Limit properties for ratios of order statistics from exponentials

Journal of Inequalities and Applications, Jan 2017

In this paper, we study the limit properties of the ratio for order statistics based on samples from an exponential distribution and obtain the expression of the density functions, the existence of the moments, the strong law of large numbers for R n i j with 1 ≤ i < j < m n = m . We also discuss other limit theorems such as the central limit theorem, the law of iterated logarithm, the moderate deviation principle, the almost sure central limit theorem for self-normalized sums of R n i j with 2 ≤ i < j < m n = m .

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Limit properties for ratios of order statistics from exponentials

Zhang and Ding Journal of Inequalities and Applications Limit properties for ratios of order statistics from exponentials Yong Zhang In this paper, we study the limit properties of the ratio for order statistics based on samples from an exponential distribution and obtain the expression of the density functions, the existence of the moments, the strong law of large numbers for Rnij with 1 ≤ i < j < mn = m. We also discuss other limit theorems such as the central limit theorem, the law of iterated logarithm, the moderate deviation principle, the almost sure central limit theorem for self-normalized sums of Rnij with 2 ≤ i < j < mn = m. exponential distribution; order statistics; strong law of large numbers; central limit theorem; law of iterated logarithm - As we know, the exponential distribution can describe the lifetimes of the equipment, and the ratios Rnij can measure the stability of equipment, it shows whether or not our system Rnj = l (j – )!(m – j)!(α + ) l= Cj– (m – l – ) © The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Later on, Miao et al. [] proved the central limit theorem and the almost sure central limit for Rn with fixed sample size, we state their results as the following theorem. (Rn – ERn) →D N (, ) as N → ∞, (Rn – ERn) ≤ x = (x) a.s. In this paper, we will make a further study on the limit properties of Rnij. In the next section, firstly, we give the expression of the density functions of Rnij for all  ≤ i < j < mn, it is more interesting that the density function is free of the sample mean λn, this allows us to change the equipment from sample to sample as long as the underlying distribution remains an exponential. Also we discuss the existence of the moments for fixed sample size mn = m. Secondly, we establish the strong law of large number for Rnij with  = i < j < m and  ≤ i < j < m, respectively. At last we give some limit theorems such as the central limit theorem, the law of iterated logarithm, the moderate deviation principle, the almost sure central limit theorem for self-normalized sums of Rnij with  ≤ i < j < m. In the following, C denotes a positive constant, which may take different values whenever it appears in different expressions. an ∼ bn means that an/bn →  as n → ∞. 2 Main results and proofs 2.1 Density functions and moments of Rnij The first theorem gives the expression of the density functions. Proof It is easy to check that the joint density function of Xn(i) and Xn(j) is Therefore the density function of Rnij is fnij(r) = f (w, r) dw mn! (i – )!(j – i – )!(mn – j)! λn k= l= ∞ we–(i–k+l)w/λn e–(mn–i–l)rw/λn dw ∞ te–[(i–k+l)+r(mn–i–l)]tdt (–)j–k–l–Cik–Cjl–i– (–)j–k–l–Cik–Cjl–i– and with  ≤ i < j ≤ m, fnj(r) = m!  (j – )!(m – j)! l= (–)j–l–Cjl– [ + l + r(m – l – )] as r → ∞, Remark . Miao et al. [] obtained the density function for Rnj for fixed sample size mn = m, they also proved that the expectation of Rnj is finite and the truncated second moment is slowly varying at ∞. Adler [] also claimed that all the Rnj have infinite expectations for fixed sample size, so our theorems extended their results. 2.2 Strong law of large numbers of Rnij From our assumptions, we know that {Rnij, n ≥ } is an independent sequence with the same distribution for fixed sample size mn = m. As Theorem . states that the Rnj do not have the expectation, so the strong law of large numbers with them is not typical. Here we give the weighted strong law of large number as follows. At first, we list the following lemma, that is, Theorem . from De la Peña et al. [], which will be used in the proof. Lemma . Let {Xn, n ≥ } be a sequence of independent random variables, denote Sn = in= Xi, if bn ∞, and i∞= Var(Xi)/bi < ∞, then (Sn – ESn)/bn →  a.s. Theorem . Let {an, n ≥ } be a sequence of positive real numbers and {bn, n ≥ } be a sequence of nondecreasing positive real numbers with limn→∞ bn = ∞ and < ∞, ∞ an n= bn bN n= anRnj = an RnjI{ ≤ Rnj ≤ cn} – ERnjI{ ≤ Rnj ≤ cn} anRnj = λm! l (–)j–l– (j – )!(m – j)! l= Cj– (m – l – ) For mn → ∞, bN n= bN n= bN n= = I + I + I. anRnjI{Rnj > cn} anERnjI{ ≤ Rnj ≤ cn} By (.) and (.), it is easy to show ∞ j– n= cn ∞ an n= bn < ∞. Then by the Borel-Cantelli lemma, we get RnjI{Rnj > cn} →  a.s. n → ∞. By (.) and (.), we can obtain Therefore combining (.) with (.), we can easily conclude I →  a.s. n → ∞. ERnjI{ ≤ Rnj ≤ cn} For I, by (.) and noting cn → ∞, we get  dr ≤ C n= cn ∞ an n= bn < ∞, ≤ C ≤ C then by Lemma ., we have I →  a.s. n → ∞. P{Rnj > cn} = then combining with (.), we show I → (j – λ)!m(m! – j)! l= Cjl– (m(––)lj––l–) , n → ∞. So the proof of (.) is completed by combining (.), (.), (.), and (.). By the same argument as in the proof of (.), we can get (.), so we omit it here. (Rnij – ERnij) =  2.3 Other limit properties for Rnij, 2 ≤ i < j ≤ m By the above discussion, we know that, for fixed sample size mn = m and  ≤ i < j ≤ m, {Rnij, n ≥ } is a sequence of independent and identically distributed random variables with finite mean, and L(r) = E(Rnij – ERnij)I{|Rnij – ERnij| ≤ r} is a slowly varying function at ∞. Therefore the limit properties of Rnij for fixed sample size can easily be established by those of the self-normalized sums. We list some of them, such as the central limit theorem (CLT), the law of iterated logarithm (LIL), the moderate deviation principle (MDP), the almost sure central limit theorem (ASCLT). Denote SN = nN=(Rnij – ERnij), VN = nN=(Rnij – ERnij). Proof By Theorem . from Giné et al. [], we can obtain the CLT for Rnij. N→∞ VN √ log log N =  Proof By Theorem  from Griffin and Kuelbs [], the LIL for Rnij holds. Theorem . (MDP) Let {xn, n ≥ } be a sequence of positive numbers with xn → ∞ and √ xn = o( n), as n → ∞, then, for fixed sample size mn = m, we conclude ≥ xN Proof By Theorem . from Shao [], we can prove the MDP for Rnij. n k= dk . Then, for fixed sample size mn = m and any x ∈ R, k→∞ Dk N= ≤ x = (x) (·) is the distribution function of the standard normal random variable. Proof By Corollary  from Zhang [], we know ASCLT for Rnij holds. p Remark . It is easy to check that ηN /VN → , then by the Slutsky lemma and Theorem ., we can get Theorem . from Miao et al. []. Competing interests The authors declares that they have no competing interests. Authors’ contributions Both authors contributed equally and significantly in writing this article. Both authors read and approved the final manuscript. Acknowledgements This work was supported by National Natural Science Foundation of China (Grant Nos. 11101180, 11201175); the Science and Technology Development Program of Jilin Province (Grant Nos. 20130522096JH, 20140520056JH). 1. Adler , A: Strong laws for ratios of order statistics from exponentials . Bull. Inst. Math. Acad. Sin . (N.S.) 10 ( 1 ), 101 - 111 ( 2015 ) 2. Miao , Y, Wang , RJ, Adler , A: Limit theorems for order statistics from exponentials . Stat. Probab. Lett . 110 , 51 - 57 ( 2016 ) 3. De la Peña , VH, Tai, TL, Shao, QM: Self-Normalized Process: Limit Theory and Statistical Applications . Springer, New York ( 2009 ) 4. Adler , A: Exact strong laws . Bull. Inst. Math. Acad. Sin . 28 ( 3 ), 141 - 166 ( 2000 ) 5. Feller , W: An Introduction to Probability Theory and Its Applications , vol. 1 , 3rd edn. Wiley, New York ( 1968 ) 6. Giné , E, Götze, F, Mason, DM: When is the Student t-statistic asymptotically standard normal? Ann . Probab. 25 , 1514 - 1531 ( 1997 ) 7. 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Yong Zhang, Xue Ding. Limit properties for ratios of order statistics from exponentials, Journal of Inequalities and Applications, 2017, 11, DOI: 10.1186/s13660-016-1287-6