Research Article Almost Sure Central Limit Theorem of Sample Quantiles

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1 Advances in Decision Sciences Volume 202, Article ID 67942, 7 pages doi:0.55/202/67942 Research Article Almost Sure Central Limit Theorem of Sample Quantiles Yu Miao, Shoufang Xu, 2 and Ang Peng 3 College of Mathematics and Information Science, Henan Normal University, Henan Province, Xinxiang , China 2 Department of Mathematics and Information Science, Xinxiang University, Henan Province, Xinxiang , China 3 Heze College of Finance and Economics, Shandong Province, Heze , China Correspondence should be addressed to Yu Miao, yumiao728@yahoo.com.cn Received 9 April 202; Accepted 2 August 202 Academic Editor: Saralees Nadaraah Copyright q 202 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 wor is properly cited. We obtain the almost sure central it theorem ASCLT of sample quantiles. Furthermore, based on the method, the ASCLT of order statistics is also proved.. 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.letf n x denote the sample distribution function, that is, F n x n Let us define the pth quantile of F by n {Xi x}, <x<.. i ξ p inf { x : F x p }, p 0,,.2 and the sample quantile ξ np by ξ np inf { x : F n x p }, p 0,..3

2 2 Advances in Decision Sciences It is well nown that ξ np is a natural estimator of ξ p. 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 ris 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,, ifξ p is the unique solution x of F x p F x, then ξ a.e. np ξ p see. In addition, if F x possesses a continuous density function f x in a neighborhood of ξ p and f ξ p > 0, then n /2 f ( ) ( ξnp ) ξ p ξ p [ ( )] /2 N 0,, as n,.4 p p where N 0, denotes the standard normal variable see, 2. Suppose that F x is twice differentiable at ξ p,withf ξ p f ξ p > 0, then Bahadur 3 proved ξ np ξ p p F ( ) n ξp f ( ) R n, a.e.,.5 ξ p where R n O n 3/4 log n 3/4,a.e,asn. Very recently, Xu and Miao 4 obtained the moderate deviation, large deviation and Bahadur asymptotic efficiency of the sample quantiles ξ np.xuetal. 5 studied the Bahadur representation of sample quantiles for negatively associated sequences under some mild conditions. Based on the above wors, in the paper, we are interested in the almost sure central it theorem ASCLT of sample quantiles ξ np. 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 σ 2, n n log n {S σx} Φ x, a.s..6 for any x R, where S denotes the partial sums S X X. 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.. Let X,X 2,...,X n be a sequence of independent identically distributed random variables from a cumulative distribution function F. Letp 0, and suppose that f ξ p : F ξ p exists and is positive. Then one has n n log n { ξ p ξ p σx} Φ x, a.s. 2. for any x R, whereσ 2 p p /f 2 ξ p.

3 Advances in Decision Sciences 3 Proof. Firstly, it is not difficult to chec where { ( ξp ξ p ) } ( σx {F ξ p σx ) } { p : Y i, ω }, 2.2 i Y i, E {Xi ξ p σx/ } {X i ξ p σx/ }, ω ) (E {Xi ξp σx/ } p. 2.3 From the Taylor s formula, it follows ( E {Xi ξ p σx/ } F ξ p σx ) F ( ) ξ p F ( ( ) ) σx ξ p o 2.4 p f ( ( ) σx ξ p o ), which implies ω f ( ξ p ) σx o. 2.5 By the Lindeberg s central it theorem, we can get Hence, 2. is equivalent to f ( ) ξ p σ Y i, i d N 0,, as. 2.6 n n log n { /f ξ p σ i Y Φ x, a.s. i, x o } 2.7 Throughout the following proof, C denotes a positive constant, which may tae different values whenever it appears in different expressions. Put that Z i, : f ( ξ p ) σ Y i,. 2.8 Let g be a bounded Lipschitz function bounded by C, then from 2.6, we have ( ) Eg Z i, Eg N, as, 2.9 i

4 4 Advances in Decision Sciences where N denotes the standard normal random variable. Next, we should notice that 2.7 is equivalent to ( ) n n log n g Z i, Eg N a.s. 2.0 i from Section 2 of Peligrad and Shao 8 and Theorem 7. of Billingsley 9. Hence, to prove 2.7, itsuffices to show that as n, [ ( ) ( )] n R n g Z i, Eg Z i, log n n : log n T 0, i a.s. i 2. It is obvious that ER 2 n n n log 2 n ET n ET T. 2.2 Since g is bounded, we have log 2 n n ET 2 2 C log n. 2.3 Furthermore, for < n, we have ET T ( ( ),g( Cov g Z i, i g( Cov i Z i, ),g )) Z i, i i Z i, C E Z i, C /2, (EZ,) 2 i g i Z i, 2.4 where EZ 2, O ( ). 2.5

5 Advances in Decision Sciences 5 Therefore, we have log 2 n n n ET T C log 2 n n n ( /2 3/2 C n ( log 2 n /2 3/2 2 ) /2 EZ 2, EZ 2, ) /2 C log n. 2.6 From the above discussions, it follows that ER 2 n C log n. 2.7 Tae n e τ, where τ>. Then by Borel-Cantelli lemma, we have R n 0, a.s. as. 2.8 Since g is bounded function, then for n <n n,weobtain [ ) )] n l l R n g( Z log n i,l Eg( Z i,l l l l i l i ( ) ) n l l log n l g Z i,l Eg( Z i,l l l l n C R n log n n l n i i 0, a.s., as n, l 2.9 where we used the fact log n log n τ τ, as 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 {X,...,X n } of observations on F x, the ordered sample values: X X 2 X n 3. are called the order statistics. For more details about order statistics, one can refer to Serfling or David and Nagaraa 0. Suppose that F is twice differentiable at ξ p with

6 6 Advances in Decision Sciences F ξ p f ξ p > 0, then the Bahadur representation for order statistics was first established by Bahadur 3,asn X n ξ p n/n F n ( ξp ) f ( ξ p ) ( O n 3/4( log n ) ) /2 δ a.e., 3.2 where ( n ( ) ) δ n np o log n, n, for some δ 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. 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 2,Hörmann 3,Tongetal. 4,etc., based on the method to deal with the sample quantile, we can also obtain the ASCLT of the order statistics. Theorem 3.. Let X,X 2,...,X n be a sequence of independent identically distributed random variables from a cumulative distribution function F. Letp 0, and suppose that f ξ p : F ξ p exists and is positive. Let n np o n, then one has n n log n { X ξ p σx} Φ x, a.s. 3.4 for any x R, whereσ 2 p p /f 2 ξ p. Proof. Firstly, it is easy to see that the following two events are equivalent: { ) } { } (X ξ p σx {Xi ξ p σx/ } i i : Y i, ω, 3.5 where Y i, E {Xi ξ p σx/ } {X i ξ p σx/ }, ω ( (F ξ p σx ) ) f ( ) ξ p σx o. 3.6 Hence, by the same proof of Theorem 2., we can obtain the desired result.

7 Advances in Decision Sciences 7 References R. J. Serfling, Approximation Theorems of Mathematical Statistics, John Wiley & Sons, New Yor, NY, USA, S. S. Wils, Mathematical Statistics, John Wiley & Sons, New Yor, NY, USA, R. R. Bahadur, A note on quantiles in large samples, Annals of Mathematical Statistics, vol. 37, pp , S. F. Xu and Y. Miao, Limit behaviors of the deviation between the sample quantiles and the quantile, Filomat, vol. 25, no. 2, pp , S. F. Xu, L. Ge, and Y. Miao, On the Bahadur representation of sample quantiles and order statistics for NAsequences, the Korean Statistical Society. In press. 6 G. A. Brosamler, An almost everywhere central it theorem, Mathematical Proceedings of the Cambridge Philosophical Society, vol. 04, no. 3, pp , P. Schatte, On strong versions of the central it theorem, Mathematische Nachrichten, vol. 37, pp , M. Peligrad and Q. M. Shao, A note on the almost sure central it theorem for wealy dependent random variables, Statistics & Probability Letters, vol. 22, no. 2, pp. 3 36, P. Billingsley, Convergence of Probability Measures, John Wiley & Sons, New Yor, NY, USA, H. A. David and H. N. Nagaraa, Order Statistics, Wiley Series in Probability and Statistics, Wiley- Interscience, John Wiley & Sons, Hoboen, NJ, USA, 3rd edition, 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., pp. 8 4, L. Peng and Y. C. Qi, Almost sure convergence of the distributional it theorem for order statistics, Probability and Mathematical Statistics, vol. 23, no. 2, pp , S. Hörmann, A note on the almost sure convergence of central order statistics, Probability and Mathematical Statistics, vol. 25, no. 2, pp , B. Tong, Z. X. Peng, and S. Nadaraah, An extension of almost sure central it theorem for order statistics, Extremes, vol. 2, no. 3, pp , 2009.

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