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The Bahadur representation for sample quantiles under strongly mixing sequence
Journal of Statistical Planning and Inference 141 (2011) 655–662
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The Bahadur representation for sample quantiles under strongly mixing sequence$ Xuejun Wang, Shuhe Hu , Wenzhi Yang School of Mathematical Science, Anhui University, Hefei, Anhui 230039, China
a r t i c l e in fo
abstract
Article history: Received 26 December 2009 Received in revised form 6 July 2010 Accepted 13 July 2010 Available online 17 July 2010
Sun [2006, The Bahadur representation for sample quantiles under weak dependence. Statist. Probab. Lett. 76, 1238–1244] established the Bahadur representation for sample quantiles under the strongly mixing sequence. But there are some problems in the proofs of the main results. In the paper, we further investigate the Bahadur representation for sample quantiles under strongly mixing sequence and get better bound than that in Sun (2006). & 2010 Elsevier B.V. All rights reserved.
Keywords: Bahadur representation Sample quantile Strongly mixing sequence
1. Introduction Let fXn ,n Z 1g be a sequence of random variables defined on a fixed probability space ðO,F ,PÞ with a common marginal distribution function FðxÞ ¼ PðX1 rxÞ. Denote F 1 ðpÞ ¼ inffx : FðxÞ Z pg as the corresponding pth quantile function of distribution function F(x), 0 op o1. For a sample X1, X2,y, Xn, n Z1, let Fn represent the empirical distribution function P based on X1, X2,y, Xn, which is defined as Fn ðxÞ ¼ ð1=nÞ ni¼ 1 IðXi rxÞ, x 2 R, where I(A) denotes the indicator function of a 1 set A and R is the real line. For 0 o p o1, we define Fn ðpÞ ¼ inffx : Fn ðxÞ Zpg as the sample pth quantile. Remark 1.1. Let F be a distribution function (continuous from the right, as usual). For 0 o p o 1, the pth quantile of F is defined as
xp ¼ inffx : FðxÞ Z pg and is alternately denoted by F 1(p). The function F 1(t), 0 ot o 1, is called the inverse function of F. It is easy to check that
xp possesses the following properties: (i) Fðxp Þ rp r Fðxp Þ; (ii) If xp is the unique solution x of FðxÞ r p rFðxÞ, then for any e 4 0, Fðxp eÞ op oFðxp þ eÞ:
$ Supported by the National Natural Science Foundation of China (Grant nos. 10871001, 60803059), Provincial Natural Science Research Project of Anhui Colleges (KJ2010A005), Talents Youth Fund of Anhui Province Universities (2010SQRL016ZD), Youth Science Research Fund of Anhui University (2009QN011A). Corresponding author. E-mail addresses: [email protected] (X. Wang), [email protected] (S. Hu), [email protected] (W. Yang).
0378-3758/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2010.07.008
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X. Wang et al. / Journal of Statistical Planning and Inference 141 (2011) 655–662
Next, we briefly introduce some dependence condition on the Xi ’s. Let F nm ¼ sðXi ,m ri r n, i 2 NÞ, the sfield generated by random variables Xm, y, Xn, 1r m rn. For n Z1, we define
aðnÞ ¼ sup
sup
jPðABÞPðAÞPðBÞj:
1 m2NA2F m 1 ,B2F m þ n
Definition 1.1. A random variable sequence fXn ,n Z 1g is said to be strongly mixing or amixing if aðnÞk0 as n-1. Throughout the paper, for a fixed p 2 ð0,1Þ, let xp ¼ F 1 ðpÞ, xp,n ¼ Fn1 ðpÞ. C and C0 denote positive constants which may be different in various places. dxe denotes the largest integer not exceeding x. a4b6minfa,bg. Bahadur (1966) firstly introduced an elegant representation for sample quantiles in terms of empirical distribution function based on independent and identically distributed (i.i.d.) random variables. The Bahadur representation allows one to express asymptotically a sample quantiles as a sample mean of certain (bounded) random variables and obtain the limiting properties of the sample quantile. At present, many researchers have obtained the Bahadur results under the case of dependent sequences, see for example, Sen (1972), Babu and Singh (1978) and Yoshihara (1995) gave the Bahadur representation for sample quantiles under fmixing sequence and strongly mixing (or amixing) sequence, respectively. Cai and Roussas (1997) studied the smooth estimate of quantiles under association. Wu (2005) established the Bahadur representation of sample quantiles for linear and some widely used nonlinear processes, and so forth. Recently, Sun (2006) established the Bahadur representation for sample quantiles under the strongly mixing (or amixing) sequence with polynomially decaying rate and obtained the following two theorems. Theorem 1.1 (cf. Sun, 2006, Theorem 1). Suppose that the strongly mixing coefficients of fXn ,n Z 1g satisfy aðnÞ rCnb for some positive constant C and for some b 40. Also suppose that the common marginal distribution function F(x) is continuously differentiable with derivative function f(x) in a neighborhood N p of xp such that 0 o d1 ¼ infff ðxÞ : x 2 N p g rd2 ¼ supff ðxÞ : x 2 N p g o 1: Then for any d 2 ð11=4ð1þ bÞ, 12Þ, b 4 4:5, supjðFn ðxÞFðxÞÞðFn ðxp ÞpÞj r ð1 þ d2 Þn3=4 þ d logn
a:s:;
ð1:1Þ
x2Dn
where Dn ¼ ½xp n1=2 þ d logn, xp þ n1=2 þ d logn. Theorem 1.2 (cf. Sun, 2006, Theorem 2). Suppose that conditions in Theorem 1.1 hold. Also suppose that f uðxÞ is bounded in a neighborhood of xp , say N p . Then for any d 2 ð11=4ð1 þ bÞ, 14Þ, b 4 10,
xp,n ¼ xp
Fn ðxp Þp þ Oðn3=4 þ d lognÞ f ðxp Þ
a:s:
ð1:2Þ
Remark 1.2. We point out that there are some mistakes in Sun (2006), which prejudices the validity of the results. Using the same notations as that in Sun (2006), Sun estimated the inequality 0 1 X 1 X 1 n P @ Vj,n 4 n3=4 þ d lognA o 1, n j ¼ 1 n¼1 and obtained that X 1 n jDr,n j ¼ Vj,n rn3=4 þ d logn ¼ en n j¼1
a:s:
by Borel–Cantelli Lemma. But it is not always true. For example, let tn ¼ 1=n and Xn ¼ ð1=n2 ÞX, where X is a standard normally distributed random variable. One can check that 1 X
PðjXn j4 tn Þ r
n¼1
1 1 X X EX 2n 1 ¼ o1: 2 2 n t n n¼1 n¼1
But ‘‘jXn jr tn a.s.’’ is uncorrect. In fact, PðjXn j rtn Þ ¼ PðjXjr nÞa1: Furthermore, by inequality (3.4) in Sun (2006), one should estimate 1 X P max jDr,n j 4 n3=4 þ d logn o1: n¼1
0 r jrj r mn
The similar problem exists in (3.16) in Sun (2006). Hence these problems above affect the proofs of Theorems 1 and 2 in Sun (2006).
X. Wang et al. / Journal of Statistical Planning and Inference 141 (2011) 655–662
657
In the paper, we will further investigate the Bahadur representation for sample quantiles under strongly mixing sequence and get better bound than that in Sun (2006). The organization of this paper is as follows. The main results are presented in Section 2, preparations for the proofs of the theorems are given in Section 3 and the proofs are provided in Section 4. 2. Main results
Theorem 2.1. Suppose that the strongly mixing coefficients of fXn ,n Z 1g satisfy aðnÞ rCnb for some positive constant C and for some b 4 0. Assume that the common marginal distribution function F(x) is continuously differentiable with derivative function f(x) in a neighborhood N p of xp such that 0 o d1 ¼ infff ðxÞ : x 2 N p g r d2 ¼ supff ðxÞ : x 2 N p g o1: Then for any d 2 ð3=ðb þ 1Þ, 12Þ, b 4 5, with probability 1 (wp1) supjðFn ðxÞFðxÞÞðFn ðxp ÞpÞj r ð1 þ d2 Þn3=4 þ d ðlognÞ3=4
ð2:1Þ
x2Dn
for all n sufficiently large, where Dn ¼ ½xp an , xp þ an and an C0 n1=2 þ d ðlognÞ1=2 as n-1. Theorem 2.2. Suppose that conditions of Theorem 2.1 are satisfied and P(Xi =Xj)=0 for any iaj. Assume that f uðxÞ is bounded in some neighborhood of xp , say N p . Then for any d 2 ð3=ðb þ 1Þ, 14, b 4 11, with probability 1,
xp,n ¼ xp
Fn ðxp Þp þOðn3=4 þ d ðlognÞ3=4 Þ, f ðxp Þ
n-1:
ð2:2Þ
Remark 2.1. Comparing our Theorem 2.2 with Theorem 2 in Sun (2006), we obtain better bound Oðn3=4 þ d ðlognÞ3=4 Þ than that in Sun (2006). Remark 2.2. We think that the equality (3.19) in Sun (2006) cannot be obtained easily by borrowing on the arguments on pp. 538–539 of Babu and Singh (1978). We have not seen the corrected proof for it in detail so for. So we add the condition ‘‘P(Xi = Xj)= 0 for any iaj’’ in Theorem 2.2, which can be applied to obtain (3.8) in the paper (see Lemma 3.4). We remark that the condition ‘‘P(Xi = Xj)= 0 for any iaj’’ is a weak condition. Here we give some examples.
(1) If random variables Xi and Xj are independent for iaj and the distribution function Fi(x) of random variable Xi is continuous, then Z Z Z þ 1 Z Z þ1 Z þ1 dF i ðxÞ dF j ðyÞ ¼ dF i ðxÞ dF j ðyÞ ¼ PðXi ¼ yÞ dF j ðyÞ ¼ 0 dF j ðyÞ ¼ 0: PðXi ¼ Xj Þ ¼ y¼x
1
x¼y
1
1
(2) If (Xi, Xj) has the joint probability density function fij(x,y) for iaj, then Z Z Z þ 1 Z Z þ1 fij ðx,yÞ dx dy ¼ fij ðx,yÞ dx dy ¼ 0 dy ¼ 0: PðXi ¼ Xj Þ ¼ y¼x
1
x¼y
1
3. Preparations The following lemmas are needed for the proofs of the main results. Lemma 3.1 (cf. Bosq, 1998). Let fXt ,t 2 Zg be a zero-mean real-valued process of strongly mixing random variables.
(i) If there exists a positive constant b such that jXt j rb, t 2 Z, then for each integer q 2 ½1,n=2 and each e 4 0, ! 1=2 n X qe2 4b n : P Xi 4 ne r4exp 2 þ 22 1 þ qa 2q e 8b
ð3:1Þ
i¼1
(ii) If there exists a positive constant c such that EjXt jk r ck2 k!EX 2t o 1,
t ¼ 1,2, . . . ,n, k ¼ 3,4, . . . ,
ð3:2Þ
658
X. Wang et al. / Journal of Statistical Planning and Inference 141 (2011) 655–662
then for each n Z2, each integer q 2 ½1,n=2, each e 4 0 and each k Z 3, ! ( )
2k=ð2k þ 1Þ n X qe2 n P Xi 4ne r a1 exp ðkÞ a , þ a 2 qþ 1 25m22 þ 5ce
ð3:3Þ
i¼1
where 2n e2 þ2 1þ a1 ¼ q 25m22 þ 5ce
! with m22 ¼ max EX 2t 1rtrn
and k=ð2k þ 1Þ
a2 ðkÞ ¼ 11n 1 þ
5mk
! with mk ¼ max ðEjXt jk Þ1=k :
e
1rtrn
Lemma 3.2 (cf. Serfling, 1980, Lemma 1.1.4). Let F(x) be a right-continuous distribution function. The inverse function F 1(t), 0 ot o 1, is nondecreasing and left-continuous, and satisfies (i) F 1 ðFðxÞÞ r x, 1 ox o 1; (ii) FðF 1 ðtÞÞ Zt, 0 ot o 1; (iii) FðxÞ Zt if and only if x Z F 1 ðtÞ. Lemma 3.3. Suppose that the strongly mixing coefficients of fXn ,n Z1g satisfy aðnÞ rCnb for some positive constant C and for some b 4 0, and the common marginal distribution function F(x) is differentiable at xp , with Fuðxp Þ ¼ f ðxp Þ 4 0. Assume that f uðxÞ is bounded in a neighborhood of xp , say N p . Then for any y 4 0 and d 2 ð9=ð10 þ8bÞ, 12Þ, b 41, with probability 1 jxp,n xp j r
yðloglognÞ1=2 f ðxp Þn1=2d
for all n sufficiently large:
ð3:4Þ
Proof. Let en ¼ yðloglognÞ1=2 =f ðxp Þn1=2d , n Z 1. Write Pðjxp,n xp j 4 en Þ ¼ Pðxp,n 4 xp þ en Þ þ Pðxp,n o xp en Þ: By Lemma 3.2(iii), Pðxp,n 4 xp þ en Þ ¼ Pðp 4 Fn ðxp þ en ÞÞ ¼ Pð1Fn ðxp þ en Þ 4 1pÞ ¼ P
n X
! IðXi 4 xp þ en Þ4 nð1pÞ
i¼1
¼P
n X
! ðVi EV i Þ 4 ndn1 ,
i¼1
where Vi ¼ IðXi 4 xp þ en Þ and dn1 ¼ Fðxp þ en Þp 40. Likewise, by Lemma 3.2(iii), ! n X ðWi EW i Þ Z ndn2 , Pðxp,n o xp en Þ r Pðp r Fn ðxp en ÞÞ ¼ P i¼1
where Wi ¼ IðXi r xp en Þ and dn2 ¼ pFðxp en Þ 40. It is easy to see that fVi EV i ,1 ri rng and fWi EW i ,1 ri r ng are still amixing sequences with jVi EV i j r1 and jWi EW i jr 1 for each i. Applying Lemma 3.1(i) to fVi EV i ,1 r ir ng and fWi EW i ,1 ri rng with q ¼ dn12d logne, we can see that ( ) 2 qdn1 4 1=2 n ð3:5Þ qa Pðxp,n 4 xp þ en Þ r4exp þ 22 1þ 2q 8 dn1 and (
) 2 qdn2 4 1=2 n : qa Pðxp,n o xp en Þ r 4exp þ 22 1 þ 2q 8 dn2
ð3:6Þ
Therefore, ( Pðjxp,n xp j 4 en Þ r 8exp
) 1=2 q½minðdn1 , dn1 Þ2 4 n : qa þ 44 1 þ minðdn1 , dn2 Þ 2q 8
ð3:7Þ
Since F(x) is continuous at xp with Fuðxp Þ 4 0, xp is the unique solution of FðxÞ r p r FðxÞ and Fðxp Þ ¼ p. By the assumption on f uðxÞ and Taylor’s expansion,
dn1 ¼ Fðxp þ en Þp ¼ f ðxp Þen þ oðen Þ ¼ yn1=2 þ d ðloglognÞ1=2 þ oðen Þ and
dn2 ¼ pFðxp en Þ ¼ f ðxp Þen þ oðen Þ ¼ yn1=2 þ d ðloglognÞ1=2 þoðen Þ:
X. Wang et al. / Journal of Statistical Planning and Inference 141 (2011) 655–662
659
So minðdn1 , dn2 Þ Z 12yn1=2 þ d ðloglognÞ1=2
for all n sufficiently large:
Together with (3.7), we have (
) 2 n12d logn y n1 þ 2d loglogn 32 !1=2 2d b 8 n n12d logn C þC 1þ 1=2 logn yn1=2 þ d ðloglognÞ
Pðjxp,n xp j 4 en Þ r Cexp
r
C CðlognÞb þ 1 þ n2 n1 þ g ðloglognÞ1=4
for all n sufficiently large, where g ¼ ðdð10 þ 8bÞ9Þ=4 4 0 when b 4 1 and d 2 ð9=ð10 þ 8bÞ, 12Þ. Therefore, 1 X
Pðjxp,n xp j4 en Þ r C þ C
n¼1
1 1 X X 1 ðlognÞb þ 1 þC o 1, 2 1 þ g ðloglognÞ1=4 n ¼ n0 n n ¼ n0 n
which implies that wp 1, the relations jxp,n xp j 4 en hold for only finitely many n by Borel–Cantelli Lemma. Thus (3.4) holds. & Lemma 3.4. Let p 2 ð0,1Þ and xp,n ¼ Fn1 ðpÞ ¼ inffx : Fn ðxÞ Z pg. Assume that P(Xi = Xj)=0 for any iaj. Then p r Fn ðxp,n Þ op þ
1 n
a:s:
ð3:8Þ
Proof. The left-hand side of (3.8) follows from the definition of xp,n and Lemma 3.2(ii) immediately. It suffices to prove the right-hand side of (3.8). If Fn ðxp,n Þ Z p þ 1=n, then xp,n ZFn1 ðp þ 1=nÞ from Lemma 3.2(iii), i.e. xp,n Z xp þ 1=n,n . Since xp,n is the nondecreasing function of p, it follows that xp,n r xp þ 1=n,n . Hence, xp,n ¼ xp þ 1=n,n and 1 ðxp,n ¼ xp þ 1=n,n Þ: ð3:9Þ Fn ðxp,n Þ Z p þ n The condition P(Xi = Xj)= 0 for any iaj implies that X1ðnÞ oX2ðnÞ o oXnðnÞ where
X(n) i
a:s:;
ð3:10Þ
is the ith order statistic of sample X1, X2, y, Xn, n Z1. We consider the following two cases:
ðnÞ ðnÞ (1) If np is an integer, then wp 1 xp,n ¼ Xdnpe and xp þ 1=n,n ¼ Xdnpe þ 1. ðnÞ ðnÞ (2) If np is not an integer, then wp 1 xp,n ¼ Xdnpe and x p þ 1=n,n ¼ Xdnpe þ 2 . þ1
In fact, if np is an integer, then by (3.10) we have that with probability 1, ðnÞ Fn ðXdnpe Þ¼
np ¼ p, n
which implies that ðnÞ Fn1 ðpÞ ¼ xp,n ¼ inffx : Fn ðxÞ Z pg rXdnpe ;
ð3:11Þ
ðnÞ , (3.10) implies that Fn ðxÞ r ðdnpe1Þ=n op, thus for x o Xdnpe ðnÞ Fn1 ðpÞ ¼ xp,n ¼ inffx : Fn ðxÞ Z pg ZXdnpe :
ð3:12Þ
Combining (3.11) and (3.12), we obtain that wp 1, ðnÞ Fn1 ðpÞ ¼ xp,n ¼ Xdnpe
and
ðnÞ ðnÞ xp þ 1=n,n ¼ Xdnðp þ 1=nÞe ¼ Xdnpe þ 1 :
Similarly, we can get the result for case (2). By (3.9) and cases (1), (2), we can see that wp 1, 1 ðnÞ ðnÞ ðnÞ ðnÞ ðXdnpe ¼ Xdnpe Fn ðxp,n Þ Z p þ þ 1 Þ [ ðXdnpe þ 1 ¼ Xdnpe þ 2 Þ: n
ð3:13Þ
660
X. Wang et al. / Journal of Statistical Planning and Inference 141 (2011) 655–662
Therefore, 0 1 [ X 1 @ rP ðXi ¼ Xj ÞA r PðXi ¼ Xj Þ ¼ 0, P Fn ðxp,n Þ Z p þ n iaj iaj which implies that Fn ðxp,n Þ o p þ 1=n a.s.. The proof is completed.
ð3:14Þ &
4. Proofs of main results Proof of Theorem 2.1. Let fbn ,n Z1g be a sequence of positive integers such that bn C0 n1=4 ðlognÞ1=2 as n-1. For integers r ¼ bn ,bn , put
Zr,n ¼ xp þ an b1 n r, ar,n ¼ FðZr þ 1,n ÞFðZr,n Þ and Gr,n ¼ Fn ðZr,n ÞFn ðxp ÞFðZr,n Þ þ p6
n 1X ðY EY i Þ, ni¼1 i
where Yi ¼ IðXi r Zr,n ÞIðXi r xp Þ. Denote gðxÞ ¼ Fn ðxÞFðxÞFn ðxp Þ þ p,
x 2 Dn :
Then for all x 2 ½Zr,n , Zr þ 1,n , gðxÞ r Fn ðZr þ 1,n ÞFðZr,n ÞFn ðxp Þ þ p ¼ Gr þ 1,n þ ar,n and gðxÞ Z Fn ðZr,n ÞFðZr þ 1,n ÞFn ðxp Þ þ p ¼ Gr,n ar,n : Therefore, Hp,n r Kn þ bn ,
ð4:1Þ
where Kn ¼ maxfjGr,n j : bn r r rbn g,
bn ¼ maxfar,n : bn r r r bn 1g: 3=4 þ d , bn r r rbn 1, we have by the Mean Value Theorem that Since Zr þ 1,n Zr,n ¼ an b1 n ¼n
ar,n rd2 ðZr þ 1,n Zr,n Þ ¼ d2 n3=4 þ d , bn r r rbn 1, thus 0 r bn rd2 n3=4 þ d : 3=4 þ d
ðlognÞ Denote en ¼ n exists N 2 N þ such that
ð4:2Þ 3=4
and zr,n ¼ jFðZr,n ÞFðxp Þj. Let C1 be some positive constant such that f ðxp Þ oC1 . Then there
Fðxp þ an ÞFðxp Þ ¼ f ðxp Þan þ oðan Þ o C1 an and Fðxp ÞFðxp an Þ ¼ f ðxp Þan þ oðan Þ oC1 an for all n 4 N. Thus zr,n r C1 an
for jrjr bn and n 4 N:
ð4:3Þ
For r = 1,2,y, bn, Yi ¼ Iðxp oXi r Zr,n Þ, i= 1,2,y, n; for r = bn,y, 1, denote Yi ¼ IðZr,n o Xi r xp Þ6Zi , i= 1,2,y, n. Then Yi follows a Bernoulli distribution with EY i ¼ FðZr,n ÞFðxp Þ ¼ zr,n . It is easy to check that EjYi EY i jk ¼ ð1zr,n Þk zr,n þj0zr,n jk ð1zr,n Þ ¼ zr,n ð1zr,n Þ½ð1zr,n Þk1 þzk1 r,n rk!EðYi EY i Þ2 42zr,n for any integer k Z2. Applying Lemma 3.1(ii) to Y1 EY1, Y2 EY2, y, Yn EYn with c =1, e ¼ en and q ¼ dn1d loglogne, we have that ! n X ðYi EY i Þ 4nen PðjGr,n j 4 en Þ ¼ P i¼1
X. Wang et al. / Journal of Statistical Planning and Inference 141 (2011) 655–662
( r a1 exp
)
2k=ð2k þ 1Þ qe2n n ðkÞ a , þ a 2 qþ 1 25m22 þ 5en
661
ð4:4Þ
where
qe2n C n1d loglogn n3=2 þ 2d ðlognÞ3=2 r 2 25m2 þ5en 25C1 C0 n1=2 þ d ðlognÞ1=2 þ5n3=4 þ d ðlognÞ3=4 r CðlognÞðloglognÞ for all n sufficiently large ! 2n e2n þ2 1þ q 25m22 þ5en en r Cnd ðloglognÞ1 þ 2 1 þ 5 r Cnd ðloglognÞ1 for all n sufficiently large
a1 ¼
and k=ð2k þ 1Þ
a2 ðkÞ ¼ 11n 1þ r11n 1 þ rCn
7=4d
5mk
! r 11n 1 þ
en
5ð2zr,n Þ1=ð2k þ 1Þ
en
5ð2C1 C0 n1=2 þ d ðlognÞ1=2 Þ1=ð2k þ 1Þ
ðlognÞ
!
!
n3=4 þ d ðlognÞ3=4 3=4
ðn1=2 þ d ðlognÞ1=2 Þ1=ð2k þ 1Þ
for all n sufficiently large:
Therefore, by the above statements, we have that PðjGr,n j 4 en Þ r Cnd ðloglognÞ1 expfCðlognÞðloglognÞg þCn7=4d ðlognÞ3=4 ðn1=2 þ d ðlognÞ1=2 Þ1=ð2k þ 1Þ ðnd ðloglognÞ1 Þ2bk=ð2k þ 1Þ C r 2 þ Cn7=4ðb þ 1Þd ðlognÞ3=4 ðn1=2 þ ðb þ 1Þd ðlognÞ1=2 ðloglognÞ1 Þ1=ð2k þ 1Þ n
ð4:5Þ
for all n sufficiently large. Likewise, (4.5) holds for r = 1, y, bn for all n sufficiently large. Therefore, 1 X
PðKn 4 en Þ r C þC
n¼1
r C þC
bn X
1 X
PðjGr,n j4 en Þ
n ¼ n0 r ¼ bn 1 X 1=4
n
ðlognÞ1=2 PðjGr,n j4 en Þ o1,
n ¼ n0
provided that k is sufficiently large such that 7=4ðb þ 1Þd þ1=4 þ
1=2 þ ðb þ 1Þd o1: 2k þ 1
Together with Borel–Cantelli Lemma, it follows that with probability 1, the relations Kn 4 en hold for only finitely many n. Hence wp 1, Kn r en
for all n sufficiently large,
i.e. wp 1, Kn r n3=4 þ d ðlognÞ3=4
for all n sufficiently large:
ð4:6Þ
Finally, (4.1), (4.2) and (4.6) yield (2.1). This completes the proof of the theorem.
&
Proof of Theorem 2.2. By Lemma 3.3, we can see that wp 1, jxp,n xp jr
yðloglognÞ1=2 f ðxp Þn1=2d
for all n sufficiently large,
ð4:7Þ
which implies that wp 1,
xp,n 2 Dn for all n sufficiently large: It follows from Theorem 2.1 that wp 1, jðFn ðxp,n ÞFðxp,n ÞÞðFn ðxp ÞpÞj r supjðFn ðxÞFðxÞÞðFn ðxp ÞpÞj x2Dn
¼ Oðn3=4 þ d ðlognÞ3=4 Þ,
n-1,
662
X. Wang et al. / Journal of Statistical Planning and Inference 141 (2011) 655–662
which implies that wp 1, Fn ðxp Þp ¼ Fn ðxp,n ÞFðxp,n Þ þ Oðn3=4 þ d ðlognÞ3=4 Þ,
n-1:
ð4:8Þ
By (4.7), (4.8) and Lemma 3.4, we can obtain that wp 1, 1 Fðxp,n Þ þOðn3=4 þ d ðlognÞ3=4 Þ Fn ðxp Þp ¼ p þ O n ¼ ðFðxp,n ÞFðxp ÞÞ þOðn3=4 þ d ðlognÞ3=4 Þ ¼ f ðxp Þðxp,n xp Þ12f uðwn Þðxp,n xp Þ2 þ Oðn3=4 þ d ðlognÞ3=4 Þ ¼ f ðxp Þðxp,n xp Þ þ Oðn3=4 þ d ðlognÞ3=4 Þ,
n-1,
where wn is a random variable between xp,n and xp . Finally, we have that wp 1,
xp,n ¼ xp
Fn ðxp Þp þ Oðn3=4 þ d ðlognÞ3=4 Þ, f ðxp Þ
The proof is completed.
n-1:
&
Acknowledgements The authors are most grateful to Executive Editor N. Balakrishnan and anonymous referee for careful reading of the manuscript and valuable suggestions which helped to improve an earlier version of this paper. References Bahadur, R.R., 1966. A note on quantiles in large samples. Ann. Math. Statist. 37, 577–580. Babu, G.J., Singh, K., 1978. On deviations between empirical and quantile processes for mixing random variables. J. Multivariate Anal. 8, 532–549. Bosq, D., 1998. Nonparametric Statistics for Stochastic Processes. Springer, New York. Cai, Z.W., Roussas, G.G., 1997. Smooth estimate of quantiles under association. Statist. Probab. Lett. 36, 275–287. Sen, P.K., 1972. On Bahadur representation of sample quantile for sequences of fmixingrandom variables. J. Multivariate Anal. 2, 77–95. Serfling, R.J., 1980. Approximation Theorems of Mathematical Statistics. John Wiley & Sons, New York. Sun, S.X., 2006. The Bahadur representation for sample quantiles under weak dependence. Statist. Probab. Lett. 76, 1238–1244. Wu, W.B., 2005. On the Bahadur representation of sample quantiles for dependent sequences. Ann. Statist. 33, 1934–1963. Yoshihara, K., 1995. The Bahadur representation of sample quantile for sequences of strongly mixing random variables. Statist. Probab. Lett. 24, 299–304.
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The Bahadur representation for sample quantiles under strongly mixing sequence$ Xuejun Wang, Shuhe Hu , Wenzhi Yang School of Mathematical Science, Anhui University, Hefei, Anhui 230039, China
a r t i c l e in fo
abstract
Article history: Received 26 December 2009 Received in revised form 6 July 2010 Accepted 13 July 2010 Available online 17 July 2010
Sun [2006, The Bahadur representation for sample quantiles under weak dependence. Statist. Probab. Lett. 76, 1238–1244] established the Bahadur representation for sample quantiles under the strongly mixing sequence. But there are some problems in the proofs of the main results. In the paper, we further investigate the Bahadur representation for sample quantiles under strongly mixing sequence and get better bound than that in Sun (2006). & 2010 Elsevier B.V. All rights reserved.
Keywords: Bahadur representation Sample quantile Strongly mixing sequence
1. Introduction Let fXn ,n Z 1g be a sequence of random variables defined on a fixed probability space ðO,F ,PÞ with a common marginal distribution function FðxÞ ¼ PðX1 rxÞ. Denote F 1 ðpÞ ¼ inffx : FðxÞ Z pg as the corresponding pth quantile function of distribution function F(x), 0 op o1. For a sample X1, X2,y, Xn, n Z1, let Fn represent the empirical distribution function P based on X1, X2,y, Xn, which is defined as Fn ðxÞ ¼ ð1=nÞ ni¼ 1 IðXi rxÞ, x 2 R, where I(A) denotes the indicator function of a 1 set A and R is the real line. For 0 o p o1, we define Fn ðpÞ ¼ inffx : Fn ðxÞ Zpg as the sample pth quantile. Remark 1.1. Let F be a distribution function (continuous from the right, as usual). For 0 o p o 1, the pth quantile of F is defined as
xp ¼ inffx : FðxÞ Z pg and is alternately denoted by F 1(p). The function F 1(t), 0 ot o 1, is called the inverse function of F. It is easy to check that
xp possesses the following properties: (i) Fðxp Þ rp r Fðxp Þ; (ii) If xp is the unique solution x of FðxÞ r p rFðxÞ, then for any e 4 0, Fðxp eÞ op oFðxp þ eÞ:
$ Supported by the National Natural Science Foundation of China (Grant nos. 10871001, 60803059), Provincial Natural Science Research Project of Anhui Colleges (KJ2010A005), Talents Youth Fund of Anhui Province Universities (2010SQRL016ZD), Youth Science Research Fund of Anhui University (2009QN011A). Corresponding author. E-mail addresses: [email protected] (X. Wang), [email protected] (S. Hu), [email protected] (W. Yang).
0378-3758/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2010.07.008
656
X. Wang et al. / Journal of Statistical Planning and Inference 141 (2011) 655–662
Next, we briefly introduce some dependence condition on the Xi ’s. Let F nm ¼ sðXi ,m ri r n, i 2 NÞ, the sfield generated by random variables Xm, y, Xn, 1r m rn. For n Z1, we define
aðnÞ ¼ sup
sup
jPðABÞPðAÞPðBÞj:
1 m2NA2F m 1 ,B2F m þ n
Definition 1.1. A random variable sequence fXn ,n Z 1g is said to be strongly mixing or amixing if aðnÞk0 as n-1. Throughout the paper, for a fixed p 2 ð0,1Þ, let xp ¼ F 1 ðpÞ, xp,n ¼ Fn1 ðpÞ. C and C0 denote positive constants which may be different in various places. dxe denotes the largest integer not exceeding x. a4b6minfa,bg. Bahadur (1966) firstly introduced an elegant representation for sample quantiles in terms of empirical distribution function based on independent and identically distributed (i.i.d.) random variables. The Bahadur representation allows one to express asymptotically a sample quantiles as a sample mean of certain (bounded) random variables and obtain the limiting properties of the sample quantile. At present, many researchers have obtained the Bahadur results under the case of dependent sequences, see for example, Sen (1972), Babu and Singh (1978) and Yoshihara (1995) gave the Bahadur representation for sample quantiles under fmixing sequence and strongly mixing (or amixing) sequence, respectively. Cai and Roussas (1997) studied the smooth estimate of quantiles under association. Wu (2005) established the Bahadur representation of sample quantiles for linear and some widely used nonlinear processes, and so forth. Recently, Sun (2006) established the Bahadur representation for sample quantiles under the strongly mixing (or amixing) sequence with polynomially decaying rate and obtained the following two theorems. Theorem 1.1 (cf. Sun, 2006, Theorem 1). Suppose that the strongly mixing coefficients of fXn ,n Z 1g satisfy aðnÞ rCnb for some positive constant C and for some b 40. Also suppose that the common marginal distribution function F(x) is continuously differentiable with derivative function f(x) in a neighborhood N p of xp such that 0 o d1 ¼ infff ðxÞ : x 2 N p g rd2 ¼ supff ðxÞ : x 2 N p g o 1: Then for any d 2 ð11=4ð1þ bÞ, 12Þ, b 4 4:5, supjðFn ðxÞFðxÞÞðFn ðxp ÞpÞj r ð1 þ d2 Þn3=4 þ d logn
a:s:;
ð1:1Þ
x2Dn
where Dn ¼ ½xp n1=2 þ d logn, xp þ n1=2 þ d logn. Theorem 1.2 (cf. Sun, 2006, Theorem 2). Suppose that conditions in Theorem 1.1 hold. Also suppose that f uðxÞ is bounded in a neighborhood of xp , say N p . Then for any d 2 ð11=4ð1 þ bÞ, 14Þ, b 4 10,
xp,n ¼ xp
Fn ðxp Þp þ Oðn3=4 þ d lognÞ f ðxp Þ
a:s:
ð1:2Þ
Remark 1.2. We point out that there are some mistakes in Sun (2006), which prejudices the validity of the results. Using the same notations as that in Sun (2006), Sun estimated the inequality 0 1 X 1 X 1 n P @ Vj,n 4 n3=4 þ d lognA o 1, n j ¼ 1 n¼1 and obtained that X 1 n jDr,n j ¼ Vj,n rn3=4 þ d logn ¼ en n j¼1
a:s:
by Borel–Cantelli Lemma. But it is not always true. For example, let tn ¼ 1=n and Xn ¼ ð1=n2 ÞX, where X is a standard normally distributed random variable. One can check that 1 X
PðjXn j4 tn Þ r
n¼1
1 1 X X EX 2n 1 ¼ o1: 2 2 n t n n¼1 n¼1
But ‘‘jXn jr tn a.s.’’ is uncorrect. In fact, PðjXn j rtn Þ ¼ PðjXjr nÞa1: Furthermore, by inequality (3.4) in Sun (2006), one should estimate 1 X P max jDr,n j 4 n3=4 þ d logn o1: n¼1
0 r jrj r mn
The similar problem exists in (3.16) in Sun (2006). Hence these problems above affect the proofs of Theorems 1 and 2 in Sun (2006).
X. Wang et al. / Journal of Statistical Planning and Inference 141 (2011) 655–662
657
In the paper, we will further investigate the Bahadur representation for sample quantiles under strongly mixing sequence and get better bound than that in Sun (2006). The organization of this paper is as follows. The main results are presented in Section 2, preparations for the proofs of the theorems are given in Section 3 and the proofs are provided in Section 4. 2. Main results
Theorem 2.1. Suppose that the strongly mixing coefficients of fXn ,n Z 1g satisfy aðnÞ rCnb for some positive constant C and for some b 4 0. Assume that the common marginal distribution function F(x) is continuously differentiable with derivative function f(x) in a neighborhood N p of xp such that 0 o d1 ¼ infff ðxÞ : x 2 N p g r d2 ¼ supff ðxÞ : x 2 N p g o1: Then for any d 2 ð3=ðb þ 1Þ, 12Þ, b 4 5, with probability 1 (wp1) supjðFn ðxÞFðxÞÞðFn ðxp ÞpÞj r ð1 þ d2 Þn3=4 þ d ðlognÞ3=4
ð2:1Þ
x2Dn
for all n sufficiently large, where Dn ¼ ½xp an , xp þ an and an C0 n1=2 þ d ðlognÞ1=2 as n-1. Theorem 2.2. Suppose that conditions of Theorem 2.1 are satisfied and P(Xi =Xj)=0 for any iaj. Assume that f uðxÞ is bounded in some neighborhood of xp , say N p . Then for any d 2 ð3=ðb þ 1Þ, 14, b 4 11, with probability 1,
xp,n ¼ xp
Fn ðxp Þp þOðn3=4 þ d ðlognÞ3=4 Þ, f ðxp Þ
n-1:
ð2:2Þ
Remark 2.1. Comparing our Theorem 2.2 with Theorem 2 in Sun (2006), we obtain better bound Oðn3=4 þ d ðlognÞ3=4 Þ than that in Sun (2006). Remark 2.2. We think that the equality (3.19) in Sun (2006) cannot be obtained easily by borrowing on the arguments on pp. 538–539 of Babu and Singh (1978). We have not seen the corrected proof for it in detail so for. So we add the condition ‘‘P(Xi = Xj)= 0 for any iaj’’ in Theorem 2.2, which can be applied to obtain (3.8) in the paper (see Lemma 3.4). We remark that the condition ‘‘P(Xi = Xj)= 0 for any iaj’’ is a weak condition. Here we give some examples.
(1) If random variables Xi and Xj are independent for iaj and the distribution function Fi(x) of random variable Xi is continuous, then Z Z Z þ 1 Z Z þ1 Z þ1 dF i ðxÞ dF j ðyÞ ¼ dF i ðxÞ dF j ðyÞ ¼ PðXi ¼ yÞ dF j ðyÞ ¼ 0 dF j ðyÞ ¼ 0: PðXi ¼ Xj Þ ¼ y¼x
1
x¼y
1
1
(2) If (Xi, Xj) has the joint probability density function fij(x,y) for iaj, then Z Z Z þ 1 Z Z þ1 fij ðx,yÞ dx dy ¼ fij ðx,yÞ dx dy ¼ 0 dy ¼ 0: PðXi ¼ Xj Þ ¼ y¼x
1
x¼y
1
3. Preparations The following lemmas are needed for the proofs of the main results. Lemma 3.1 (cf. Bosq, 1998). Let fXt ,t 2 Zg be a zero-mean real-valued process of strongly mixing random variables.
(i) If there exists a positive constant b such that jXt j rb, t 2 Z, then for each integer q 2 ½1,n=2 and each e 4 0, ! 1=2 n X qe2 4b n : P Xi 4 ne r4exp 2 þ 22 1 þ qa 2q e 8b
ð3:1Þ
i¼1
(ii) If there exists a positive constant c such that EjXt jk r ck2 k!EX 2t o 1,
t ¼ 1,2, . . . ,n, k ¼ 3,4, . . . ,
ð3:2Þ
658
X. Wang et al. / Journal of Statistical Planning and Inference 141 (2011) 655–662
then for each n Z2, each integer q 2 ½1,n=2, each e 4 0 and each k Z 3, ! ( )
2k=ð2k þ 1Þ n X qe2 n P Xi 4ne r a1 exp ðkÞ a , þ a 2 qþ 1 25m22 þ 5ce
ð3:3Þ
i¼1
where 2n e2 þ2 1þ a1 ¼ q 25m22 þ 5ce
! with m22 ¼ max EX 2t 1rtrn
and k=ð2k þ 1Þ
a2 ðkÞ ¼ 11n 1 þ
5mk
! with mk ¼ max ðEjXt jk Þ1=k :
e
1rtrn
Lemma 3.2 (cf. Serfling, 1980, Lemma 1.1.4). Let F(x) be a right-continuous distribution function. The inverse function F 1(t), 0 ot o 1, is nondecreasing and left-continuous, and satisfies (i) F 1 ðFðxÞÞ r x, 1 ox o 1; (ii) FðF 1 ðtÞÞ Zt, 0 ot o 1; (iii) FðxÞ Zt if and only if x Z F 1 ðtÞ. Lemma 3.3. Suppose that the strongly mixing coefficients of fXn ,n Z1g satisfy aðnÞ rCnb for some positive constant C and for some b 4 0, and the common marginal distribution function F(x) is differentiable at xp , with Fuðxp Þ ¼ f ðxp Þ 4 0. Assume that f uðxÞ is bounded in a neighborhood of xp , say N p . Then for any y 4 0 and d 2 ð9=ð10 þ8bÞ, 12Þ, b 41, with probability 1 jxp,n xp j r
yðloglognÞ1=2 f ðxp Þn1=2d
for all n sufficiently large:
ð3:4Þ
Proof. Let en ¼ yðloglognÞ1=2 =f ðxp Þn1=2d , n Z 1. Write Pðjxp,n xp j 4 en Þ ¼ Pðxp,n 4 xp þ en Þ þ Pðxp,n o xp en Þ: By Lemma 3.2(iii), Pðxp,n 4 xp þ en Þ ¼ Pðp 4 Fn ðxp þ en ÞÞ ¼ Pð1Fn ðxp þ en Þ 4 1pÞ ¼ P
n X
! IðXi 4 xp þ en Þ4 nð1pÞ
i¼1
¼P
n X
! ðVi EV i Þ 4 ndn1 ,
i¼1
where Vi ¼ IðXi 4 xp þ en Þ and dn1 ¼ Fðxp þ en Þp 40. Likewise, by Lemma 3.2(iii), ! n X ðWi EW i Þ Z ndn2 , Pðxp,n o xp en Þ r Pðp r Fn ðxp en ÞÞ ¼ P i¼1
where Wi ¼ IðXi r xp en Þ and dn2 ¼ pFðxp en Þ 40. It is easy to see that fVi EV i ,1 ri rng and fWi EW i ,1 ri r ng are still amixing sequences with jVi EV i j r1 and jWi EW i jr 1 for each i. Applying Lemma 3.1(i) to fVi EV i ,1 r ir ng and fWi EW i ,1 ri rng with q ¼ dn12d logne, we can see that ( ) 2 qdn1 4 1=2 n ð3:5Þ qa Pðxp,n 4 xp þ en Þ r4exp þ 22 1þ 2q 8 dn1 and (
) 2 qdn2 4 1=2 n : qa Pðxp,n o xp en Þ r 4exp þ 22 1 þ 2q 8 dn2
ð3:6Þ
Therefore, ( Pðjxp,n xp j 4 en Þ r 8exp
) 1=2 q½minðdn1 , dn1 Þ2 4 n : qa þ 44 1 þ minðdn1 , dn2 Þ 2q 8
ð3:7Þ
Since F(x) is continuous at xp with Fuðxp Þ 4 0, xp is the unique solution of FðxÞ r p r FðxÞ and Fðxp Þ ¼ p. By the assumption on f uðxÞ and Taylor’s expansion,
dn1 ¼ Fðxp þ en Þp ¼ f ðxp Þen þ oðen Þ ¼ yn1=2 þ d ðloglognÞ1=2 þ oðen Þ and
dn2 ¼ pFðxp en Þ ¼ f ðxp Þen þ oðen Þ ¼ yn1=2 þ d ðloglognÞ1=2 þoðen Þ:
X. Wang et al. / Journal of Statistical Planning and Inference 141 (2011) 655–662
659
So minðdn1 , dn2 Þ Z 12yn1=2 þ d ðloglognÞ1=2
for all n sufficiently large:
Together with (3.7), we have (
) 2 n12d logn y n1 þ 2d loglogn 32 !1=2 2d b 8 n n12d logn C þC 1þ 1=2 logn yn1=2 þ d ðloglognÞ
Pðjxp,n xp j 4 en Þ r Cexp
r
C CðlognÞb þ 1 þ n2 n1 þ g ðloglognÞ1=4
for all n sufficiently large, where g ¼ ðdð10 þ 8bÞ9Þ=4 4 0 when b 4 1 and d 2 ð9=ð10 þ 8bÞ, 12Þ. Therefore, 1 X
Pðjxp,n xp j4 en Þ r C þ C
n¼1
1 1 X X 1 ðlognÞb þ 1 þC o 1, 2 1 þ g ðloglognÞ1=4 n ¼ n0 n n ¼ n0 n
which implies that wp 1, the relations jxp,n xp j 4 en hold for only finitely many n by Borel–Cantelli Lemma. Thus (3.4) holds. & Lemma 3.4. Let p 2 ð0,1Þ and xp,n ¼ Fn1 ðpÞ ¼ inffx : Fn ðxÞ Z pg. Assume that P(Xi = Xj)=0 for any iaj. Then p r Fn ðxp,n Þ op þ
1 n
a:s:
ð3:8Þ
Proof. The left-hand side of (3.8) follows from the definition of xp,n and Lemma 3.2(ii) immediately. It suffices to prove the right-hand side of (3.8). If Fn ðxp,n Þ Z p þ 1=n, then xp,n ZFn1 ðp þ 1=nÞ from Lemma 3.2(iii), i.e. xp,n Z xp þ 1=n,n . Since xp,n is the nondecreasing function of p, it follows that xp,n r xp þ 1=n,n . Hence, xp,n ¼ xp þ 1=n,n and 1 ðxp,n ¼ xp þ 1=n,n Þ: ð3:9Þ Fn ðxp,n Þ Z p þ n The condition P(Xi = Xj)= 0 for any iaj implies that X1ðnÞ oX2ðnÞ o oXnðnÞ where
X(n) i
a:s:;
ð3:10Þ
is the ith order statistic of sample X1, X2, y, Xn, n Z1. We consider the following two cases:
ðnÞ ðnÞ (1) If np is an integer, then wp 1 xp,n ¼ Xdnpe and xp þ 1=n,n ¼ Xdnpe þ 1. ðnÞ ðnÞ (2) If np is not an integer, then wp 1 xp,n ¼ Xdnpe and x p þ 1=n,n ¼ Xdnpe þ 2 . þ1
In fact, if np is an integer, then by (3.10) we have that with probability 1, ðnÞ Fn ðXdnpe Þ¼
np ¼ p, n
which implies that ðnÞ Fn1 ðpÞ ¼ xp,n ¼ inffx : Fn ðxÞ Z pg rXdnpe ;
ð3:11Þ
ðnÞ , (3.10) implies that Fn ðxÞ r ðdnpe1Þ=n op, thus for x o Xdnpe ðnÞ Fn1 ðpÞ ¼ xp,n ¼ inffx : Fn ðxÞ Z pg ZXdnpe :
ð3:12Þ
Combining (3.11) and (3.12), we obtain that wp 1, ðnÞ Fn1 ðpÞ ¼ xp,n ¼ Xdnpe
and
ðnÞ ðnÞ xp þ 1=n,n ¼ Xdnðp þ 1=nÞe ¼ Xdnpe þ 1 :
Similarly, we can get the result for case (2). By (3.9) and cases (1), (2), we can see that wp 1, 1 ðnÞ ðnÞ ðnÞ ðnÞ ðXdnpe ¼ Xdnpe Fn ðxp,n Þ Z p þ þ 1 Þ [ ðXdnpe þ 1 ¼ Xdnpe þ 2 Þ: n
ð3:13Þ
660
X. Wang et al. / Journal of Statistical Planning and Inference 141 (2011) 655–662
Therefore, 0 1 [ X 1 @ rP ðXi ¼ Xj ÞA r PðXi ¼ Xj Þ ¼ 0, P Fn ðxp,n Þ Z p þ n iaj iaj which implies that Fn ðxp,n Þ o p þ 1=n a.s.. The proof is completed.
ð3:14Þ &
4. Proofs of main results Proof of Theorem 2.1. Let fbn ,n Z1g be a sequence of positive integers such that bn C0 n1=4 ðlognÞ1=2 as n-1. For integers r ¼ bn ,bn , put
Zr,n ¼ xp þ an b1 n r, ar,n ¼ FðZr þ 1,n ÞFðZr,n Þ and Gr,n ¼ Fn ðZr,n ÞFn ðxp ÞFðZr,n Þ þ p6
n 1X ðY EY i Þ, ni¼1 i
where Yi ¼ IðXi r Zr,n ÞIðXi r xp Þ. Denote gðxÞ ¼ Fn ðxÞFðxÞFn ðxp Þ þ p,
x 2 Dn :
Then for all x 2 ½Zr,n , Zr þ 1,n , gðxÞ r Fn ðZr þ 1,n ÞFðZr,n ÞFn ðxp Þ þ p ¼ Gr þ 1,n þ ar,n and gðxÞ Z Fn ðZr,n ÞFðZr þ 1,n ÞFn ðxp Þ þ p ¼ Gr,n ar,n : Therefore, Hp,n r Kn þ bn ,
ð4:1Þ
where Kn ¼ maxfjGr,n j : bn r r rbn g,
bn ¼ maxfar,n : bn r r r bn 1g: 3=4 þ d , bn r r rbn 1, we have by the Mean Value Theorem that Since Zr þ 1,n Zr,n ¼ an b1 n ¼n
ar,n rd2 ðZr þ 1,n Zr,n Þ ¼ d2 n3=4 þ d , bn r r rbn 1, thus 0 r bn rd2 n3=4 þ d : 3=4 þ d
ðlognÞ Denote en ¼ n exists N 2 N þ such that
ð4:2Þ 3=4
and zr,n ¼ jFðZr,n ÞFðxp Þj. Let C1 be some positive constant such that f ðxp Þ oC1 . Then there
Fðxp þ an ÞFðxp Þ ¼ f ðxp Þan þ oðan Þ o C1 an and Fðxp ÞFðxp an Þ ¼ f ðxp Þan þ oðan Þ oC1 an for all n 4 N. Thus zr,n r C1 an
for jrjr bn and n 4 N:
ð4:3Þ
For r = 1,2,y, bn, Yi ¼ Iðxp oXi r Zr,n Þ, i= 1,2,y, n; for r = bn,y, 1, denote Yi ¼ IðZr,n o Xi r xp Þ6Zi , i= 1,2,y, n. Then Yi follows a Bernoulli distribution with EY i ¼ FðZr,n ÞFðxp Þ ¼ zr,n . It is easy to check that EjYi EY i jk ¼ ð1zr,n Þk zr,n þj0zr,n jk ð1zr,n Þ ¼ zr,n ð1zr,n Þ½ð1zr,n Þk1 þzk1 r,n rk!EðYi EY i Þ2 42zr,n for any integer k Z2. Applying Lemma 3.1(ii) to Y1 EY1, Y2 EY2, y, Yn EYn with c =1, e ¼ en and q ¼ dn1d loglogne, we have that ! n X ðYi EY i Þ 4nen PðjGr,n j 4 en Þ ¼ P i¼1
X. Wang et al. / Journal of Statistical Planning and Inference 141 (2011) 655–662
( r a1 exp
)
2k=ð2k þ 1Þ qe2n n ðkÞ a , þ a 2 qþ 1 25m22 þ 5en
661
ð4:4Þ
where
qe2n C n1d loglogn n3=2 þ 2d ðlognÞ3=2 r 2 25m2 þ5en 25C1 C0 n1=2 þ d ðlognÞ1=2 þ5n3=4 þ d ðlognÞ3=4 r CðlognÞðloglognÞ for all n sufficiently large ! 2n e2n þ2 1þ q 25m22 þ5en en r Cnd ðloglognÞ1 þ 2 1 þ 5 r Cnd ðloglognÞ1 for all n sufficiently large
a1 ¼
and k=ð2k þ 1Þ
a2 ðkÞ ¼ 11n 1þ r11n 1 þ rCn
7=4d
5mk
! r 11n 1 þ
en
5ð2zr,n Þ1=ð2k þ 1Þ
en
5ð2C1 C0 n1=2 þ d ðlognÞ1=2 Þ1=ð2k þ 1Þ
ðlognÞ
!
!
n3=4 þ d ðlognÞ3=4 3=4
ðn1=2 þ d ðlognÞ1=2 Þ1=ð2k þ 1Þ
for all n sufficiently large:
Therefore, by the above statements, we have that PðjGr,n j 4 en Þ r Cnd ðloglognÞ1 expfCðlognÞðloglognÞg þCn7=4d ðlognÞ3=4 ðn1=2 þ d ðlognÞ1=2 Þ1=ð2k þ 1Þ ðnd ðloglognÞ1 Þ2bk=ð2k þ 1Þ C r 2 þ Cn7=4ðb þ 1Þd ðlognÞ3=4 ðn1=2 þ ðb þ 1Þd ðlognÞ1=2 ðloglognÞ1 Þ1=ð2k þ 1Þ n
ð4:5Þ
for all n sufficiently large. Likewise, (4.5) holds for r = 1, y, bn for all n sufficiently large. Therefore, 1 X
PðKn 4 en Þ r C þC
n¼1
r C þC
bn X
1 X
PðjGr,n j4 en Þ
n ¼ n0 r ¼ bn 1 X 1=4
n
ðlognÞ1=2 PðjGr,n j4 en Þ o1,
n ¼ n0
provided that k is sufficiently large such that 7=4ðb þ 1Þd þ1=4 þ
1=2 þ ðb þ 1Þd o1: 2k þ 1
Together with Borel–Cantelli Lemma, it follows that with probability 1, the relations Kn 4 en hold for only finitely many n. Hence wp 1, Kn r en
for all n sufficiently large,
i.e. wp 1, Kn r n3=4 þ d ðlognÞ3=4
for all n sufficiently large:
ð4:6Þ
Finally, (4.1), (4.2) and (4.6) yield (2.1). This completes the proof of the theorem.
&
Proof of Theorem 2.2. By Lemma 3.3, we can see that wp 1, jxp,n xp jr
yðloglognÞ1=2 f ðxp Þn1=2d
for all n sufficiently large,
ð4:7Þ
which implies that wp 1,
xp,n 2 Dn for all n sufficiently large: It follows from Theorem 2.1 that wp 1, jðFn ðxp,n ÞFðxp,n ÞÞðFn ðxp ÞpÞj r supjðFn ðxÞFðxÞÞðFn ðxp ÞpÞj x2Dn
¼ Oðn3=4 þ d ðlognÞ3=4 Þ,
n-1,
662
X. Wang et al. / Journal of Statistical Planning and Inference 141 (2011) 655–662
which implies that wp 1, Fn ðxp Þp ¼ Fn ðxp,n ÞFðxp,n Þ þ Oðn3=4 þ d ðlognÞ3=4 Þ,
n-1:
ð4:8Þ
By (4.7), (4.8) and Lemma 3.4, we can obtain that wp 1, 1 Fðxp,n Þ þOðn3=4 þ d ðlognÞ3=4 Þ Fn ðxp Þp ¼ p þ O n ¼ ðFðxp,n ÞFðxp ÞÞ þOðn3=4 þ d ðlognÞ3=4 Þ ¼ f ðxp Þðxp,n xp Þ12f uðwn Þðxp,n xp Þ2 þ Oðn3=4 þ d ðlognÞ3=4 Þ ¼ f ðxp Þðxp,n xp Þ þ Oðn3=4 þ d ðlognÞ3=4 Þ,
n-1,
where wn is a random variable between xp,n and xp . Finally, we have that wp 1,
xp,n ¼ xp
Fn ðxp Þp þ Oðn3=4 þ d ðlognÞ3=4 Þ, f ðxp Þ
The proof is completed.
n-1:
&
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