Let F (x) be a probability distribution,the p(0 < p < 1)-quantile ofF is defined as ξ_{p} = inf{x, F (x) ≥ p},and denoted by F^{−1}(p).Corresponding to a simple random sample X_{1}, X_{2}, ⋯, X_{n} of observations on F,the sample pth quantile is defined as the p- quantile of the sample distribution function F_{n},that is F_{n}^{−1}(p),denoted by ξ̂_{n}.Furthermore,set ξ̂_{n}^{∗} = F_{n}^{∗}^{−1}(p)the bootstrap sample p- quantile,whereF_{n}^{∗} is the empirical distribution based on a bootstrap sample X_{1}^{∗}, X_{2}^{∗}, ⋯, X_{n}^{∗}i.i.d from F_{n}.

1. Bahadur representation for bootstrap case

Theorem 1. Suppose F is twice differentiable at ξ_{p},with F′(ξ_{p}) = f(ξ_{p}) > 0,then

ξ̂_{n}^{∗} = ξ̂_{n} + \frac{F_{n}(ξ_{p}) − F_{n}^{∗}(ξ_{p})}{f(ξ_{p})} + R̃_{n}  (1)

where with probability 1

R̃_{n} = O(n^{−3/4} log n)

Theorem 2. Suppose F is twice differentiable at ξ_{p},with F′(ξ_{p}) = f(ξ_{p}) > 0,let k_{n} be a sequence of positive integers (1 ≤ k_{n} ≤ n) such that

\frac{k_{n}}{n} = p + o(\frac{log n}{n^{1/2}})  (2)

Then

X_{n(k_{n})}^{∗} = ξ_{p} + \frac{\frac{k_{n}}{n} − F_{n}^{∗}(ξ_{p})}{f(ξ_{p})} + R̃_{n}  (3)

where X_{n(k_{n})}^{∗} is k_{n}th order statistics in the bootstrap sample of size n,and with probability 1

R̆_{n} = O(n^{−3/4} log n)

The proof of these two theorems are based on the following lemmas. Let

G_{n}^{∗}(x, ω) = [F_{n}^{∗}(x, ω) − F_{n}^{∗}(ξ_{p}, ω)] − [F (x) − F (ξ_{p})]

{a_{n}} be a sequence of positive constants,such that

a_{n} ∼ \frac{log n}{n^{1/2}} asn → ∞

Let I_{n} = (ξ_{p} − a_{n}, ξ_{p} + a_{n}),and let

H_{n}^{∗}(ω) = sup{|G_{n}(x, ω)| : x in I_{n}}

P_{∗} be the probability taken under the bootstrap distribution conditional on X_{1}, X_{2}, ⋯, X_{n}.

Lemma 3. With probability one, H_{n}^{∗}(ω) = O(n^{−3/4} log n) as n → ∞.

Proof: Let {b_{n}, n = 1, 2, ⋯ } be a sequence of positive integers such that

b_{n} ∼ n^{1/4} as n → ∞.

Consider a particular n,for any integer r,let η_{r,n} = ξ_{p}+a_{n}b_{n}^{−1}r,,J_{r,n} = [η_{r,n}, η_{r+1,n}]. For x in J_{r,n},fixed ω

|G_{n}^{∗}(x)| ≤ |[F_{n}^{∗}(x) − F_{n}^{∗}(ξ_{p})] − [F_{n}(x) − F_{n}(ξ_{p})]| + |[F_{n}(x) − F_{n}(ξ_{p})] − [F (x) − F (ξ_{p})]|

≤ H̃_{n}^{∗} + H_{n} (4)

where H_{n} = max_{x∈I_{n}} |F_{n}(x) − F_{n}(ξ_{p}) − F (x) + F (ξ_{p})| = O(n^{−3/4} log n) by lemma1 in [1][1](1996); H̃_{n}^{∗} = max_{x∈I_{n}} |G̃^{∗}(x)|, where G̃^{∗}(x) = F_{n}^{∗}(x) − F_{n}^{∗}(ξ_{p}) − [F_{n}(x) − F_{n}(ξ_{p})]. Since F_{n}^{∗} and F_{n} are non-decreasing in x,then

G̃^{∗}(x) ≤F_{n}^{∗}(η_{r+1,n}) − F_{n}^{∗}(ξ_{p}) − (F_{n}(η_{r+1,n}) − F_{n}(η_{p})) + F_{n}(η_{r+1,n}) − F_{n}(η_{r,n})
≤ K_{n}^{∗} + β_{n}^{∗}

where

K_{n}^{∗} = max_{−b_{n}≤r≤b_{n}} |G̃^{∗}(η_{r,n})|
β_{n}^{∗} = max_{−b_{n}≤r≤b_{n}−1} |F_{n}(η_{r+1,n}) − F_{n}(η_{n})|.

Similarly, G̃^{∗}(x) ≥ K_{n}^{∗} + β_{n}^{∗} for x ∈ J_{r,n} which implies

H̃_{n}^{∗} ≤ K_{n}^{∗} + β_{n}^{∗} (5)
Claim  β_{n}^{∗} = O(\frac{log n}{n^{3/4}})  (6)

in fact,let B(n, z) denote a Binomial random variable with parameters n and Z ,using Bernstein inequality,for constant C_{1} > 0

P (|F_{n}(η_{r+1,n}) − F_{n}(η_{r,n}) − [F (η_{r+1,n}) − F (η_{r,n})]| ≥ C_{1}n^{−3/4} log n)
=P (|B(n, z_{n}) − nz_{n}| ≥ n · C_{1}n^{−3/4} log n)
≤2 exp(−h_{n})

where z_{n} = F (η_{r+1,n}) − F (η_{r,n}) = O(n^{−3/4} log n) (7) by the proof of lemma1 in Bahadur(1966)[1][1],since F is sufficiently smooth in the neighborhood of ξ_{p}.

Therefore,there exists C_{2},such that z_{n} < C_{2}n^{−3/4} log n for n large enough,and

h_{n} = \frac{nC_{1}^{2}n^{− \frac{3}{2}} (log n)^{2}}{2(z_{n}) + C_{1}n^{− \frac{3}{4}} log n} ≥ \frac{nC_{1}^{2}n^{− \frac{3}{2}} (log n)^{2}}{2(C_{2}n^{− \frac{3}{4}} log n) + C_{1}n^{− \frac{3}{4}} log n} > \frac{C_{2}^{2}n^{− \frac{1}{4}} log n}{2(C_{2} + C_{1})} > Cn^{1/5}

with n sufficiently large and C = \frac{C_{2}^{2}}{2(C_{2}+C_{1})}. It follows that

∑ P (|F_{n}(η_{r+1,n})−F_{n}(η_{r,n})−[F (η_{r+1,n})−F (η_{r,n})]| ≥ C_{1}n^{−3/4} log n) ≤ ∑ 2 exp^{−Cn^{1/5}} < ∞

Borel-Cantelli lemma implies

|F_{n}(η_{r+1,n}) − F_{n}(η_{r,n}) − [F (η_{r+1,n}) − F (η_{r,n})]| = O(n^{−3/4} log n)

Associated with (7),it follows

|F_{n}(η_{r+1,n})−F_{n}(η_{r,n})| ≤ O(n^{−3/4} log n)+|F (η_{r+1,n})−F (η_{r,n})| = O(n^{−3/4} log n)

The claim (6) is therefore established.

On the other hand,for constant C_{3}

P_{∗}(|G̃_{n}^{∗}(η_{r,n})| > C_{3}n^{−3/4} log n) =P_{∗}(|nG̃_{n}^{∗}(η_{r,n})| > n · C_{3}n^{−3/4} log n)
≤P_{∗}(|B(n, z̃_{n}(η_{r,n}) − nz̃_{n}(η_{r,n})| > n · C_{3}n^{−3/4} log n)
≤2exp(−h̃_{n})

where z̃_{n}(x) = F_{n}(x)−F_{n}(ξ_{p}) = O(n^{−1/2} log n),forx ∈ Ī_{n} (8), i.e.∃C_{5} such that z̃_{n}(x) ≤ C_{5}n^{−1/2} log n,since

|F_{n}(x)−F_{n}(ξ_{p})| ≤ H_{n}+|F (x)−F (ξ_{p})| ≤ O(n^{−3/4} log n)+C_{4}a_{n} = O(n^{−1/2} log n)

Here note that F is twice differentiable at ξ_{p},a_{n} = O(n^{−1/2} log n),so there exists C_{4} > 0,such that |F (x) − F (ξ_{p})| ≤ C_{4}a_{n} as n sufficiently large.

Using the same computation in the proof of Lemma1 in Bahadur(1966)[1][1], we have

h̃_{n} = \frac{n(C_{3}n^{−3/4} log n)^{2}}{2z̃_{n} + n^{−3/4} log n} > \frac{C_{3}^{2} log n}{2C_{5} + 2C_{3}n^{−1/4}}

Hence

P_{∗}(K̃_{n}^{∗} > C_{3}n^{−3/4} log n) ≤ 4b_{n} exp(\frac{C_{3}^{2} log n}{2C_{5} + 2C_{3}n^{−1/4}} ) ˆ=λ_{n}

Notice that

\frac{log λ_{n}}{log n} → \frac{1}{4} − \frac{C_{3}}{2C_{5}} (9)

The limit on (9) is less than -2 if given C_{5}, C_{3} is chosen sufficiently large,then ∑ λ_{n} < ∞. Taking expectation on both sides of (8) gives

P (K̃_{n}^{∗} > C_{3}n^{−3/4} log n) ≤ λ_{n}

Using Borel-Cantelli lemma again yields

K̃_{n}^{∗} = O(n^{−3/4} log n) (10) Combining (5),(6)and (10) shows

H̃_{n}^{∗} = O(n^{−3/4} log n) (11) This associated with (4)completes the proof.

Lemma 4.

∑_{j=1}^{n} I_{{X_{j}^{∗}=X_{i}}} = O(log n), for i = 1, 2, ⋯, n

Proof: For constant C_{6} > 5, n > 3, C_{7} = C_{6} − 1,

P_{∗}(∑_{j=1}^{n} I_{{X_{j}^{∗}=X_{i}}} ≥ C_{6} log n) =P_{∗}(∑_{j=1}^{n} I_{{X_{j}^{∗}=X_{i}}} − 1 ≥ C_{6} log n − 1)
≤P_{∗}(∑_{j=1}^{n} I_{{X_{j}^{∗}=X_{i}}} − 1 ≥ (C_{6} − 1) log n)
=P_{∗}(B(n, 1/n) − 1 ≥ C_{1} \frac{log n}{n} · n)
≤2 exp(−h̆_{n})

where

h̆_{n} = \frac{n · (\frac{C_{7} log n}{n})^{2}}{2(C_{1} \frac{log n}{n} + \frac{1}{n})} = \frac{C_{7}^{2} log n}{2(C_{1} + \frac{1}{log n})}

Since \frac{C_{7}^{2} log n}{2(C_{1}+ \frac{1}{log n})} → C_{7}/2,h̆_{n} > 2 log n for n sufficiently large. It follows that

P_{∗}(∑_{j=1}^{n} I_{{X_{j}^{∗}=X_{i}}} ≥ C_{6} log n) ≤ 2n^{−2}

Taking expectation on both sides gives

P (∑_{j=1}^{n} I_{{X_{j}^{∗}=X_{i}}} ≥ C_{6} log n) ≤ 2n^{−2}

Note that ∑ 2n^{−2} < ∞,Borel-Cantelli lemma implies

P (∑_{j=1}^{n} I_{{X_{j}^{∗}=X_{i}}} ≥ C_{6} log ni.o.) = 0

i.e. ∑_{j=1}^{n} I_{{X_{j}^{∗}=X_{i}}} = O(log n) lemma4 is proved.

Lemma 5. With probability one,|X_{nk_{n}}^{∗} − ξ_{p}| = O(n^{−1/2} log n)

Proof: Without loss of generality,denote n^{−1/2} log n as a_{n}.

From the Bahadur representation theorem in [1][1] and the fact ξ_{p} − ξ̂_{n} = O(n^{−1/2} log n) (lemmaB 2.5.4 in [?][2]),we have

\frac{F_{n}(ξ_{p}) − p}{f(ξ_{p})} = ξ_{p}−ξ̂_{n}+O(n^{− \frac{3}{4}} log n) = O(n^{− \frac{1}{2}} log n)+O(n^{− \frac{3}{4}} log n) = O(n^{− \frac{1}{2}} log n)

which implies

F_{n}(ξ_{p}) = p + O(n^{− \frac{1}{2}} log n).  (4)

Let ž_{n} = F_{n}(ξ_{p} − a_{n}), then ž_{n} = p + O(n^{− \frac{1}{2}} log n),since (7) and (12) validate

F_{n}(ξ_{p} − a_{n}) =F_{n}(ξ_{p} − a_{n}) − F_{n}(ξ_{p}) + F_{n}(ξ_{p})
≤|F_{n}(ξ_{p} − a_{n}) − F_{n}(ξ_{p})| + p + O(n^{− \frac{1}{2}} log n)
=|z̃_{n}(ξ_{p} − a_{n})| + p + O(n^{− \frac{1}{2}} log n)
=p + O(n^{− \frac{1}{2}} log n)

For constant 1,consider

P_{∗}(X_{nk_{n}}^{∗} − ξ_{p} ≤ −a_{n}) =P_{∗}(X_{nk_{n}}^{∗} ≤ ξ_{p} − a_{n})
≤P_{∗}(B(n, ž_{n}) ≥ k_{n} + O(log n))
=P_{∗}(B(n, ž_{n}) − nž_{n} ≥ k_{n} − nž_{n} + O(log n))
≤2 exp(−\frac{n(\frac{k_{n}}{n} − ž_{n} + O(\frac{log n}{n}))^{2}}{2(ž_{n} + \frac{k_{n}}{n} − ž_{n} + O(\frac{log n}{n}))})

where the second inequality follows from lemma4.

Note that for n sufficiently large

\frac{n(\frac{k_{n}}{n} − ž_{n} + O(\frac{log n}{n}))^{2}}{2(\frac{k_{n}}{n} + O(\frac{log n}{n}))} = \frac{n(p + o(\frac{log n}{\sqrt{n}}) + O(\frac{log n}{n}) − p − O(\frac{log n}{\sqrt{n}}))^{2}}{2(p + O(\frac{log n}{n}) + o(\frac{log n}{\sqrt{n}}))} > 3(log n)^{2}

Then there exists C_{8} such that

P_{∗}(X_{nk_{n}}^{∗} − ξ_{p} ≤ −a_{n}) ≤ 2 exp(−C_{8}(log n)^{2})

Taking expectation on both sides gives

P (X_{nk_{n}}^{∗} − ξ_{p} ≤ −a_{n}) ≤ 2 exp(−C_{8}(log n)^{2})

It follows that

∑_{n} P (X_{nk_{n}}^{∗} ≤ ξ_{p} − a_{n}) < ∞

Similar arguments shows that ∑_{n} P (X_{nk_{n}}^{∗} ≥ ξ_{p} + a_{n}) < ∞. This completes the proof of lemma5 by Borel Cantelli lemma.

Proof of Theorem2: Under the conditions of the theorem,we may apply lemma5. By Tylor theorem

F (X_{nk_{n}}^{∗})−F (ξ_{p}) = F′(ξ_{p})(X_{nk_{n}}^{∗} −ξ_{p})+O(\frac{(log n)^{2}}{n}) (13)

Utilizing lemma3 and again appealing to lemma5 we may pass (13) to :With probability one F_{n}^{∗}(X_{nk_{n}}^{∗})−F_{n}^{∗}(ξ_{p}) = F′(ξ_{p})(X_{nk_{n}}^{∗}−ξ_{p})+O(n^{−3/4} log n) (14) Finally,since F_{n}^{∗}(X_{nk_{n}}^{∗}) = \frac{k_{n}}{n} + O(\frac{log n}{n}) due to lemma4. Solving X_{nk_{n}}^{∗} gives

X_{n(k_{n})}^{∗} = ξ_{p} + \frac{\frac{k_{n}}{n} − F_{n}^{∗}(ξ_{p})}{f(ξ_{p})} + O(\frac{log n}{n}) + O(n^{−3/4} log n)

i.e.

X_{n(k_{n})}^{∗} = ξ_{p} + \frac{\frac{k_{n}}{n} − F_{n}^{∗}(ξ_{p})}{f(ξ_{p})} + O(n^{−3/4} log n)

Theorem2 therefore followed. Proof of Theorem1: Note that

ξ_{n}^{∗} = (1 − ψ_{n})X_{[np]}^{∗} + ψ_{n}X_{[np]}^{∗} where ψ_{n} = np − [np], 0 ≤ ψ_{n} < 1 Let k_{n}^{(1)} = [np], then \frac{k_{n}^{(1)}}{n} = p+O(1/n) and k_{n}^{(2)} = [np]+1,then \frac{k_{n}^{(2)}}{n} = p+O(1/n). Theorem2 implies

X_{[np]}^{∗} =ξ_{p} + \frac{p + O(1/n) − F_{n}^{∗}(ξ_{p})}{f(ξ_{p})} + O(n^{−3/4} log n)  (15)
X_{[np]+1}^{∗} =ξ_{p} + \frac{p + O(1/n) − F_{n}^{∗}(ξ_{p})}{f(ξ_{p})} + O(n^{−3/4} log n)  (16)

(15) × (1 − ψ_{n}) + ψ_{n} × (16) gives

ξ̂_{n}^{∗} = ξ_{p} + \frac{p − F_{n}^{∗}(ξ_{p})}{f(ξ_{p})} + O(n^{−3/4} log n) (17)

On the other hand,based on the Bahadur representation theorem in[1][1]

ξ̂_{n} = ξ_{p} + \frac{p − F_{n}(ξ_{p})}{f(ξ_{p})} + O(n^{−3/4} log n) (18)

(17)-(18) yields the theorem1.

2. Comparasion of two types of Confidence interval(CI)

2.1 Accuracy of asymptotic confidence sets (I) The hybrid bootstrap CI [\underline{ξ}_{HB}, ξ̄_{HB}]with confidence coefficient 1-2α. where

\underline{ξ}_{HB} =ξ̂_{n} − n^{−1/2}Ĥ_{B}^{−1}(1 − α)  (19)
ξ̄_{HB} =ξ̂_{n} − n^{−1/2}Ĥ_{B}^{−1}(α)  (20)

whereĤ_{B}(x) = P_{∗}(\sqrt{n}ξ̂_{n}^{∗} − ξ̂_{n}) ≤ xwhich is the bootstrap estimator of the distribution function H_{n} of \sqrt{n}(ξ̂_{n} − ξ_{p}).

Proposition 6. P (\underline{ξ}_{HB} < ξ < ξ̄_{HB}) = 1 − 2α + O(n^{−1/4} log n) Proof: By the Bahadur representation theorems for plain sample (17)and bootstrap sample (1) cases, we have

\sqrt{n}(ξ̂_{n} − ξ_{p}) =\sqrt{n}(\frac{p − F_{n}(ξ_{p})}{f(ξ_{p})}) + O(n^{−1/4} log n)  (21)
\sqrt{n}(ξ̂_{n}^{∗} − ξ̂_{n}) =\sqrt{n}(\frac{F_{n}(ξ_{p}) − F_{n}^{∗}(ξ_{p})}{f(ξ_{p})}) + O(n^{−1/4} log n)  (22)

Let

W_{n} =\sqrt{n}(\frac{p − F_{n}(ξ_{p})}{f(ξ_{p})})/\frac{\sqrt{p(1 − p)}}{f(ξ_{p})}
W̃_{n} =\sqrt{n}(\frac{F_{n}(ξ_{p}) − F_{n}^{∗}(ξ_{p})}{f(ξ_{p})})/\frac{\sqrt{F_{n}(ξ_{p})(1 − F_{n}(ξ_{p}))}}{f(ξ_{p})}

By Berry Eesseen theorem

sup_{x} |F_{W_{n}}(x) − Φ(x)| =O(1/\sqrt{n})  (23)
and sup_{x} |F_{W_{n}}(x) − Φ(x)| =O(1/\sqrt{n}) conditional onX_{1}, X_{2} ⋯ X_{n}  (24)

note that \frac{1}{f(ξ_{p})}I_{\{X_{i}≤ξ_{p}\}} is bounded and

\frac{\sqrt{F_{n}(ξ_{p})(1 − F_{n}(ξ_{p}))}}{f(ξ_{p})} = \frac{\sqrt{p(1 − p)}}{f(ξ_{p})} +O(n^{−1/2} log n) (25)

In fact (25) follows from (12) which implies

\frac{F_{n}(ξ_{p})(1 − F_{n}(ξ_{p}))}{f^{2}(ξ_{p})} = (\frac{p}{f(ξ_{p})} + O(n^{−1/2} log n))(\frac{1 − p}{f(ξ_{p})} + O(n^{−1/2} log n))

= \frac{p(1 − p)}{f^{2}(ξ_{p})} + O(n^{−1/2} log n)

Then it can be shown that

sup_{x} |H_{n}(x) − Ĥ_{B}(x)| = O(n^{−1/4} log n)

since

sup_{x} |H_{n}(x) − Ĥ_{B}(x)| = sup_{x}[P (\sqrt{n}(\frac{p − F_{n}(ξ_{p})}{f(ξ_{p})}) + O(n^{−1/4} log n) ≤ x)

− P (\sqrt{n}(\frac{F_{n}(ξ_{p}) − F_{n}^{∗}(ξ_{p})}{f(ξ_{p})}) + O(n^{−1/4} log n) ≤ x)

= sup_{x} |P (W_{n} ≤ \frac{x + O(n^{−1/4} log n)}{\frac{\sqrt{p(1−p)}}{f(ξ_{p})}}) − P (W̃_{n} ≤ \frac{x + O(n^{−1/4} log n)}{\frac{\sqrt{F_{n}(ξ_{p})(1−F_{n}(ξ_{p}))}}{f(ξ_{p})}})|
= sup_{x} |P (W_{n} ≤ \frac{x + O(n^{−1/4} log n)}{\frac{\sqrt{p(1−p)}}{f(ξ_{p})}})

− E(P_{∗}(W̃_{n} ≤ \frac{x + O(n^{−1/4} log n)}{\frac{\sqrt{F_{n}(ξ_{p})(1−F_{n}(ξ_{p}))}}{f(ξ_{p})}})|X_{1}, ⋯, X_{n})|

= sup_{x} |Φ(\frac{x + O(n^{−1/4} log n)}{\frac{\sqrt{p(1−p)}}{f(ξ_{p})}}) + O(n^{−1/2})

− E(Φ(\frac{x + O(n^{−1/4} log n)}{\frac{\sqrt{F_{n}(ξ_{p})(1−F_{n}(ξ_{p}))}}{f(ξ_{p})}}) + O(n^{−1/2}))|

= sup_{x} |Φ( \frac{x}{\frac{\sqrt{p(1−p)}}{f(ξ_{p})}} + O(n^{−1/4} log n)) + O(n^{−1/2})

− Φ( \frac{x}{\frac{\sqrt{p(1−p)}}{f(ξ_{p})}} + O(n^{−1/4} log n)) + O(n^{−1/2})|

= sup_{x} |Φ( \frac{x}{\frac{\sqrt{p(1−p)}}{f(ξ_{p})}} ) + Φ′( \frac{x}{\frac{\sqrt{p(1−p)}}{f(ξ_{p})}} )O(n^{−1/4} log n)

− Φ( \frac{x}{\frac{\sqrt{p(1−p)}}{f(ξ_{p})}} − Φ′( \frac{x}{\frac{\sqrt{p(1−p)}}{f(ξ_{p})}} )O(n^{−1/4} log n) + O(n^{−1/2})|

= sup_{x}(Φ′( \frac{x}{\frac{\sqrt{p(1−p)}}{f(ξ_{p})}} ))O(n^{−1/4} log n) + O(n^{−1/2})
=O(n^{−1/4} log n)

where the first equality follows from(21)(22),the fourth equality holds for (23)(24),the next equality is true because (25)implies

\frac{x + O(n^{−1/4} log n)}{\frac{\sqrt{F_{n}(ξ_{p})(1−F_{n}(ξ_{p}))}}{f(ξ_{p})}} = \frac{x}{\frac{\sqrt{p(1−p)}}{f(ξ_{p})}} + O(n^{−1/4} log n).

Hence

P (\underline{ξ}_{HB} < ξ < ξ̄_{HB}) =P (ξ̂_{n} − n^{− \frac{1}{2}} Ĥ_{B}^{−1}(1 − α) ≤ ξ ≤ ξ̂_{n} − n^{− \frac{1}{2}} Ĥ_{B}^{−1}(α))
=P (n^{− \frac{1}{2}} Ĥ_{B}^{−1}(α) ≤ ξ̂_{n} − ξ ≤ n^{− \frac{1}{2}} Ĥ_{B}^{−1}(1 − α))
=P (Ĥ_{B}^{−1}(α) ≤ \sqrt{n}(ξ̂_{n} − ξ) ≤ Ĥ_{B}^{−1}(1 − α))
=H_{n}(Ĥ_{B}^{−1}(1 − α)) − H_{n}(Ĥ_{B}^{−1}(α))
=Ĥ_{B}(Ĥ_{B}^{−1}(1 − α)) − Ĥ_{B}(Ĥ_{B}^{−1}(α)) + O(n^{−1/4} log n)
=1 − 2α + O(n^{−1/4} log n)

The proposition6 is therefore established.

(II)Woodruff’s interval Let {K_{1n}},{K_{2n}} be sequences of integers satisfying 1 ≤ K_{1n} < K_{2n} ≤ n and

\frac{K_{1}n}{n} − p ∼ −\frac{z_{1−α}[p(1 − p)]^{1/2}}{n^{1/2}}  (26)
\frac{K_{2}n}{n} − p ∼ \frac{z_{1−α}[p(1 − p)]^{1/2}}{n^{1/2}}  (27)

where z_{1−α} = Φ^{−1}(1 − α),∼ means ratio of two sides tends to 1. Confidence interval I_{Q_{n}} = (X_{K_{1n}}, X_{K_{2n}}) is so called Woodruff’s interval with confidence level 1 − 2α

Proposition 7. P (X_{K_{1n}} > ξ_{p}) = α+o(1)+O(n^{−1/4} log n) (28) Proof: Notice that (26) is equivalent to say

\frac{K_{1}n}{n} − p = \frac{K_{1}}{n^{1/2}} + o(n^{−1/2})

where K_{1} = −z_{1−α}[p(1 − p)]^{1/2}. By Bahadur representation theorems in [1][1],we have (18)and

X_{K_{1n}} = ξ_{p} + \frac{\frac{K_{1n}}{n} − F_{n}(ξ_{p})}{f(ξ_{p})} + O(n^{−3/4} log n)

= ξ_{p} + \frac{p + \frac{K_{1}}{n^{1/2}} o(\frac{1}{n^{1/2}} ) − F_{n}(ξ_{p})}{f(ξ_{p})} + O(n^{−3/4} log n)

Therefore

\sqrt{n}(X_{K_{1n}} − ξ̂_{n}) = \sqrt{n}(\frac{K_{1}}{n^{\frac{1}{2}} f(ξ_{p})} + o(\frac{1}{n^{1/2}} ) + O(n^{−3/4} log n))

It follows that

P (X_{K_{1n}} > ξ_{p}) = P (\sqrt{n}[ξ̂_{n} − \frac{z_{1−α}[p(1 − p)]^{1/2}}{n^{1/2}f(ξ_{p})} ] + o(1) + O(n^{−1/4} log n) > \sqrt{n}ξ_{p})

= P (\sqrt{n}[ξ̂_{n} − ξ_{p}] + o(1) + O(n^{−1/4} log n) > \frac{z_{1−α}[p(1 − p)]^{1/2}}{f(ξ_{p})} )

= P (\frac{\sqrt{n}(p − F_{n})}{\sqrt{p(1 − p)}/f(ξ_{p})} + o(1) + O(n^{−1/4} log n) > z_{1−α})

= Φ(−z_{1−α} + o(1) + O(n^{−1/4} log n) + O(n^{−1/2}))

= α + o(1) + O(n^{−1/4} log n)

where the last second equality follows from Berry Esseen theorem. By analogous method we can show

P (X_{K_{2n}} > ξ_{p}) = α + o(1) + O(n^{−1/4} log n)

This and (28) implies

P (X_{K_{1n}} ≤ ξ_{p} ≤ X_{K_{2n}}) = 1 − 2α + o(1) + O(n^{−1/4} log n) (29)

Remark 8. If K_{1n} = np+\sqrt{n}K_{1}+O(n^{1/4+ϵ}), K_{2n} = np+\sqrt{n}K_{2}+O(n^{1/4+ϵ})whereK_{2} = z_{1−α}[p(1 − p)]^{1/2}, ϵ > 0,which satisfy (26) and (27),then

P (X_{K_{1n}} ≤ ξ_{p} ≤ X_{K_{2n}}) = 1 − 2α + O(n^{−1/4+ϵ})

It is clear that the asymptotic accuracy order of Woodruff’s CI is worse than that of Hybrid Bootstrap CI in this case.In word the Woodruff’s CI is not better than Hybrid Bootstrap CI in terms of accuracy order.

2.2The asymptotic relative efficiencies of confidence sets According to the definition in [2][?](page 105),this concept is to take as an efficiency criterion the (almost sure) rate at which the length of the confidence interval tends to 0 while the confidence coefficient tends to a limit sayγ, 0 < γ < 1 (∗). (i)Woodruff’s interval: The length of interval I_{Q_{n}} ∼ \frac{2z_{1−α}[p(1−p)]^{1/2}}{n^{1/2}f(ξ_{p})} .(Refer to 2.6.3 in [2][?]) (ii)Hybrid bootstrap CI:

Length of interval[\underline{ξ}_{HB}, ξ̄_{HB}]

=n^{−1/2}(Ĥ_{B}^{−1}(1 − α) − Ĥ_{B}^{−1}(α))
=n^{−1/2}(σ̂_{F} H̃_{B}^{−1}(1 − α) − H̃_{B}^{−1}(α)  (30)

where H̃_{B}(x) = P_{∗}(\frac{\sqrt{n}(ξ̂_{n}^{∗}−ξ̂_{n})}{σ̂_{F}} ≤ x),σ̂_{F} = ^{\frac{\sqrt{F_{n}(ξ_{p})(1−F_{n}(ξ_{p}))}}{f(ξ_{p})}}which is ^{\frac{\sqrt{p(1−p)}}{f(ξ_{p})}} + O(n^{−1/2} log n) by(25). ClearlyĤ_{B}^{−1}(1 − α) = σ̂_{F} H̃_{B}^{−1}(1 − α) Note that

H̃_{B}^{−1}(t) = z_{t} + O(n^{−1/4} log n) (31)

In fact in the proof of proposition6,we have shown H̃_{B}(x) = Φ(x)+O(n^{−1/4} log n),so it follows that Φ(H̃_{B}^{−1}(x)) = H̃_{B}(H̃_{B}^{−1}(x))+O(n^{−1/4} log n) (32) On the other hand

Φ(H̃_{B}^{−1}(x)) = Φ(Φ^{−1}(x))+Φ′(Φ^{−1}(x))(H̃_{B}^{−1}(x)−Φ^{−1}(x))+Φ′′(ζ(x))(H̃_{B}^{−1}(x)−Φ^{−1}(x))^{2} (33)

where ζ(x) lies between H̃_{B}^{−1}(x) and Φ^{−1}(x). Then (31) follows from (32),(33). Hence the LHS of (30)

= n^{−1/2}(\frac{\sqrt{p(1 − p)}}{f(ξ_{p})} + O(n^{−1/2} log n))(z_{1−α} + O(n^{−1/4} log n) − z_{α} − O(n^{−1/4} log n))

= n^{−1/2}(\frac{\sqrt{p(1 − p)}}{f(ξ_{p})} + O(n^{−1/2} log n))(O(n^{−1/4} log n))

= O(n^{−3/4} log n)

Consequently the latter CI is more efficient in sense of (∗),since

\frac{length of interval[\underline{ξ}_{HB}, ξ̄_{HB}]}{length of intervalI_{Q_{n}}} = \frac{n^{− \frac{3}{4}} log n}{O(\frac{1}{\sqrt{n}})} = O(n^{−1/4} log n) → 0

References

[1] Bahadur.R.R.(1966). A note on quantiles in large samples. Ann. Math. Statist. 37 577-580

Figure 1: (a) Scatter plot of CC1 and CC2. (b) Histogram of CC1.

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