Advances in Decision Sciences

Advances in Decision Sciences / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 671942 | https://doi.org/10.1155/2012/671942

Yu Miao, Shoufang Xu, Ang Peng, "Almost Sure Central Limit Theorem of Sample Quantiles", Advances in Decision Sciences, vol. 2012, Article ID 671942, 7 pages, 2012. https://doi.org/10.1155/2012/671942

Almost Sure Central Limit Theorem of Sample Quantiles

Academic Editor: Saralees Nadarajah
Received19 Apr 2012
Accepted12 Aug 2012
Published04 Sep 2012

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(π‘₯) with a continuous probability density function 𝑓(π‘₯). Let 𝐹𝑛(π‘₯) denote the sample distribution function, that is, 𝐹𝑛1(π‘₯)=𝑛𝑛𝑖=11{𝑋𝑖≀π‘₯},βˆ’βˆž<π‘₯<∞.(1.1) Let us define the 𝑝th quantile of 𝐹 by πœ‰π‘=inf{π‘₯∢𝐹(π‘₯)β‰₯𝑝},π‘βˆˆ(0,1),(1.2) and the sample quantile Μ‚πœ‰π‘›π‘ by Μ‚πœ‰π‘›π‘ξ€½=infπ‘₯βˆΆπΉπ‘›ξ€Ύ(x)β‰₯𝑝,π‘βˆˆ(0,1).(1.3) It is well known that Μ‚πœ‰π‘›π‘ is a natural estimator of πœ‰π‘. 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 π‘βˆˆ(0,1), if πœ‰π‘ is the unique solution π‘₯ of 𝐹(π‘₯βˆ’)≀𝑝≀𝐹(π‘₯), then Μ‚πœ‰π‘›π‘a.e.β†’πœ‰π‘ (see [1]). In addition, if 𝐹(π‘₯) possesses a continuous density function 𝑓(π‘₯) in a neighborhood of πœ‰π‘ and 𝑓(πœ‰π‘)>0, then 𝑛1/2π‘“ξ€·πœ‰π‘Μ‚πœ‰ξ€Έξ€·π‘›π‘βˆ’πœ‰π‘ξ€Έ[]𝑝(1βˆ’π‘)1/2βŸΆπ‘(0,1),asπ‘›βŸΆβˆž,(1.4) where 𝑁(0,1) denotes the standard normal variable (see [1, 2]). Suppose that 𝐹(π‘₯) is twice differentiable at πœ‰π‘, with πΉξ…ž(πœ‰π‘)=𝑓(πœ‰π‘)>0, then Bahadur [3] proved Μ‚πœ‰π‘›π‘=πœ‰π‘+π‘βˆ’πΉπ‘›ξ€·πœ‰π‘ξ€Έπ‘“ξ€·πœ‰π‘ξ€Έ+𝑅𝑛,a.e.,(1.5) where 𝑅𝑛=𝑂(π‘›βˆ’3/4(log𝑛)3/4), a.e, as π‘›β†’βˆž. Very recently, Xu and Miao [4] obtained the moderate deviation, large deviation and Bahadur asymptotic efficiency of the sample quantiles Μ‚πœ‰π‘›π‘. 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 Μ‚πœ‰π‘›π‘. 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, limπ‘›β†’βˆž1logπ‘›π‘›ξ“π‘˜=11π‘˜1{π‘†π‘˜β‰€βˆšπ‘˜πœŽπ‘₯}=Ξ¦(π‘₯),a.s.(1.6) for any π‘₯βˆˆβ„, where π‘†π‘˜ denotes the partial sums π‘†π‘˜=𝑋1+β‹―+π‘‹π‘˜. 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 π‘βˆˆ(0,1) and suppose that 𝑓(πœ‰π‘)∢=πΉξ…ž(πœ‰π‘) exists and is positive. Then one has limπ‘›β†’βˆž1logπ‘›π‘›ξ“π‘˜=11π‘˜1{βˆšπ‘˜(Μ‚πœ‰π‘˜π‘βˆ’πœ‰π‘)β‰€πœŽπ‘₯}=Ξ¦(π‘₯),a.s.(2.1) for any π‘₯βˆˆβ„, where 𝜎2=𝑝(1βˆ’π‘)/𝑓2(πœ‰π‘).

Proof. Firstly, it is not difficult to check ξ‚†βˆšπ‘˜ξ€·Μ‚πœ‰π‘˜π‘βˆ’πœ‰π‘ξ€Έξ‚‡=ξƒ―πΉβ‰€πœŽπ‘₯π‘˜ξƒ©πœ‰π‘+𝜎π‘₯βˆšπ‘˜ξƒͺξƒ°ξƒ―1β‰₯𝑝=βˆΆβˆšπ‘˜π‘˜ξ“π‘–=1π‘Œπ‘–,π‘˜β‰€πœ”π‘˜ξƒ°,(2.2) where π‘Œπ‘–,π‘˜=𝔼1{π‘‹π‘–β‰€πœ‰π‘βˆš+𝜎π‘₯/π‘˜}βˆ’1{π‘‹π‘–β‰€πœ‰π‘βˆš+𝜎π‘₯/π‘˜},πœ”π‘˜=βˆšπ‘˜ξ‚€π”Ό1{π‘‹π‘–β‰€πœ‰π‘βˆš+𝜎π‘₯/π‘˜}.βˆ’π‘(2.3) From the Taylor's formula, it follows 𝔼1{π‘‹π‘–β‰€πœ‰π‘βˆš+𝜎π‘₯/π‘˜}ξƒ©πœ‰=𝐹𝑝+𝜎π‘₯βˆšπ‘˜ξƒͺξ€·πœ‰=𝐹𝑝+πΉξ…žξ€·πœ‰π‘ξ€ΈπœŽπ‘₯βˆšπ‘˜ξƒ©1+π‘œβˆšπ‘˜ξƒͺξ€·πœ‰=𝑝+π‘“π‘ξ€ΈπœŽπ‘₯βˆšπ‘˜ξƒ©1+π‘œβˆšπ‘˜ξƒͺ,(2.4) which implies πœ”π‘˜ξ€·πœ‰=π‘“π‘ξ€ΈπœŽπ‘₯+π‘œ(1).(2.5) By the Lindeberg's central limit theorem, we can get 1π‘“ξ€·πœ‰π‘ξ€ΈπœŽβˆšπ‘˜π‘˜ξ“π‘–=1π‘Œπ‘‘π‘–,π‘˜βŸΆπ‘(0,1),asπ‘˜βŸΆβˆž.(2.6) Hence, (2.1) is equivalent to limπ‘›β†’βˆž1logπ‘›π‘›ξ“π‘˜=11π‘˜1{(1/𝑓(πœ‰π‘βˆš)πœŽβˆ‘π‘˜)π‘˜π‘–=1π‘Œπ‘–,π‘˜β‰€π‘₯+π‘œ(1)}=Ξ¦(π‘₯),a.s.(2.7)
Throughout the following proof, 𝐢 denotes a positive constant, which may take different values whenever it appears in different expressions.
Put that 𝑍𝑖,π‘˜1∢=π‘“ξ€·πœ‰π‘ξ€ΈπœŽπ‘Œπ‘–,π‘˜.(2.8) Let 𝑔 be a bounded Lipschitz function bounded by 𝐢, then from (2.6), we have 1π”Όπ‘”βˆšπ‘˜π‘˜ξ“π‘–=1𝑍𝑖,π‘˜ξƒͺβŸΆπ”Όπ‘”(𝑁),asπ‘˜βŸΆβˆž,(2.9) where 𝑁 denotes the standard normal random variable. Next, we should notice that (2.7) is equivalent to limπ‘›β†’βˆž1logπ‘›π‘›ξ“π‘˜=11π‘˜π‘”ξƒ©1βˆšπ‘˜π‘˜ξ“π‘–=1𝑍𝑖,π‘˜ξƒͺ=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 π‘›β†’βˆž, 𝑅𝑛=1logπ‘›π‘›ξ“π‘˜=11π‘˜ξƒ¬π‘”ξƒ©1βˆšπ‘˜π‘˜ξ“π‘–=1𝑍𝑖,π‘˜ξƒͺ1βˆ’π”Όπ‘”βˆšπ‘˜π‘˜ξ“π‘–=1𝑍𝑖,π‘˜1ξƒͺξƒ­=∢logπ‘›π‘›ξ“π‘˜=11π‘˜π‘‡π‘˜βŸΆ0,a.s.(2.11) It is obvious that 𝔼𝑅2𝑛=1log2π‘›ξƒ¬π‘›ξ“π‘˜=11π‘˜2𝔼𝑇2π‘˜+2π‘›βˆ’1ξ“π‘›π‘˜=1𝑗=π‘˜+11π‘˜π‘—π”Όπ‘‡π‘˜π‘‡π‘—ξƒ­.(2.12) Since 𝑔 is bounded, we have 1log2π‘›π‘›ξ“π‘˜=11π‘˜2𝔼𝑇2π‘˜β‰€πΆ.log𝑛(2.13) Furthermore, for 1β‰€π‘˜<𝑗≀𝑛, we have ||π”Όπ‘‡π‘˜π‘‡π‘—||=|||||𝑔1Covβˆšπ‘˜k𝑖=1𝑍𝑖,π‘˜ξƒͺ1,π‘”βˆšπ‘—π‘—ξ“π‘–=1𝑍𝑖,𝑗|||||=|||||ξƒ©π‘”ξƒ©βˆ‘ξƒͺξƒͺCovπ‘˜π‘–=1𝑍𝑖,π‘˜βˆšπ‘˜ξƒͺξƒ©βˆ‘,𝑔𝑗𝑖=1𝑍𝑖,π‘—βˆšπ‘—ξƒͺξƒ©βˆ‘βˆ’π‘”π‘—π‘–=π‘˜+1𝑍𝑖,π‘—βˆšπ‘—|||||≀𝐢ξƒͺξƒͺβˆšπ‘—π”Ό|||||π‘˜ξ“π‘–=1𝑍𝑖,𝑗|||||β‰€πΆβˆšπ‘˜βˆšπ‘—ξ‚€π”Όπ‘21,𝑗1/2,(2.14) where 𝔼𝑍21,𝑗1=1+π‘‚βˆšπ‘—ξƒͺ.(2.15) Therefore, we have 1log2π‘›π‘›βˆ’1ξ“π‘›π‘˜=1𝑗=π‘˜+11||π‘˜π‘—π”Όπ‘‡π‘˜π‘‡π‘—||≀𝐢log2π‘›π‘›βˆ’1ξ“π‘›π‘˜=1𝑗=π‘˜+11π‘˜1/2𝑗3/2𝔼𝑍21,𝑗1/2=𝐢log2𝑛𝑛𝑗=2π‘—βˆ’1ξ“π‘˜=11π‘˜1/2𝑗3/2𝔼𝑍21,𝑗1/2≀𝐢.log𝑛(2.16) From the above discussions, it follows that 𝔼𝑅2𝑛≀𝐢.log𝑛(2.17) Take π‘›π‘˜=π‘’π‘˜πœ, where 𝜏>1. Then by Borel-Cantelli lemma, we have π‘…π‘›π‘˜βŸΆ0,a.s.asπ‘˜βŸΆβˆž.(2.18) Since 𝑔 is bounded function, then for π‘›π‘˜<π‘›β‰€π‘›π‘˜+1, we obtain ||𝑅𝑛||≀1logπ‘›π‘˜|||||π‘›π‘˜ξ“π‘™=11𝑙𝑔1βˆšπ‘™π‘™ξ“π‘–=1𝑍𝑖,𝑙ξƒͺ1βˆ’π”Όπ‘”βˆšπ‘™π‘™ξ“π‘–=1𝑍𝑖,𝑙|||||+1ξƒͺξƒ­logπ‘›π‘˜π‘›π‘˜+1𝑙=π‘›π‘˜+11𝑙|||||𝑔1βˆšπ‘™π‘™ξ“π‘–=1𝑍𝑖,𝑙ξƒͺ1βˆ’π”Όπ‘”βˆšπ‘™π‘™ξ“π‘–=1𝑍𝑖,𝑙ξƒͺ|||||≀||π‘…π‘›π‘˜||+𝐢logπ‘›π‘˜π‘›π‘˜+1𝑙=π‘›π‘˜+11π‘™βŸΆ0,a.s.,asπ‘›βŸΆβˆž,(2.19) where we used the fact logπ‘›π‘˜+1logπ‘›π‘˜=(π‘˜+1)πœπ‘˜πœβŸΆ1,asπ‘˜βŸΆβˆž.(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(π‘₯), 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 πœ‰π‘ with πΉξ…ž(πœ‰π‘)=𝑓(πœ‰π‘)>0, then the Bahadur representation for order statistics was first established by Bahadur [3], as π‘›β†’βˆžπ‘‹(π‘˜π‘›)=πœ‰π‘+ξ€·π‘˜π‘›ξ€Έ/π‘›βˆ’πΉπ‘›ξ€·πœ‰π‘ξ€Έπ‘“ξ€·πœ‰π‘ξ€Έξ€·π‘›+π‘‚βˆ’3/4(log𝑛)(1/2)(𝛿+1)ξ€Έa.e.,(3.2) where π‘˜π‘›ξ‚€βˆš=𝑛𝑝+π‘œπ‘›(log𝑛)𝛿1,π‘›βŸΆβˆž,forsome𝛿β‰₯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], HΓΆrmann [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 π‘βˆˆ(0,1) and suppose that 𝑓(πœ‰π‘)∢=πΉξ…ž(πœ‰π‘) exists and is positive. Let π‘˜π‘›βˆš=𝑛𝑝+π‘œ(𝑛), then one has limπ‘›β†’βˆž1log𝑛𝑛𝑗=11𝑗1{βˆšπ‘—(𝑋𝑗)(π‘˜βˆ’πœ‰π‘)β‰€πœŽπ‘₯}=Ξ¦(π‘₯),a.s.(3.4) for any π‘₯βˆˆβ„, where 𝜎2=𝑝(1βˆ’π‘)/𝑓2(πœ‰π‘).

Proof. Firstly, it is easy to see that the following two events are equivalent: ξ‚†βˆšπ‘—ξ‚€π‘‹(π‘˜π‘—)βˆ’πœ‰π‘ξ‚ξ‚‡=ξƒ―β‰€πœŽπ‘₯𝑗𝑖=11{π‘‹π‘–β‰€πœ‰π‘βˆš+𝜎π‘₯/𝑗}β‰₯π‘˜π‘—ξƒ°βŽ§βŽͺ⎨βŽͺβŽ©βˆ‘=βˆΆπ‘—βˆ’π‘–=1π‘Œπ‘–,π‘—βˆšπ‘—β‰€βˆ’πœ”π‘—βŽ«βŽͺ⎬βŽͺ⎭,(3.5) where βˆ’π‘Œπ‘–,𝑗=𝔼1{π‘‹π‘–β‰€πœ‰π‘βˆš+𝜎π‘₯/𝑗}βˆ’1{π‘‹π‘–β‰€πœ‰π‘βˆš+𝜎π‘₯/βˆ’π‘—},πœ”π‘—=1βˆšπ‘—ξƒ©ξƒ©πœ‰π‘—πΉπ‘+𝜎π‘₯βˆšπ‘—ξƒͺβˆ’π‘˜π‘—ξƒͺξ€·πœ‰=π‘“π‘ξ€ΈπœŽπ‘₯+π‘œ(1).(3.6) Hence, by the same proof of Theorem 2.1, we can obtain the desired result.

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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.


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