and Singh Thus [ˆθn n 1/2 H 1 1 α, ˆθn n 1/2 H 1 α ] 9 is a confidence interval for θ with an approximate significance level 1 2α for large sample size n. It is called the hybrid bootstrap confidence interval. This bootstrap procedure in essence provides a consistent estimator for the variance of ˆθ n. It is an approximation to I Qn see 3 in our quantile case. In some cases and for small n H x can be calculated directly given F n ; see, e.g., Efron This is true for our quantile functional case. But in general and for large n it is difficult to calculate. We thus consider its empirical version based on bootstrap samples F n1,, F nm, Ĥ B x = 1 m m j=1 n I T F nj T F n x, for any x R. which converges to H x a.s. and uniformly in x as m conditional on F n. Thus for large m and n, an approximation to 9 or 8 is the confidence interval [ˆθn n 1/2 Ĥ 1 B 1 α, ˆθn n 1/2 Ĥ 1 B α]. 10 One may also simply consider a bootstrap percentile confidence interval defined as [ ˆK 1 B α, ˆK 1 B 1 α], 11 which is closed related to 10, where the bootstrap sample distribution of T F n is ˆK B x = 1 m I T F m nj x, for any x R. j=1 We remark that the accuracy of the bootstrap confidence interval 9 but 10 or 11 has been studied for very smooth mean functional T thoroughly by, e.g., Hall 1992 and for the quantile functional by Falk and Kaufmann Bahadur representations of bootstrap sample quantiles We start with the strong consistency before establishing the Bahadur representations. 4

6 3.1 Strong consistency and asymptotic normality of bootstrap sample quantiles Naturally we focus on the continuous point p of F 1 p. That is, we assume that A1 x = ξ p is the unique solution for F x p F x. The assumption simply excludes the value p with F ξ p + ε = p for ε [0, r for some r > 0. It is also sufficient and necessary for ˆξ pn ξ p a.s. see Theorem of Serfling This is also true for ˆξ pn ξ p a.s. as shown below. Lemma 3.1 Under A1, ˆξ pn ξ p a.s. as n. Clearly the lemma holds if and only if ˆξ pn ˆξ pn 0 a.s. However, A1 no longer holds if F and ξ p in it are replaced by F n and ˆξ pn. The key point in the proof of ˆξ pn ξ p a.s. is the fact that δ ±ε = F ξ p ± ε p > 0 and nδ ±ε as n for any fixed ε > 0. We have to show that this fact is true for F n replacing F. Proof: Denote by P the conditional probability given X 1,, X n. For a given small ε > 0, P ˆξ pn ξ p > ε = P ˆξ pn > ξ p + ε + P ˆξ pn < ξ p ε. Now P ˆξ pn > ξ p + ε = P Fn ξ p + ε Fnξ p + ε > F n ξ p + ε p. By the Glivenko-Cantelli theorem, F n ξ p + ε p > δ +ε / 2 a.s. for large n. Hoeffding s inequality implies that P ˆξ pn > ξ p + ε < exp nδ+ε 2. Likewise, we can show that P ˆξ pn < ξ p ε < exp nδ ε 2. Note that the right hand sides of the last two inequalities are independent of X 1,, X n. Thus the two inequalities hold with P replacing P. The Borel-Cantelli lemma gives the desired result. Remark 3.2 The necessity of A1 for ˆξ pn ξ p a.s. follows from the fact that P ˆξ pn > ξ p + ε = P Fn ξ p + ε F nξ p + ε > 0 1/2, as n a.s. by the central limit theorem if F ξ p + 2ε = p, that is, if A1 is violated. Although A1 guarantees the strong convergence of ˆξ pn to ξ p a.s., it does not tell the rate of the convergence. The latter, on the other hand, is crucial for the Bahadur type representation of ˆξ pn. We now strength A1 and make the following assumption, which is exactly the one for the Ghosh 1971 representation for ˆξ pn. A2 For a given 0 < p < 1, F ξ p = fξ p exists and is positive. Clearly, A2 implies A1 and hence ˆξ pn ξ p a.s. It also implies that F ξ p = p. Furthermore it is also sufficient for a Ghosh type representation of ˆξ pn as shown 5

7 below. Note that X i in the original sample can appear in the bootstrap sample multiple times. The multiplicity however is bounded by log n for large n. Lemma 3.3 n I {X j =X i } < log n a.s. for large n and for i = 1, 2,, n. j=1 Proof: Denote by Bn, z the number of successes in n independent Bernoulli trials, each with success probability z. Let a n = 1 1/ log n. For n 3, we have P n j=1 I {X j =X i } log n P n j=1 I {X j =X i } 1 a n log n = P Bn, 1/n 1 a n log n n n 2 exp h n by Bernstein s inequality see, e.g., Lemma A of Serfling 1980, where h n := n a n log n/n 2 2a n /3 log n/n + 1/n = a 2 n log n 2a n /3 + 1/ log n > b log n for some constant b > 1 b does not depend on n and large n. It follows that P n j=1 I {X j =X i } log n 2 exp h n 2n b, which clearly is also true if P is replaced by P since the right side is independent of X 1,, X n. The Borel-Cantelli lemma now gives the desired result. We remark the upper bound in Lemma 3.3 could be improved to C log n/ log log n see Csorgo and Wu This result turns out to be vital for the establishment of the theoretical results in the sequel. Lemma 3.4 Let integer k n = np + on 1/2. Then under A2, we have X k n = ˆξ pn + F nξ p F nξ p fξ p + o p 1 = ξ p + F ξ p Fnξ p + o p n fξ p n Proof: The first representation follows from the second and Ghosh s representation of ˆξ pn. So we focus on the second representation. Let wn = n Xk ξ n p, ξ n t = ξ p + tn 1/2 and znt = n F ξ n t Fnξ n t/fξ p. We show that wn zn0 = o p

8 By the lemma in Ghosh 1971, we need only to show that for any t R and ε > 0 P w n t, z n0 t + ε 0, P w n t + ε, z n0 t and n F ξp F nξ p /fξ p = z n0 = O p For 14, it suffices to show its first part. Let Y n t = n F ξ n t F nx k n /fξ p. If we can show that Y n t t and z nt z n0 = o p 1 as n, then P w n t, z n0 t + ε = P z nt Y n t, z n0 t + ε P z nt z n0 > ε/2 + P Y n t t > ε/ By Lemma 3.3, F nx k n = k n/n + Olog /n = p + on 1/2. This and A2 yield Y n t = n F ξ n t p + p F nx k n /fξ p = t + o1. 17 To show that z nt z n0 = o p 1, write V n t = F ξ n t p F n ξ n t + F n ξ p, and V n t = F n ξ n t F n ξ p F nξ n t + F nξ p. Then it suffices to show that V n t = o p n 1/2, V n t = o p n 1/2, 18 since z nt z n0 = n V n t + V n t/fξ p. Let p n = F ξ n t p. Then P V n t > 2 t log n 1/2 n 3/4 = P Bn, p n n p n > 2n t log n 1/2 /n 3/4 2 t log n/n 1/2 2 exp p n + t log n 1/2 /n 3/4 2/n b, for some b > 1 and large n by Bernstein s inequality and the fact that p n = fξ p tn 1/2 + otn 1/2. The Borel-Cantelli lemma leads to for large n V n t 2 t log n 1/2 /n 3/4, a.s. 19 Hence V n t = o p n 1/2. Let q n = F n ξ n t F n ξ p. Then 19 implies that q n p n + 2 t log n 1/2 /n 3/4. Now a conditional expectation argument leads to V n t 2 t log n 1/2 /n 3/4, a.s. 20 for large n and hence V n t = o p n 1/2. Now 18, 17 and 16 lead to 14. 7

9 For 15, it suffices to show that U n + Un := n F ξ p F n ξ p + n F n ξ p Fnξ p = O p By the SLLN, F n ξ p 1 F n ξ p p1 p a.s. as n. By the CLT, for any ε > 0, P U n > M < ε/2 and P Un > M < ε/2 for some M, M > 0. Hence P U n + Un > M + M P U n + Un > M + M P U n > M + P Un > M ε/2 + E[P Un > M ] ε, which gives 21 and hence we have 15. This completes the proof. The asymptotic normality of n ˆξ pn ˆξ pn, conditional on F n, follows immediately from the representation. When p = 1/2, it recovers Proposition 5.1 of Bickel and Freedman Under special survey data setting and assumptions, Shao and Chen 1998 also obtained Ghosh type representation of ˆξ pn. The proof above see 15 implies the n-consistency of ˆξ pn ξ p. We now provide a precise upper bound for ˆξ pn ξ p. We first impose a condition on an integer k n. C Integer 1 k n n satisfies k n = np + on 1/2 log n δ, δ 1/2, as n. Theorem 3.5 Under A2 and C, X k n ξ p 2log nδ fξ p n for large n, a.s. Note that ˆξ pn = X np, where is the ceiling function. Clearly np satisfies C. Theorem 3.5 thus implies the strong consistency of ˆξ pn. A similar result for ˆξ pn is given by Lemmas B and C of Serfling Note that A2 is more than what needed for the strong consistency of ˆξ pn. But Theorem 3.5 also offers more. It provides the rate of convergence of ˆξ pn to ξ p that is not given in Lemma 3.1. Proof: Let d n = log n δ / n, k = 2/fξ p, ž n = F n ξ p kd n, and p n = F ξ p kd n. By Hoeffding s inequality Lemma of Serfling 1980, we have P ž n p n d n = P Bn, p n np n nd n exp 2nd 2 n exp 2 log n The Borel-Cantelli lemma implies that for large n with probability one ž n = F n ξ p kd n < p n + d n = p d n + od n. 8

10 Now for large n, employing Hoeffding s inequality agian we have P X kn ξ p < kd n = P X kn < ξ p kd n P Bn, žn k n = P Bn, žn nž n nk n /n ž n P Bn, žn nž n nod n + d n exp 2 log n1 + o1 2 < exp 3/2 log n. Taking the expectation at both side we have P X k n ξ p < kd n n 3/2. Likewise we can show that P X k n ξ p > kd n n 3/2 with the help of Lemma 3.3. The Borel-Cantelli lemma now leads to the desired result. Remark 3.6 Consider k n = np + Olog n δ n 1/2. If 0 < δ < 1/2, then the theorem covers this case. If δ = 0, then with the representation of Xk n given later in Theorem 3.12 and the law of the iterated logarithm, the optimal upper bound is C log log n/n 1/2 for a C > 0 see Bahadur Theorem 3.5 and other later results are valid uniformly for k n in a neighborhood of np. 3.2 A preliminary lemma for the Bahadur representation of ˆξ pn To establish the Bahadur representation of ˆξ pn, we make the following assumption: A3 F is twice differentiable at ξ p with F ξ p = fξ p > 0, which is the standard one for the Bahadur representation of ˆξ pn. Clearly, A3 implies A2 which in turn implies A1. For the lemma below, A3 can be weakened to A2 plus bounded F x in a neighborhood of ξ p. We start with a preliminary lemma before establishing the representation theorem for ˆξ pn. First define G nx, ω := F nx, ω F nξ p, ω F x F ξ p. 22 Let constants a n > 0, n = 1, 2, and a n log n/n 1/2, that is, a n / log n/n 1/2 1, as n. Let I n = ξ p a n, ξ p + a n and define We now show that it is bounded by n 3/4 log n. H nω := sup x I n G nx, ω. 23 Borel-Cantelli lemma are still the main technical tools in the proof. Bernstein inequality and the Lemma 3.7 Under A3, H nω = On 3/4 log n as n, a.s. 9

11 Proof: Let b n > 0 n = 1, 2, be positive integers such that b n n 1/4 as n. Consider a particular n. For any integer r, define η r,n := ξ p + a n b 1 n r, J r,n := [η r,n, η r+1,n ]. For convenience, we suppress ω in the following. Note that H n sup x I n F nx F nξ p F n x F n ξ p + sup x I n F n x F n ξ p F x F ξ p := H n + H n where H n := sup F n x F n ξ p F x + F ξ p and H n := sup G x with x I n x I n G x := Fnx Fnξ p [F n x F n ξ p ]. By Lemma 1 of Bahadur 1966 H n = On 3/4 log n as n, a.s. 24 Thus we need only to show that this is also true for H n. Observe that for x J r,n G x G η r+1,n + F n η r+1,n F n η r,n K n + β n for n sufficiently large and b n r < b n, where K n = max G η r,n, βn = max F nη r+1,n F n η r,n. b n r b n b n r b n 1 Similarly, we can show that G x K n β n for x J r,n which implies H n K n + β n 25 We now show that both K n and β n are Olog n / n 3/4 a.s. First we show that β n = Olog n / n 3/4, a.s. as n 26 Let z n r := F η r+1,n F η r,n. Then for any constant C 4 > 0, b n r < b n P Fn η r+1,n F n η r,n F η r+1,n F η r,n C4 n 3/4 log n = P Bn, z n r nz n r n C 4 n 3/4 log n 2 exp h n r by Bernstein s inequality see, e. g., Theorem of Serfling 1980 with h n r = nc 2 4 n 3/2 log n 2/ 2 zn r + C 4 n 3/4 log n. 27 Clearly, z n r < C 5 n 3/4 log n for all b n r < b n, C 5 = 2F ξ p, and large n and h n r > nc4 2n 3 2 log n 2 2 C 5 n 3 4 log n + C 4 n 3 4 log n > C2 4 n 1 4 log n 2C 5 + C 4 > 9 4 logn 10

12 for all b n r < b n and n sufficiently large. Since b n < 2n 1/4 for large n, thus P max Fn η r+1,n F n η r,n F η r+1,n F η r,n C 4 n 3/4 log n b n r<b n 4b n exp 9 4 log n < 8n1/4 n 9/4 = 8/n 2. The Borel-Cantelli lemma implies that βn On 3/4 log n + sup z n r = On 3/4 log n, as n, b n r<b n a.s. which gives 26. To show that Kn is Olog n / n 3/4 a.s., we observe that for constant C 6 > 0 P G n η r,n > C 6 n 3/4 log n = P n G n η r,n > n C 6 n 3/4 log n P Bn, zn η r,n n z n η r,n > n C 6 n 3/4 log n 2 exp h n r by Bernstein s inequality, where z n x = F n x F n ξ p for x I n and h n r := nc 6 n 3/4 log n 2/ 2 z n η r,n + C 6 n 3/4 log n. By 24 it follows that for sufficiently large n and C 7 2F ξ p sup z n x H n + sup F x F ξ p On 3/4 log n + C 7 a n = On 1/2 log n. x I n x I n Hence there is a C 8 > 0 such that sup x In z n x < C 8 n 1/2 log n for large n. Thus / h n r > C6 2 log n 2C 8 + 2C 6 n 1/4 > 9 log n, 4 uniformly for r b n for a sufficient large C 6 > 0 and large n. Hence P Kn > C 6 n 3/4 log n 4b n exp 9 log n < 8/n2 4 for large n such that b n < 2n 1/4. Note that the right hand side is independent of X 1,, X n. The above inequality holds for P. The Borel-Cantelli lemma yields Kn = On 3/4 log n, as n, a.s. 28 This, together with 25 and 26, yields H n = On 3/4 log n, as n, a.s

13 which, combined with 24, completes the proof. Remark 3.8 From the proof above and the one in Bahadur 1966, it is seen that Hn Cn 3/4 log n a.s. for large n and for some constant C > 0. Furthermore, the result in the lemma and in Bahadur 1966 can be improved to, e.g., On k log n α log log n β by setting a n log n α log log n β /n k ε and b n n ε with ε, α, β 0, k ε and either k + ε = 1, 2α α = 1, 2β β = 0 or k + ε < 1, 2α α 0, 2β β 0. In the later application of this result, however, Remark 3.8 dictates that the optimal choice under this setting is k ε = 1/2, α = 0 and β = 1/2, which then leads to an optimal choice: k = 3/4, α = 1/2 and β = 1/ Representation Theorems With the preliminary lemma in the last subsection we now are in a position to establish the following representation results. Theorem 3.9 Let 0 < p < 1. Under A3, we have ˆξ pn = ˆξ pn + F nξ p F nξ p fξ p + R 1n = ξ p + p F nξ p fξ p where, with probability 1, R in = On 3/4 log n, i = 1, 2, as n. + R 2n 30 This representation offers more than what the Ghosh type one does: it provides an absolute not probability upper bound for the difference between ˆξ pn ˆξ pn and the average of i.i.d sum F n ξ p F nξ p /fξ p, conditional on F n. It also leads immediately to a main theorem Theorem 5.1 in Bickel and Freedman 1981 about weak convergence of n ˆξ pn ˆξ pn as a process indexed in p under their assumptions. It allows one to employ the law of the iterated logarithm conditional on F n. Before proving the theorem we first establish a more general result. Theorem 3.10 Let 0 < p < 1. Under A3 and C with 1 2 δ 1, we have X k n = X k n + F nξ p F nξ p fξ p + R 1n = ξ p + k n/n F nξ p fξ p where, with probability 1, R in = On 3/4 log n, i = 1, 2, as n. + R 2n 31 Remark 3.11 The reminders in the above theorems are bounded by Cn 3 4 log n a.s. for large n for a constant C > 0. By Remark 3.8, they can be improved to On 3 4 log n 1 2 log log n 1 4. With Kiefer 1967 s approach, it seems they can be further improved to On 3 4 log log n 3 4. Details will not be pursued here though. 12

14 Proof of Theorem 3.9: Theorem 3.5 and the Taylor theorem imply that F Xk F ξ n p = F ξ p Xk ξ n p + Olog n 2δ /n, a.s. 32 as n. This, together with Theorem 3.5 and Lemma 3.7, yields as n FnX k F n nξ p = F ξ p Xk ξ n p + On 3/4 log n, a.s. 33 By Lemma 3.3, FnX k = k n n/n + Olog n/n. This and the last display imply Xk n = ξ p + k n /n Fnξ p / fξ p + Olog n/n + On 3/4 log n = ξ p + k n /n Fnξ p / fξ p + On 3/4 log n which, combining with the result in Bahadur 1966, completes the proof. Proof of Theorem 3.10: This follows in a straightforward fashion from Theorem 2.9 by letting k n = np and noticing k n /n = p + O1/n. 4. Concluding remarks This paper establishes Bahadur type representations of the bootstrap quantiles under a very general setting. These representations not only provide consistency and asymptotic normality results about sample quantiles but also reveals the rate of convergence of sample quantiles to population counterparts. Furthermore, one could further take advantage of these representations in examining the performance of the Woodruff and the bootstrap confidence interval 11 for quantiles. A natural question raised here is: How well does 11 perform compared with the classical Woodruff s one. It is found details will be pursued elsewhere that the bootstrap procedure 11 can be more accurate than Woodruff s one for most choices of k in and is as accurate as the Woodruff one with optimal k in, i = 1, 2. Acknowledgment The author thanks his RA Juan Du for the initial 13 pages Latex typing. The research was partially supported by NSF grants DMS and DMS

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16 [16] Reiss, R. D On the accuracy of the normal approximation for quantiles. Ann. Probab [17] Sen, P. K Asymptotic normality of sample quantiles for m-dependent processes. Ann. Math. Statist [18] Sen, P. K On the Bahadur representation of sample quantile for sequences of φ mixing random variables. J. Multivariate Anal [19] Serfling, R. J Approximation theorems of mathematical statistics. Wiley, New York. [20] Shao, J. and Chen, Y Bootstrapping sample quantiles based on complex survey data under hot deck imputation. Statist. Sinica [21] Shao, J. and Wu, C. F. J Asymptotic properties of the balanced repeated replication method for sample quantiles. Ann. Statist [22] Singh, K On the asymptotic accuracy of Efron s boostrap. Ann. Statist [23] Smirnov, N. V Limit Distributions for the Terms of a Variational Series. Amer. Math. Soc., Translation No. 67. [24] Woodruff, R. S Confidence intervals for medians and other positive measures. J. Amer. Statist. Assoc [25] Wu, W. B On the Bahadur representation of sample quantiles for dependent sequences. Ann. Statist

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