Hindawi Publishing Corporation
Advances in Decision Sciences
Volume 2012, Article ID 671942, 7 pages
doi:10.1155/2012/671942
Research Article
Almost Sure Central Limit Theorem of
Sample Quantiles
Yu Miao,1 Shoufang Xu,2 and Ang Peng3
1
College of Mathematics and Information Science, Henan Normal University, Henan Province,
Xinxiang 453007, China
2
Department of Mathematics and Information Science, Xinxiang University, Henan Province,
Xinxiang 453000, China
3
Heze College of Finance and Economics, Shandong Province, Heze 274000, China
Correspondence should be addressed to Yu Miao, yumiao728@yahoo.com.cn
Received 19 April 2012; Accepted 12 August 2012
Academic Editor: Saralees Nadarajah
Copyright q 2012 Yu Miao et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We obtain the almost sure central limit theorem ASCLT of sample quantiles. Furthermore, based
on the method, the ASCLT of order statistics is also proved.
1. Introduction
To describe the results of the paper, suppose that we have an independent and identically
distributed sample of size n from a distribution function Fx with a continuous probability
density function fx. Let Fn x denote the sample distribution function, that is,
Fn x n
1
1{Xi ≤x} ,
n i1
−∞ < x < ∞.
1.1
Let us define the pth quantile of F by
ξp inf x : Fx ≥ p ,
p ∈ 0, 1,
1.2
and the sample quantile ξnp by
ξnp inf x : Fn x ≥ p ,
p ∈ 0, 1.
1.3
2
Advances in Decision Sciences
It is well known that ξnp is a natural estimator of ξp . Since the quantile can be used for
describing some properties of random variables, and there are not the restrictions of moment
conditions, it is being widely employed in diverse problems in finance, such as quantilehedging, optimal portfolio allocation, and risk management.
In practice, the large sample theory which can give the asymptotic properties of
sample estimator is an important method to analyze statistical problems. There are numerous
literatures to study the sample quantiles. Let p ∈ 0, 1, if ξp is the unique solution x of
a.e.
Fx− ≤ p ≤ Fx, then ξnp −−−→ ξp see 1. In addition, if Fx possesses a continuous
density function fx in a neighborhood of ξp and fξp > 0, then
n1/2 f ξp ξnp − ξp
−→ N0, 1,
1/2
p 1−p
as n −→ ∞,
1.4
where N0, 1 denotes the standard normal variable see 1, 2. Suppose that Fx is twice
differentiable at ξp , with F ξp fξp > 0, then Bahadur 3 proved
p − Fn ξp
n,
ξnp ξp R
f ξp
1.5
a.e.,
n On−3/4 log n3/4 , a.e, as n → ∞. Very recently, Xu and Miao 4 obtained the
where R
moderate deviation, large deviation and Bahadur asymptotic efficiency of the sample quantiles ξnp . Xu et al. 5 studied the Bahadur representation of sample quantiles for negatively
associated sequences under some mild conditions.
Based on the above works, in the paper, we are interested in the almost sure central
limit theorem ASCLT of sample quantiles ξnp . The theory of ASCLT has been first introduced independently by Brosamler 6 and Schatte 7. The classical ASCLT states that when
EX 0, VarX σ 2 ,
n
1 1
1{Sk ≤√kσx} Φx,
n → ∞ log n
k
k1
lim
a.s.
1.6
for any x ∈ R, where Sk denotes the partial sums Sk X1 · · · Xk . Moreover, from the
method to prove the ASCLT of sample quantiles, in Section 3, we obtain the ASCLT of order
statistics.
2. Main Results
Theorem 2.1. Let X1 , X2 , . . . , Xn be a sequence of independent identically distributed random
variables from a cumulative distribution function F. Let p ∈ 0, 1 and suppose that fξp : F ξp exists and is positive. Then one has
n
1 1 √
1{ kξkp −ξp ≤σx} Φx,
n → ∞ log n
k
k1
lim
for any x ∈ R, where σ 2 p1 − p/f 2 ξp .
a.s.
2.1
Advances in Decision Sciences
3
Proof. Firstly, it is not difficult to check
σx
k ξkp − ξp ≤ σx Fk ξp √
≥ p :
k
k
1 Yi,k ≤ ωk ,
√
k i1
2.2
where
Yi,k E1{Xi ≤ξp σx/√k} − 1{Xi ≤ξp σx/√k},
ωk k E1{Xi ≤ξp σx/√k} − p .
2.3
From the Taylor’s formula, it follows
σx
E1{Xi ≤ξp σx/√k} F ξp √
k
σx
1
F ξp F ξp √ o √
k
k
σx
1
p f ξp √ o √ ,
k
k
2.4
ωk f ξp σx o1.
2.5
which implies
By the Lindeberg’s central limit theorem, we can get
k
1
d
Yi,k −→ N0, 1,
√
f ξp σ k i1
2.6
as k −→ ∞.
Hence, 2.1 is equivalent to
n
1 1
1{1/fξp σ √k k Yi,k ≤xo1} Φx,
i1
n → ∞ log n
k
k1
lim
a.s.
2.7
Throughout the following proof, C denotes a positive constant, which may take different values whenever it appears in different expressions.
Put that
Zi,k :
1
Yi,k .
f ξp σ
2.8
Let g be a bounded Lipschitz function bounded by C, then from 2.6, we have
Eg
k
1 Zi,k
√
k i1
−→ EgN,
as k −→ ∞,
2.9
4
Advances in Decision Sciences
where N denotes the standard normal random variable. Next, we should notice that 2.7 is
equivalent to
n
1 1
g
lim
n → ∞ log n
k
k1
k
1 Zi,k
√
k i1
EgN a.s.
2.10
from Section 2 of Peligrad and Shao 8 and Theorem 7.1 of Billingsley 9. Hence, to prove
2.7, it suffices to show that as n → ∞,
n
k
k
1 1 1
1 g √
Zi,k − Eg √
Zi,k
Rn log n k1 k
k i1
k i1
n
1
1 Tk −→ 0,
:
log n k1 k
2.11
a.s.
It is obvious that
ER2n
⎡
⎤
n
n−1 n
1
1
2
⎣
ETk Tj ⎦.
ETk 2
kj
log2 n k1 k2
k1 jk1
1
2.12
Since g is bounded, we have
n
1
C
.
ETk2 ≤
2
2
log
n
k
log n k1
1
2.13
Furthermore, for 1 ≤ k < j ≤ n, we have
j
k
1
1
ETk Tj Cov g √
Zi,k , g Zi,j
j i1
k i1
⎛ ⎞
⎛
⎞⎞
⎛ j
j
k
Z
Z
Z
i,j
i,j
i,k
ik1
i1
i1
⎝
⎠
⎝
⎠
⎠
⎝
Cov g
−g
,g
√
j
j
k
2.14
k
C √k 1/2
C 2
≤ E Zi,j ≤ EZ1,j
,
j i1
j
where
2
EZ1,j
1O
1
j
.
2.15
Advances in Decision Sciences
5
Therefore, we have
n
n
n−1 n−1 1/2
1 1
2
≤ C
EZ
ET
T
k
j
1,j
log2 n k1 jk1 kj
log2 n k1 jk1 k1/2 j 3/2
1
1/2
1
C
2
.
EZ
≤
1,j
2
1/2
3/2
log
n
log n j2 k1 k j
j−1
n C 2.16
From the above discussions, it follows that
ER2n ≤
C
.
log n
2.17
τ
Take nk ek , where τ > 1. Then by Borel-Cantelli lemma, we have
Rnk −→ 0,
a.s. as k −→ ∞.
2.18
Since g is bounded function, then for nk < n ≤ nk1 , we obtain
nk
l
l
1 1 1
1 g √
Zi,l − Eg √
Zi,l
|Rn | ≤
log nk l1 l
l i1
l i1
n
l
l
k1
1
1 1 1 Zi,l − Eg √
Zi,l g √
log nk ln 1 l l i1
l i1
2.19
k
≤ |Rnk | n
k1
1
C
−→ 0,
log nk ln 1 l
a.s., as n −→ ∞,
k
where we used the fact
log nk1 k 1τ
−→ 1,
log nk
kτ
as k −→ ∞.
2.20
So, the proof of the theorem is completed.
3. Further Results
Another method to estimate the quantile is to use the order statistics. Based on the sample
{X1 , . . . , Xn } of observations on Fx, the ordered sample values:
X1 ≤ X2 ≤ · · · ≤ Xn
3.1
are called the order statistics. For more details about order statistics, one can refer to
Serfling 1 or David and Nagaraja 10. Suppose that F is twice differentiable at ξp with
6
Advances in Decision Sciences
F ξp fξp > 0, then the Bahadur representation for order statistics was first established
by Bahadur 3, as n → ∞
Xkn kn /n − Fn ξp
1/2δ1 ξp O n−3/4 log n
f ξp
a.e.,
3.2
where
√ δ kn np o n log n
,
n −→ ∞, for some δ ≥
1
.
2
3.3
From the idea of the Bahadur representation for order statistics, many important properties
of order statistics can be easily proved. For example, Miao et al. 11 proved asymptotic
properties of the deviation between order statistics and pth quantile, which included large
and moderate deviation, Bahadur asymptotic efficiency.
Though there are some papers to study the ASCLT for the order statistics e.g., Peng
and Qi 12, Hörmann 13, Tong et al. 14, etc., based on the method to deal with the sample
quantile, we can also obtain the ASCLT of the order statistics.
Theorem 3.1. Let X1 , X2 , . . . , Xn be a sequence of independent identically distributed random
variables from a cumulative distribution function F. Let p ∈ 0, 1 and suppose that fξp : F ξp √
exists and is positive. Let kn np o n, then one has
n
1 1 √
1
Φx,
n → ∞ log n
j { jXkj −ξp ≤σx}
j1
lim
a.s.
3.4
for any x ∈ R, where σ 2 p1 − p/f 2 ξp .
Proof. Firstly, it is easy to see that the following two events are equivalent:
$ j
√
j Xkj − ξp ≤ σx 1{Xi ≤ξp σx/ j} ≥ kj
i1
⎧
⎫
⎨ j Y i,j
⎬
i1
:
≤ ωj ,
⎩
⎭
j
3.5
where
Y i,j E1{Xi ≤ξp σx/√j} − 1{Xi ≤ξp σx/√j},
1
ωj j
σx
jF ξp j
− kj
f ξp σx o1.
Hence, by the same proof of Theorem 2.1, we can obtain the desired result.
3.6
Advances in Decision Sciences
7
References
1 R. J. Serfling, Approximation Theorems of Mathematical Statistics, John Wiley & Sons, New York, NY,
USA, 1980.
2 S. S. Wilks, Mathematical Statistics, John Wiley & Sons, New York, NY, USA, 1962.
3 R. R. Bahadur, “A note on quantiles in large samples,” Annals of Mathematical Statistics, vol. 37, pp.
577–580, 1966.
4 S. F. Xu and Y. Miao, “Limit behaviors of the deviation between the sample quantiles and the
quantile,” Filomat, vol. 25, no. 2, pp. 197–206, 2011.
5 S. F. Xu, L. Ge, and Y. Miao, “On the Bahadur representation of sample quantiles and order statistics
for NAsequences,” Journal of the Korean Statistical Society. In press.
6 G. A. Brosamler, “An almost everywhere central limit theorem,” Mathematical Proceedings of the
Cambridge Philosophical Society, vol. 104, no. 3, pp. 561–574, 1988.
7 P. Schatte, “On strong versions of the central limit theorem,” Mathematische Nachrichten, vol. 137, pp.
249–256, 1988.
8 M. Peligrad and Q. M. Shao, “A note on the almost sure central limit theorem for weakly dependent
random variables,” Statistics & Probability Letters, vol. 22, no. 2, pp. 131–136, 1995.
9 P. Billingsley, Convergence of Probability Measures, John Wiley & Sons, New York, NY, USA, 1968.
10 H. A. David and H. N. Nagaraja, Order Statistics, Wiley Series in Probability and Statistics, WileyInterscience, John Wiley & Sons, Hoboken, NJ, USA, 3rd edition, 2003.
11 Y. Miao, Y.-X. Chen, and S.-F. Xu, “Asymptotic properties of the deviation between order statistics
and p-quantile,” Communications in Statistics, vol. 40, no. 1, pp. 8–14, 2011.
12 L. Peng and Y. C. Qi, “Almost sure convergence of the distributional limit theorem for order statistics,”
Probability and Mathematical Statistics, vol. 23, no. 2, pp. 217–228, 2003.
13 S. Hörmann, “A note on the almost sure convergence of central order statistics,” Probability and
Mathematical Statistics, vol. 25, no. 2, pp. 317–329, 2005.
14 B. Tong, Z. X. Peng, and S. Nadarajah, “An extension of almost sure central limit theorem for order
statistics,” Extremes, vol. 12, no. 3, pp. 201–209, 2009.
Advances in
Operations Research
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Mathematical Problems
in Engineering
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Algebra
Hindawi Publishing Corporation
http://www.hindawi.com
Probability and Statistics
Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of
Advances in
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com
Mathematical Physics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
Hindawi Publishing Corporation
http://www.hindawi.com
Stochastic Analysis
Abstract and
Applied Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014