Table of Contents Author Guidelines Submit a Manuscript
Advances in Decision Sciences
Volume 2012, Article ID 671942, 7 pages
http://dx.doi.org/10.1155/2012/671942
Research Article

Almost Sure Central Limit Theorem of Sample Quantiles

1College of Mathematics and Information Science, Henan Normal University, Henan Province, Xinxiang 453007, China
2Department of Mathematics and Information Science, Xinxiang University, Henan Province, Xinxiang 453000, China
3Heze College of Finance and Economics, Shandong Province, Heze 274000, China

Received 19 April 2012; Accepted 12 August 2012

Academic Editor: Saralees Nadarajah

Copyright © 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.

Abstract

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 𝐹(𝑥) with a continuous probability density function 𝑓(𝑥). Let 𝐹𝑛(𝑥) denote the sample distribution function, that is, 𝐹𝑛1(𝑥)=𝑛𝑛𝑖=11{𝑋𝑖𝑥},<𝑥<.(1.1) Let us define the 𝑝th quantile of 𝐹 by 𝜉𝑝=inf{𝑥𝐹(𝑥)𝑝},𝑝(0,1),(1.2) and the sample quantile ̂𝜉𝑛𝑝 by ̂𝜉𝑛𝑝=inf𝑥𝐹𝑛(x)𝑝,𝑝(0,1).(1.3) 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 quantile-hedging, 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 𝐹(𝑥)𝑝𝐹(𝑥), then ̂𝜉𝑛𝑝a.e.𝜉𝑝 (see [1]). In addition, if 𝐹(𝑥) possesses a continuous density function 𝑓(𝑥) in a neighborhood of 𝜉𝑝 and 𝑓(𝜉𝑝)>0, then 𝑛1/2𝑓𝜉𝑝̂𝜉𝑛𝑝𝜉𝑝[]𝑝(1𝑝)1/2𝑁(0,1),as𝑛,(1.4) where 𝑁(0,1) denotes the standard normal variable (see [1, 2]). Suppose that 𝐹(𝑥) is twice differentiable at 𝜉𝑝, with 𝐹(𝜉𝑝)=𝑓(𝜉𝑝)>0, then Bahadur [3] proved ̂𝜉𝑛𝑝=𝜉𝑝+𝑝𝐹𝑛𝜉𝑝𝑓𝜉𝑝+𝑅𝑛,a.e.,(1.5) where 𝑅𝑛=𝑂(𝑛3/4(log𝑛)3/4), a.e, as 𝑛. Very recently, Xu and Miao [4] obtained the 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,  Var(𝑋)=𝜎2, lim𝑛1log𝑛𝑛𝑘=11𝑘1{𝑆𝑘𝑘𝜎𝑥}=Φ(𝑥),a.s.(1.6) for any 𝑥, 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 𝑓(𝜉𝑝)=𝐹(𝜉𝑝) exists and is positive. Then one has lim𝑛1log𝑛𝑛𝑘=11𝑘1{𝑘(̂𝜉𝑘𝑝𝜉𝑝)𝜎𝑥}=Φ(𝑥),a.s.(2.1) for any 𝑥, where 𝜎2=𝑝(1𝑝)/𝑓2(𝜉𝑝).

Proof. Firstly, it is not difficult to check 𝑘̂𝜉𝑘𝑝𝜉𝑝=𝐹𝜎𝑥𝑘𝜉𝑝+𝜎𝑥𝑘1𝑝=𝑘𝑘𝑖=1𝑌𝑖,𝑘𝜔𝑘,(2.2) where 𝑌𝑖,𝑘=𝔼1{𝑋𝑖𝜉𝑝+𝜎𝑥/𝑘}1{𝑋𝑖𝜉𝑝+𝜎𝑥/𝑘},𝜔𝑘=𝑘𝔼1{𝑋𝑖𝜉𝑝+𝜎𝑥/𝑘}.𝑝(2.3) From the Taylor's formula, it follows 𝔼1{𝑋𝑖𝜉𝑝+𝜎𝑥/𝑘}𝜉=𝐹𝑝+𝜎𝑥𝑘𝜉=𝐹𝑝+𝐹𝜉𝑝𝜎𝑥𝑘1+𝑜𝑘𝜉=𝑝+𝑓𝑝𝜎𝑥𝑘1+𝑜𝑘,(2.4) which implies 𝜔𝑘𝜉=𝑓𝑝𝜎𝑥+𝑜(1).(2.5) By the Lindeberg's central limit theorem, we can get 1𝑓𝜉𝑝𝜎𝑘𝑘𝑖=1𝑌𝑑𝑖,𝑘𝑁(0,1),as𝑘.(2.6) Hence, (2.1) is equivalent to lim𝑛1log𝑛𝑛𝑘=11𝑘1{(1/𝑓(𝜉𝑝)𝜎𝑘)𝑘𝑖=1𝑌𝑖,𝑘𝑥+𝑜(1)}=Φ(𝑥),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,𝑘1=𝑓𝜉𝑝𝜎𝑌𝑖,𝑘.(2.8) Let 𝑔 be a bounded Lipschitz function bounded by 𝐶, then from (2.6), we have 1𝔼𝑔𝑘𝑘𝑖=1𝑍𝑖,𝑘𝔼𝑔(𝑁),as𝑘,(2.9) where 𝑁 denotes the standard normal random variable. Next, we should notice that (2.7) is equivalent to lim𝑛1log𝑛𝑛𝑘=11𝑘𝑔1𝑘𝑘𝑖=1𝑍𝑖,𝑘=𝔼𝑔(𝑁)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, 𝑅𝑛=1log𝑛𝑛𝑘=11𝑘𝑔1𝑘𝑘𝑖=1𝑍𝑖,𝑘1𝔼𝑔𝑘𝑘𝑖=1𝑍𝑖,𝑘1=log𝑛𝑛𝑘=11𝑘𝑇𝑘0,a.s.(2.11) It is obvious that 𝔼𝑅2𝑛=1log2𝑛𝑛𝑘=11𝑘2𝔼𝑇2𝑘+2𝑛1𝑛𝑘=1𝑗=𝑘+11𝑘𝑗𝔼𝑇𝑘𝑇𝑗.(2.12) Since 𝑔 is bounded, we have 1log2𝑛𝑛𝑘=11𝑘2𝔼𝑇2𝑘𝐶.log𝑛(2.13) Furthermore, for 1𝑘<𝑗𝑛, we have ||𝔼𝑇𝑘𝑇𝑗||=|||||𝑔1Cov𝑘k𝑖=1𝑍𝑖,𝑘1,𝑔𝑗𝑗𝑖=1𝑍𝑖,𝑗|||||=|||||𝑔Cov𝑘𝑖=1𝑍𝑖,𝑘𝑘,𝑔𝑗𝑖=1𝑍𝑖,𝑗𝑗𝑔𝑗𝑖=𝑘+1𝑍𝑖,𝑗𝑗|||||𝐶𝑗𝔼|||||𝑘𝑖=1𝑍𝑖,𝑗|||||𝐶𝑘𝑗𝔼𝑍21,𝑗1/2,(2.14) where 𝔼𝑍21,𝑗1=1+𝑂𝑗.(2.15) Therefore, we have 1log2𝑛𝑛1𝑛𝑘=1𝑗=𝑘+11||𝑘𝑗𝔼𝑇𝑘𝑇𝑗||𝐶log2𝑛𝑛1𝑛𝑘=1𝑗=𝑘+11𝑘1/2𝑗3/2𝔼𝑍21,𝑗1/2=𝐶log2𝑛𝑛𝑗=2𝑗1𝑘=11𝑘1/2𝑗3/2𝔼𝑍21,𝑗1/2𝐶.log𝑛(2.16) From the above discussions, it follows that 𝔼𝑅2𝑛𝐶.log𝑛(2.17) Take 𝑛𝑘=𝑒𝑘𝜏, where 𝜏>1. Then by Borel-Cantelli lemma, we have 𝑅𝑛𝑘0,a.s.as𝑘.(2.18) Since 𝑔 is bounded function, then for 𝑛𝑘<𝑛𝑛𝑘+1, we obtain ||𝑅𝑛||1log𝑛𝑘|||||𝑛𝑘𝑙=11𝑙𝑔1𝑙𝑙𝑖=1𝑍𝑖,𝑙1𝔼𝑔𝑙𝑙𝑖=1𝑍𝑖,𝑙|||||+1log𝑛𝑘𝑛𝑘+1𝑙=𝑛𝑘+11𝑙|||||𝑔1𝑙𝑙𝑖=1𝑍𝑖,𝑙1𝔼𝑔𝑙𝑙𝑖=1𝑍𝑖,𝑙|||||||𝑅𝑛𝑘||+𝐶log𝑛𝑘𝑛𝑘+1𝑙=𝑛𝑘+11𝑙0,a.s.,as𝑛,(2.19) where we used the fact log𝑛𝑘+1log𝑛𝑘=(𝑘+1)𝜏𝑘𝜏1,as𝑘.(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 𝐹(𝑥), the ordered sample values: 𝑋(1)𝑋(2)𝑋(𝑛)(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 𝐹(𝜉𝑝)=𝑓(𝜉𝑝)>0, then the Bahadur representation for order statistics was first established by Bahadur [3], as 𝑛𝑋(𝑘𝑛)=𝜉𝑝+𝑘𝑛/𝑛𝐹𝑛𝜉𝑝𝑓𝜉𝑝𝑛+𝑂3/4(log𝑛)(1/2)(𝛿+1)a.e.,(3.2) where 𝑘𝑛=𝑛𝑝+𝑜𝑛(log𝑛)𝛿1,𝑛,forsome𝛿2.(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 𝑓(𝜉𝑝)=𝐹(𝜉𝑝) exists and is positive. Let 𝑘𝑛=𝑛𝑝+𝑜(𝑛), then one has lim𝑛1log𝑛𝑛𝑗=11𝑗1{𝑗(𝑋𝑗)(𝑘𝜉𝑝)𝜎𝑥}=Φ(𝑥),a.s.(3.4) for any 𝑥, where 𝜎2=𝑝(1𝑝)/𝑓2(𝜉𝑝).

Proof. Firstly, it is easy to see that the following two events are equivalent: 𝑗𝑋(𝑘𝑗)𝜉𝑝=𝜎𝑥𝑗𝑖=11{𝑋𝑖𝜉𝑝+𝜎𝑥/𝑗}𝑘𝑗=𝑗𝑖=1𝑌𝑖,𝑗𝑗𝜔𝑗,(3.5) where 𝑌𝑖,𝑗=𝔼1{𝑋𝑖𝜉𝑝+𝜎𝑥/𝑗}1{𝑋𝑖𝜉𝑝+𝜎𝑥/𝑗},𝜔𝑗=1𝑗𝜉𝑗𝐹𝑝+𝜎𝑥𝑗𝑘𝑗𝜉=𝑓𝑝𝜎𝑥+𝑜(1).(3.6) Hence, by the same proof of Theorem 2.1, we can obtain the desired result.

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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  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. View at Publisher · View at Google Scholar
  5. S. F. Xu, L. Ge, and Y. Miao, “On the Bahadur representation of sample quantiles and order statistics for NA sequences,” 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. P. Schatte, “On strong versions of the central limit theorem,” Mathematische Nachrichten, vol. 137, pp. 249–256, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  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, Wiley-Interscience, John Wiley & Sons, Hoboken, NJ, USA, 3rd edition, 2003. View at Publisher · View at Google Scholar
  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. View at Publisher · View at Google Scholar
  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. View at Google Scholar · View at Zentralblatt MATH
  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. View at Google Scholar · View at Zentralblatt MATH
  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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH