Limit Distributions of Extreme Order Statistics under Power Normalization and Random Index

Transcription

1 Limit Distributions of Extreme Order tatistics under Power Normalization and Random Index Zuoxiang Peng, Qin Jiang & aralees Nadarajah First version: 3 December 2 Research Report No. 2, 2, Probability and tatistics Group chool of Mathematics, The University of Manchester

2 Limit distributions of extreme order statistics under power normalization and random index a Zuoxiang Peng a Qin Jiang b aralees Nadarajah a chool of Mathematics and tatistics, outhwest University, Chongqing, China b chool of Mathematics, University of Manchester, Manchester, United Kingdom Abstract. Let (X n be a sequence of independent and identically distributed random variables and let M n denote the sth largest order statistic of X, n. In this note, the limiting distributions of M under power normalization are derived as the integer valued random index follows the shifted negative binomial, shifted Poisson and shifted binomial distributions. Keywords. Limiting distribution; Order statistic; Power normalization; Random index; hifted distribution. AM 2 subject classifications. Primary 62G32; econdary 65C5. Introduction Let (X n be a sequence of independent and identically distributed (iid random variables with common cumulative distribution function (cdf F. Define the partial maxima M n = max(x, X 2,..., X n and let M n be the sth largest order statistic of (X, n. The limiting distributions of order statistics under linear normalization have attracted considerable attention. For details, see Gnedeno (943, Resnic (987 and de Haan and Ferreira (26. The wea convergence of the exceedances point process formed by (X n and studied by Leadbetter et al. (983 implies the asymptotic distribution of M n for fixed s. A cdf F (x is said to belong to the domain of attraction of extreme value cdfs, written as F D l (G, if there exist some normalizing constants >, b n R such that P M n x + b n w G(x as n, where w denotes wea convergence. The cdf G(x is called a max stable cdf under linear normalization or simply a l-max stable cdf. It is well nown that G(x belongs to one of the following three classes of extreme value cdfs: Type I Gumbel: Λ (x = exp exp( x, x R;, x <, Type II Fréchet: Φ α (x = exp x α, x exp ( x α, x <, Type III Weibull: Ψ α (x = x for some α > ; for some α >. For more details, see Leadbetter et al. (983, Resnic (987 and de Haan and Ferreira (26. Here, cdfs G and G 2 are said to belong to the same class if there exist constants a > and b R

3 such that G (x = G 2 (ax + b for all x R. Pancheva (985, 986, 994 and Mohan and Ravi (992 extended the above results to M n under power normalization, i.e. a cdf F belongs to the p-max domain of attraction of H(x under power normalization if there exist normalizing constants > and b n > such that M n P ( bn (Mn x = F n w x (x bn H(x (. as n, where (x is the sign function given by (x = if x <, (x = if x =, and (x = if x >. In this case, we write F D p (H and call H(x a p-max stable cdf under power normalization or simply a p-max stable cdf. Pancheva (985, 986, 994 proved that H(x belongs to one of the following six classes of extreme value cdfs:, Type I : H β (x = exp (log x β,, Type II : H 2,β (x = exp ( log x β,, Type III : H 3,β (x = exp exp Type IV : H 4,β (x =, Type V : H 5,β (x = exp ( x, exp(x, Type VI : H 6,β (x = ( log ( x β, (log ( x β, x x > x, < x x > x < x, x > x x > x, x > ; x, x >. for some β > ; for some β > ; for some β > ; for some β > ; Here, cdfs H and H 2 are said to belong to the same class if there exist constants A > and B > such that H (x = H 2 (A x B (x for all x R. The necessary and sufficient conditions for a cdf F to belong to D p (H for each of the six p-max stable laws were studied by Mohan and Ravi (992 and ubramanya (994. Baraat and Nigm (22 studied the limiting distribution of extreme order statistics under power normalization and random index. ome results of Gnedeno (943 and de Haan (97 concerning linear normalization were extended to p-stable laws. The following result is due to Baraat and Nigm (22. Theorem A. Let (X n be a sequence of iid random variables. For suitable normalizing constants > and b n >, and for some non-degenerate cdf H(x, we have M bn ( n s P M n x = P M n ( log H(x j x bn (x H(x (.2 (j! as n for all continuity points x of H if and only if ( ( n F x bn (x log H(x as n. j= 2

4 The aim of this note is to study the limiting distribution of extremes under power normalization and some time shifted integer random indices (which will be defined later. Let M Nn = max(x, X 2,..., X Nn denote the partial maxima with integer random index. The random variable M Nn arises in many applied areas: chneider and chultz (982 suggest the shifted negative binomial distribution to model lengths of dry periods measured in days. o, if X i represents the economic cost for day i then M Nn will represent the daily maximum cost incurred during the period. In queuing theory, the batch size is often modeled by the shifted negative binomial distribution (see, for example, avariappan et al. (29. o, if X i represents the time taen to process the ith item in a batch then M Nn will represent the maximum processing time. Also in queuing theory, the service time is often modeled by the shifted binomial distribution (see, for example, Mitrou (24, page 342. o, if X i represents the number of new arrivals in the ith unit time of a service period for a single sever model then M Nn will represent the maximum new arrivals. hifted Poisson distributions are commonly used to model the number of insurance claims made over fixed periods. o, if X i represents the ith claim size then M Nn will represent the largest claim size. Rothschild (986 has shown that lengths of English words follow the shifted Poisson distribution. o, if X i represents the frequency of words of length i on a page, say, then M Nn will represent the highest frequency. Many more examples can be stated. For p-max stable distributions, Baraat and Nigm (22 proved the following result: Theorem B. Let (X n be a sequence of iid random variables. Assume that the integer valued random variable satisfies /n P τ, where τ is a positive random variable and the convergence is in probability. Then for H(x > and fixed s P M bn as n if (.2 holds. ( M s x ( log H(x y H y (xd P (τ y For the limiting distribution of maximum with integer random index, see Galambos (2. For the case of following some time shifted discrete distributions, we will show that the limiting distributions of M Nn are different according to the distribution of. To the best of our nowledge, there has not been any study about M under power normalization and time shifted integer. This note aims to fill the gap. The contents of this note are organized as follows. In ection 2, we give three possible models for : the time shifted negative binomial, the time shifted Poisson and the time shifted binomial distributions. The corresponding main results are given in ection 3. The corresponding proofs are provided in ection 4. 3

5 2 Time shifted integer random variables In this section, we state definitions of the time shifted negative binomial, the time shifted Poisson and the time shifted binomial distributions. Definition 2.. An integer valued random variable follows the time shifted negative binomial distribution if its probability mass function (pmf is ( r P = = p r n ( q n rm (2. rm for rm, where m is on-negative integer, r is a fixed integer, p n >, q n = p n >, and ( x m means x(x (x m + /m! for any positive integer m. Remar 2.. The time shifted negative binomial distribution defined by (2. is also called the time shifted Pascal distribution. For r = follows the time shifted geometric distribution, i.e. P = = p n q m n (2.2 for m. The Pascal and geometric distributions are special cases of (2. and (2.2, respectively. Definition 2.2. An integer valued random variable follows the time shifted Poisson distribution if its pmf is P = = for m and non-negative integer m, where λ n may depend on n. Next, we define the time shifted binomial distribution. λ n m ( m! exp ( λ n (2.3 Definition 2.3. An integer valued random variable follows the time shifted binomial distribution if its pmf is ( ln P = = p m n qn ln +m (2.4 m for = m, m +..., m + l n and m a fixed integer, where p n >, q n = p n > may depend on n. 3 Main results For the time shifted negative binomial distribution defined by (2., we have the following result. Theorem 3.. Let (X n be a sequence of iid random variables such that (. holds. uppose that follows the time shifted negative binomial pmf defined by (2.. Additionally, assume that and X n, n are independent. Then for H(x > and fixed s provided np n as n. P M x bn (x s ( r + ( log H(x ( log H(x +r 4

6 For the case of r = in Theorem 3. we have the following result. Corollary 3.. Let (X n be a sequence of iid random variables such that (. holds. Let follow the time shifted geometric pmf defined by (2.2. Additionally, assume that and X n are independent. Then for H(x > and fixed s provided np n as n. P M s x bn (x ( log H(x ( log H(x + For the time shifted Poisson and time shifted binomial distributions, we have the following results. Theorem 3.2. Let (X n be a sequence of iid random variables such that (. holds. uppose that is a time shifted Poisson random variable with pmf defined by (2.3 with λ n /n as n. Then for H(x > and fixed s as n. P M x bn (x s H(x ( log H(x Theorem 3.3. Let (X n be a sequence of iid random variables such that (. holds. uppose that is a time shifted binomial random variable with pmf defined by (2.4 with l n p n /n as n. Then for H(x > and fixed s as n. P M x bn (x s H(x ( log H(x 4 Proofs of the main results In this section, we prove the results provided in ection 3. Before proving the results, we consider asymptotic properties of /n as the integer valued random index follows the time shifted negative binomial, the time shifted Poisson and the time shifted binomial distributions. For the time shifted negative binomial distribution, we have the following result. Lemma 4.. Let be a time shifted negative binomial random variable with pmf defined by (2.. Assume that np n as n. Then /n converges in distribution to τ, a gamma random variable with shape parameter r and scale parameter i.e. the probability density function (pdf of τ is given by where Γ( denotes the gamma function. P (τ dy = Γ(r exp( yyr, y >, (4. 5

7 Proof. First note np n implies q n as n. Let f n (t denote the characteristic function of /n. o, by definition of the time shifted negative binomial distribution, one can chec ( Nn f n (t = E exp n it ( r = exp(it/n p r n ( q n rm rm =rm = exp (rmit/n p r n [ q n exp(it/n] r [ = exp(rmit/n q n it + o (/ (np n np n ( it r =: f(t by virtue of np n and q n as n. Note that f(t is the characteristic function of a gamma random variable with shape parameter r and scale parameter. The proof is complete. Remar 4.. By Lemma 4. for the time shifted geometric random variable with pmf defined by (2.2, /n converges in distribution to τ, a standard exponential random variable. By arguments similar to the proof of Lemma 4. one obtains the following results. Lemma 4.2. Let be a time shifted Poisson random variable with pmf defined by (2.3. Assume that λ n /n as n. Then /n in probability. Lemma 4.3. Let be a time shifted binomial random variable with pmf defined by (2.4. Assume that l n p n /n as n. Then /n in probability. Proof of Theorem 3.. Let η n = /n. Lemma 4. shows that η n converges in distribution to τ with pdf given by (4.. o, by horohod representation theorem, there exist random variables η n = Ñn/n d = η n and τ d = τ such that η n converges to τ almost surely. There also exists an iid random sequence ( X n such that (X n = d ( X n as (X n is a sequence of iid random variables. We can assume independence of ( X n and Ñn since (X n is independent of. Let M n denote the sth maximum of X for n. Now we have M /b n ( M d= M /b n ] r 6

8 since o, by Theorem B, P = P = P = P = P M M /b n M M M x /b n /b n /b n ( /b n M ( M x. M /b n / x Ñn = P (Ñn = x P (Ñn = x P ( = converges almost surely to a random variable M with cdf for H(x >. o, P M /b n s ( log H(x ( M x s y H y (xd P (τ y ( log H(x y H y (xd P (τ y. Note y H y (xdp (τ y = = = Γ(r Γ(r Γ( + r Γ(r y H y (x exp( yy r dy y +r exp [ y ( log H(x] dy ( log H(x +r. The result of Theorem 3. follows. The proof is complete. Proof of Corollary 3.. The result follows directly from Theorem 3. and Remar 4.. Proof of Theorem 3.2. By applying Theorem B and Lemma 4.2, we obtain the desired result. Proof of Theorem 3.3. By applying Theorem B and Lemma 4.3, we obtain the desired result. 7

9 References [] Baraat, H. M. and Nigm, E. M. (22. Extreme order statistics under power normalization and random sample size. Kuwait Journal of cience and Engineering, 29, [2] de Haan, L. (97. A form of regular variation and its application to the domain of attraction of the double exponential distribution. Zeitschift fur Wahrscheinlicheitstheorie and Verwandte Gebiete, 7, [3] de Haan, L. and Ferreira, A. (26. Extreme Value Theory. pringer, New Yor. [4] Galambos, J. (2. The Asymptotic Theory of Extreme Order tatistics. China cience Press, Beijing. [5] Gnedeno, B. V. (943. ur la distribution limit du treme maximum d une serie aleatorie. Annals of Mathematics, 44, [6] Leadbetter, M. R., Lindgren, G. and Rootzén, H. (983. Extremes and Related Properties of Random equences and Processes. pringer, New Yor. [7] Mitrou, N. (24. Networing 24: Networing Technologies, ervices, and Protocols; Performance of Computer and Communication Networs, Mobile and Wireless Communication. Proceedings of the Third International IFIP-TC6 Networing Conference, Athens, Greece, 9-4 May 24. [8] Mohan, N. R. and Ravi,. (992. Max domains of attraction of univariate and multivariate p-max stable laws. Theory of Probability and Its Applications, 37, [9] Pancheva, E. (985. Limit theorems for extreme order statistics under nonlinear normalization. Lecture Notes in Mathematics, 5, [] Pancheva, E. (986. General limit theorems for maxima of independent random variables. Theory of Probability and Its Applications, [] Pancheva, E. (994. Extreme value theory with non-linear normalization. In: Extreme Value Theory and Applications, editors J. Galambos et al., Kluwer Academic Publishers, pp [2] Resnic,. I. (987. Extreme Values, Regular Variation, and Point Processes. pringer, New Yor. [3] Rothschild, L. (986. The distribution of English dictionary word lengths. Journal of tatistical Planning and Inference, 4, [4] avariappan, P., Chandrasehar, P. and Vaidyanathan, V.. (29. tatistical inference for a bularrival queue. In: Proceedings of the Fifth Asian Mathematical Conference, pp [5] chneider, K. and chultz, G. A. (982. A multisite data generation model for daily discharges. In: Optimal Allocation of Water Resources, IAH publication number 35, pp [6] ubramanya, U. R. (994. On max domains of attraction of univariate p-max stable laws. tatistics and Probability Letters, 9,