Extreme value theory of mixture generalized order statistics

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1 ProbStat Forum, Volume, July 208, Pages 04 6 ISSN ProbStat Forum is an e-journal. For details please visit.probstat.org.in Extreme value theory of mixture generalized order statistics A M Elsaah a, Gajendra K Vishaarma b, Zhongquan Tan c a Department of Mathematics, Faculty of Science, Zagazig University, Zagazig 4459, Egypt and Division of Science and Technology, BNU-HKBU United International College, Zhuhai 59085, China. b Department of Applied Mathematics, Indian Institute of Technology, Dhanbad , India. c College of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing 3400, China. Abstract. Most statistical approaches assume that all of the data points come from the same distribution. Hoever, in real-life applications the data points come from more than one distribution ith no information to identify hich observation goes ith hich distribution. In such cases, the classical extreme value theory can not help us. In this paper, e investigate the extreme value theory of data from more than one distribution based on generalized order statistics under continuous strictly monotone normalization. This paper investigates the asymptotic behaviors of upper and loer extremes generalized order statistics based on a random sample dran from a finite mixture of distributions normalized by the same continuous strictly monotonic sequence or a mixture of continuous strictly monotonic sequences. For illustration of the usage of our theoretical results, these asymptotic behaviors under linear and poer normalization are studied ith examples.. Introduction.. Generalized order statistics Let X j : j N} be a sequence of independent and identically distributed random variables ith common probability density function f and distribution function F. If the first n random variables are arranged in ascending order of magnitude and ritten as X :n < X 2:n <... < X n:n, e call them ordinary order statistics. Generalized order statistics have been introduced by Kamps (995 as a unification of several models of ascendingly ordered random variables. The generalized order statistics X ( m, :n, X ( m, 2:n,..., X n:n ( m, are defined by their probability density function, hich is given on the cone (x,..., x n : x 0 = F (0 < x... x n < F ( = x 0 } as follos n f ( m,,2,...,n:n (x,..., x n = f(x n ( F(x n θ i f(x i ( F(x i mi, here θ,..., θ n are defined by θ n = > 0 and θ j m = (m,..., m n R n. = + n j + n i=j m i > 0, j =, 2,..., n and 200 Mathematics Subject Classification. 62G30, 60F05, 62E20. Keyords. Mixture distributions; Extreme value theory; Generalizes order statistics; Linear normalization; Poer normalization; Domains of attraction. Received: 3 December 207; Revised: 03 April 208, Re-revised 07 May 208; Accepted: 3 June 208. Elsaah-UIC Grants (Nos: R20409, R2072 and R2080 & the Zhuhai Premier Discipline Grant and Tan-National Science Foundation of China (No & Natural Science Foundation of Zhejiang Province of China (No. LQ4A address: a_elsaah85@yahoo.com, amelsaah@uic.edu.h, a.elsaah@zu.edu.eg (A M Elsaah

2 Elsaah, Vishaarma and Tan / ProbStat Forum, Volume, July 208, Pages Generalized order statistics have been idely used and they attracted much attention since Kamps (995 first introduced them. Generalized order statistics have the advantage of including important ell-non models hich are discussed separately in literature, such as the ordinary order statistics, sequential order statistics, Progressive type II censored order statistics, record values, th record values and Pfeifer s records. Particular choice of the parameters θ,..., θ n leads to different models, e.g., ordinary order statistics (m =... = m n = 0, = ; order statistics ith non-integral sample size (m =... = m n = 0, = τ n +, and τ > n ; th record values (m =... = m n = and is any positive integer and sequential order statistics (m i = (n i + φ i (n iφ i+, i n, = φ n and φ, φ 2,..., φ n > 0. In this paper, e consider a ide subclass of generalized order statistics by assuming θ j θ j+ = m+ > 0, i.e., m =... = m n = m. This subclass is non as m-generalized order statistics, e call it symmetric generalized order statistics throughout this paper. Many important practical models, such as ordinary order statistics, record values, sequential order statistics and order statistics ith non-integral sample size, are included in the symmetric generalized order statistics. The distribution function of the lth loer and the lth upper symmetric generalized order statistics are represented by and (x = I ( F(x m+ ( l, m + + n l n l+:n (x = I ( F(x (n m+ l +, m + + l, (2 x respectively, here I x (n, m = B(n,m 0 tn ( t m dt, B(n, m = (n(m (n+m function (c.f. Nasri-Roudsari, Classical extreme value theory ( and (. is the gamma The main problem of the classical extreme value theory of ordinary order statistics is to find conditions on the distribution function F under hich there is a strongly monotone continuous transformation B n (x, such that for the lth upper extreme ordinary order statistic the distribution function of Bn (X n l+:n (for the lth loer extreme ordinary order statistic the distribution function of Bn (X converges ealy to a non-degenerate distribution (i.e., the distribution is not degenerate at 0 or, to obtain the classes of possible limit distribution functions and to give methods for calculating the normalizing transformation B n (x. Several authors have studied this problem for the linear transformation B n (x = a n x + b n, such as Gnedeno (943, Balema and de Haan (972 and Smirnov (952. The central result of the classical extreme value theory turns out that the class of possible limit distributions has essentially three different types of distributions. For some suitable normalizing constants c n, a n > 0 and d n, b n R, e have Ξ (0, (c nx + d n H l(0, i,α (x = (l U i,α(x t l e t dt (3 and Ξ (0, n l+:n (a nx + b n H u(0, j,β (x = t l e t dt, (4 (l V j,β (x if, and only if, nf(c n x + d n U i,α (x and n( F(a n x + b n V j,β (x, respectively at all continuity points of U i,α (x and V j,β (x, here denotes the ea convergence, as n, means the limit as n, U,α (x = ( x α, x < 0, α > 0,, x 0, x U 2,α (x = α, x 0, α > 0, 0, x < 0,

3 Elsaah, Vishaarma and Tan / ProbStat Forum, Volume, July 208, Pages x V,β (x = β, x > 0, β > 0, ( x V, x 0, 2,β (x = β, x 0, β > 0, 0, x > 0 and U 3,0 (x = V 3,0 ( x = e x, x. In this case, e say that the distribution function F belongs to the domain of attraction of each of the limits H l(0, i,α and H u(0, j,β under linear normalization, ritten F LDA(H l(0, and F LDA(Hu(0,, respectively. i,α j,β The folloing theorem, due to Nasri-Roudsari (996 (cf. also Nasri-Roudsari and Cramer, 999, extends the above results to the symmetric generalized order statistics. Theorem.. For any symmetric generalized order statistics, let m >,, c n, a n > 0, d n, b n R, A n(m+ (x = c n(m+ x+d n(m+, B n /(m+(x = a n /(m+x+b n /(m+ and l be a fixed integer l n. Then, e have and ( An(m+ (x n n H l(m, i,α (x = t l e t dt (l U i,α(x H u(m, j,β (x = ( m+ + l if, and only if, F LDA(H l(0, and F LDA(Hu(0,, respectively. i,α j,β t m+ +l 2 e t dt, V m+ j,β (x Pantcheva (985 (cf. also Mohan and Ravi, 992 introduced a non-linear normalization for the maximal ordinary order statistics B n (x = α n x βn S(x non as the poer normalization, here S(x =, 0,, if x < 0, x = 0, x > 0, respectively. The limit distributions obtained ith non-linear normalization attract more distributions than the linear normalization. The folloing slight generalization of the results in Pantcheva (985 for the lth upper extreme ordinary order statistics under poer normalization is given by Baraat and Nigm (2002. For some suitable normalizing constants α n > 0 and β n > 0, e have Ξ (0, n l+:n (α n x βn S(x = I F(Bn(x(n l +, l Q (0, j,δ = (l ϕ j,δ (x t l e t dt, if, and only if, n ( F ( α n x βn S(x ϕ j,δ (x at all continuity points of ϕ j,δ (x, here, x 0,, x, ϕ,δ (x = (log x δ ϕ, x >, δ > 0, 2,δ (x = ( log x δ, 0 < x, δ > 0, 0, x >,, x, (log( x ϕ 3,δ (x = ( log( x δ, < x 0, δ > 0, ϕ 4,δ (x = δ, x, δ > 0, 0, x >, 0, x > 0, ϕ 5,δ (x = ϕ 5 (x = V, (x and ϕ 6,δ (x = ϕ 6 (x = V 2, (x. In this case, e say that the distribution function F belongs to the domain of attraction of Q (0, j,δ under poer normalization, ritten F PDA(Q (0, j,δ. The folloing theorem, due to Nasri-Roudsari (999, extends the above results to the symmetric generalized order statistics. Theorem.2. For any symmetric generalized order statistics, let m >,, α n, β n > 0 and l be a fixed integer l n. Then, e have ( n l+:n αn /(m+ x β n /(m+ S(x n Q (m, j,δ = ( m+ + l if, and only if, the distribution function F satisfies F PDA(Q (0, j,δ. ϕ m+ j,δ (x t m+ +l 2 e t dt,

4 Elsaah, Vishaarma and Tan / ProbStat Forum, Volume, July 208, Pages Finite mixture distribution The classical extreme value theory assumes that all the observations of a sample come from the same distribution (i.e., homogeneous population. Hoever, in real-life projects the observations come from more than one distribution (i.e., heterogeneous population ith no information to identify hich observation goes ith hich distribution. In this mixture, the ith component (distribution function of the ith subpopulation is F i (x and the mixing proportions P i > 0, i ζ are such that ζ P i =. In such cases, the distribution function is called a finite mixture distribution and is given by F(x = ζ P if i (x. The finite mixture distribution provides a natural representation of heterogeneity in many real-life data from a mixture of distributions. The finite mixture distribution has received increasing attention in recent years and has proven to be a useful approach in modeling heterogeneous data. The reader can refer to Everitt and Hand (98, Titterington et al. (985, Maclachlan and Basford (988, Lindsay (995 and Elsaah (204 for some properties and applications of finite mixture distribution. In the last fe years, much attention has been paid in exploring the potential application of the extreme value theory to the finite mixture distribution. AL-Hussaini and El-Adll (2004 studied the asymptotic distribution of the maximum ordinary order statistics under finite mixture distribution. Subsequently, Sreehari and Ravi (200 gave a closer loo at the asymptotic distribution of the maximum ordinary order statistics under finite mixture distribution ith some generalizations. Ga et al. (206 studied the asymptotic distribution of the maximum ordinary order statistics under finite mixture distribution and a statistical method to control the possible bias. Finally, Alaady et al. (206 investigated the asymptotic behavior of the appropriately linear normalized coordinateise maximum and minimum under multivariate finite mixture distribution from independent, but not obligatory identically distributed random vectors. In this paper, e extend the classical extreme value theory to the case of finite mixture distribution based on symmetric generalized order statistics under continuous strictly monotone normalization. This paper provides the possible limit distributions of upper and loer extremes symmetric generalized order statistics based on a random sample dran from a finite of mixture distributions normalized by the same continuous strictly monotonic sequence or different continuous strictly monotonic sequences. For illustration of the usage of our theoretical results, the asymptotic behaviors of mixture symmetric generalized order statistics under linear and poer normalization are studied ith some examples. 2. Mixture extreme value theory The possible non-degenerate limit distribution functions of the lth upper extreme and the lth loer extreme symmetric generalized order statistics based on a random sample dran from a finite of mixture distributions normalized by the same continuous strictly monotonic sequence, as ell as sufficient conditions for the existence of these limit distribution functions are given in Theorems 2.6 and 2.7, respectively. Theorems 2.3 and 2.4 are intended to extend the results of Theorems 2.6 and 2.7, by considering a finite of mixture distributions here the mixing distributions for the purpose of limiting distributions could be normalized by different continuous strictly monotonic sequences. The proof of the results for the lth upper extreme depends on the folloing Lemmas 2. and 2.2, hich are simple applications of Theorem.5. in Leadbetter et al. (983 (cf. also Lemma 3. and Corollary 3.2 in Nasri-Roudsari, 996 and Theorem 3 in Smirnov (952 (cf. also Lemma 3. in Baraat, 997, respectively. The proof of the results for the lth loer extreme depends on the folloing Lemmas 2.3 and 2.4, hich are simple extensions of Lemmas 2. and 2.2 respectively. Lemma 2.. For any distribution function F and non-degenerate distribution function G, let a n > 0, b n R, n N, a(t = a t, b(t = b t and t be the integral part of t. Then, the folloing statements are equivalent: F n (a n x + b n G(x for all continuity points of G.

5 Elsaah, Vishaarma and Tan / ProbStat Forum, Volume, July 208, Pages n[ F(a n x + b n ] log(g(x for all continuity points of G. F t (a(tx + b(t t G(x for all continuity points of G. t[ F(a(tx + b(t] t log(g(x for all continuity points of G. Lemma 2.2. For any distribution function F, let R be a fixed integer R N, N ( F N λ <, (x, y = y tx e t dt and σ N and ρ N N 0. Then, for large values of N e have (R (R, N ( F(x σ N I F(x (N R +, R (R (R, N ( F(x + ρ N. Lemma 2.3. For any distribution function F and non-degenerate distribution function J, let c n > 0, d n R, n N, c(t = c t, d(t = d t and t is the integral part of t. Then, the folloing statements are equivalent: [ F(c n x + d n ] n n J (x for all continuity points of J. nf(c n x + d n log(j (x for all continuity points of J. [ F(c(tx + d(t] t t J (x for all continuity points of J. tf(c(tx + d(t t log(j (x for all continuity points of J. Lemma 2.4. For any distribution function F, let N F N γ < and Θ N and Ω N N 0. Then, for large values of N e have (R (R, N F(x Θ N I F(x (R, N R + (R (R, N F(x + Ω N. Remar 2.5. The same results in Lemmas 2. and 2.3 hold ith a n x + b n and c n x + d n replaced by any continuous strictly monotonic sequence B n (x. 2.. Mixture extreme value theory via the same normalization Theorem 2.6. For any symmetric generalized order statistics from a mixture of ζ distributions F i (x ith mixture distribution function F(x = ζ P if i (x, 0 < P i < and ζ P i =, let m >, > 0, l be a fixed integer l n and L(K i (x and R(K i (x be the left and the right end points of a non-degenerate distribution function K i (x, respectively. If F i (x are normalized by the same strongly monotone continuous sequence B n (x} n= and F n i (B n (x K i (x, (5 then hen max i ζ L(K i (x < x < max i ζ R(K i (x e get n ( m+ + l Proof. Since (5 holds, from Lemma 2. and Remar 2.5 for i ζ e have ( m+ ( t ζ log(ki(xp m+ i n /(m+ ( F i (B n /(m+(x log K i (x. (6 It then follos, since ζ P i = and F(x = ζ P if i (x that F(B n /(m+(x = P i ( F i (B n /(m+(x. (7

6 Elsaah, Vishaarma and Tan / ProbStat Forum, Volume, July 208, Pages Therefore, from (6 and (7 e get ( m + + n ( F(B n /(m+(x m+ n( F(B n /(m+(x m+ [ ] m+ n /(m+ ( F(B n /(m+(x [ P i n /(m+ ( F i (B n /(m+(x ( P i log K i (x m+ ]m+. (8 From (2 and (8 and by taing F = ( F(B n /(m+(x m+, N = m+ + n and R = m+ + l in Lemma 2.2, the proof can be completed. Theorem 2.7. For any symmetric generalized order statistics from a mixture of ζ distributions F i (x ith mixture distribution function F(x = ζ P if i (x, 0 < P i < and ζ P i =, let m >, > 0, l be a fixed integer l n and L(E i (x and R(E i (x be the left and the right end points of a non-degenerate distribution function E i (x, respectively. If F i (x are normalized by the same strongly monotone continuous sequence A n (x} n= and ( F i (A n (x n n E i (x, (9 then hen max i ζ L(E i (x < x < max i ζ R(E i (x e have (A n(m+ (x (l ζ log(ei(x P i t l e t dt. Proof. Since (9 holds, from Lemma 2.3 and Remar 2.5 e get n(m + F i (A n(m+ (x log E i (x. (0 It then follos, since ζ P i = and F(A n(m+ (x = ζ P if i (A n(m+ (x that F(A n(m+ (x = ( P i Fi (A n(m+ (x. ( From Maclaurin series, e have ( F(A n(m+ (x m+ = (m + F(A n(m+ (x( + ( (m + F(A n(m+ (x, here ( 0. Therefore, from (0, ( and (2 e get ( ( m + + n ( F(An(m+(x m+ n ( ( F(A n(m+(x m+ n(m + F(A n(m+ (x P i n(m + F i (A n(m+ (x (2 P i log E i (x. (3 From ( and (3 and by taing F = ( F(A n(m+ (x m+, N = m+ + n and R = l in Lemma 2.4, the proof can be completed.

7 Elsaah, Vishaarma and Tan / ProbStat Forum, Volume, July 208, Pages Corollary 2.8. From Theorems 2.6 and 2.7, e have Ξ (0, n l+:n (B n(x and 0, if x max i ζ L(K i (x, n, if x max i ζ R(K i (x and Ξ (0, (A n(x and (A n(m+ (x 0, if x max i ζ L(E i (x,, if x max i ζ R(E i (x. Corollary 2.9. From Theorems 2.6 and 2.7, e have the folloing special cases. It is orth noting that, the special cases (iii in (I and (II can be found in AL-Hussaini and El-Adll (2004 and Sreehari and Ravi (200. (I For ordinary order statistics, i.e., m = 0 and =, e get (i Ξ (0, n l+:n (B n(x (ii Ξ (0, (A n(x (iii Ξ (0, n:n (B n (x (iv Ξ (0, :n (A n(x ζ (l (l log(ki(x P i tl e t dt. ζ ζ (K i(x Pi. ζ (E i(x Pi. log(ei(x P i tl e t dt. (II If the observations come from the same distribution, i.e., K i (x = K(x i and E i (x = E(x i, then (i (ii Ξ (0, (iii Ξ (0, n:n (B n (x ( (A n(x and (A n(m+ (x K(x. (iv Ξ (0, :n (A n(x and :n (A n(m+(x +l ( m+ (log K(x t m+ m+ (l E(x. log E(x tl e t dt. Corollary 2.0. For any symmetric generalized order statistics from a mixture of ζ distributions normalized by the same linear sequence B n (x = a n x + b n, let m >,, a n > 0, b n R, l be a fixed integer l n and L(K i (x and R(K i (x be the left and the right end points of a non-degenerate distribution function K i (x, respectively. Then, hen max i ζ L(K i (x < x < max i ζ R(K i (x, e get (I Let F(x = PF (x+( PF 2 (x, here 0 < P <, F LDA(H u(0,,β (x and F 2 LDA(H u(0, 2,β (x. Then, e get F(x LDA(H u(0,,β (x. That is, for x > 0 e get ( +l (Px β t m+ m+ (II Let F(x = PF (x+( PF 2 (x, here 0 < P <, F LDA(H u(0,,β (x and F 2 LDA(H u(0, 3,β (x. Then for x > 0, e get the folloing non-degenerate distribution ( +l (P(x β e x +e x t m+ m+ (III Let F(x = PF (x+( PF 2 (x, here 0 < P <, F LDA(H u(0, 2,β (x and F 2 LDA(H u(0, 3,β (x. Then, e have n l+:n (B n /(m+(x +l (e x +P(( x β e x t m+ m+ +l (( Pe x t m+ m+ ( ( m+ +l 2 e t dt, x 0, m+ +l 2 e t dt, x > 0.

8 Elsaah, Vishaarma and Tan / ProbStat Forum, Volume, July 208, Pages 04 6 (IV Let F(x = P F (x + P 2 F 2 (x + P 3 F 3 (x, here P + P 2 + P 3 =, 0 < P, P 2, P 3 <, F (m, LDA( H,β (x, F 2 LDA(H u(0, 2,β (x and F 3 LDA(H u(0, 3,β (x. Then for x > 0, ( +l (P m+ x β +P 3e x t m+ Proof. The proof is obvious from Theorems. and 2.6 ith simple algebra. We ill provide the proof for the first case (I and the proof for the other cases ill be by the same technique. Let F(x = PF (x + ( PF 2 (x, here 0 < P <, F LDA(H u(0,,β (x and F 2 LDA(H u(0, 2,β (x. Then, from Theorems. and 2.6 e get n ( m+ + l ( m+ ( PV,β (x ( PV 2,β (x m+ t From (4, e get Thus, e get PV,β (x + ( PV 2,β (x = Px β, x > 0, β > 0,, x 0, PV,β (x + ( PV 2,β (x = ( P( x + β, x 0, β > 0, 0, x > 0 Px β, x > 0, β > 0,, x 0. Corollary 2.. For any symmetric generalized order statistics from a mixture of ζ distributions normalized by the same strongly monotone continuous poer sequence B n (x = α n x βn S(x, let m >,, α n, β n > 0, l be a fixed integer l n and L(K i (x and R(K i (x be the left and the right end points of a nondegenerate distribution function K i (x, respectively. Then, hen max i ζ L(K i (x < x < max i ζ R(K i (x, e get (I Let F(x = 5 P if i (x, here 5 P i =, 0 < P i <, F PDA(Q (0, PDA(Q (0, t,δ (x, t 2, 3, 4, 6}, 2 j 5. Then for x >, e get ( +l (P m+ (log x δ t m+,δ (x and F j (II Let F(x = 6 P if i (x, here 6 P i =, 0 < P i <, F PDA(Q (0,,δ (x, F 5 PDA(Q (0, 5,δ (x and F j PDA(Q (0, t,δ (x, t 2, 3, 4, 6}, j = 2, 3, 4, 6. Then for x >, e get ( (P(log +l x δ +P 5 m+ x m+ t (III Let F(x = 4 P if i (x, here 4 P i =, 0 < P i <, F 2 PDA(Q (0, PDA(Q (0, t,δ (x, t 3, 4, 6}, j =, 3, 4. Then for 0 < x, e get ( +l (( δ P m+ 2(log x δ t m+ 2,δ (x and F j (IV Let F(x = 5 P if i (x, here 5 P i =, 0 < P i <, F 2 PDA(Q (0, 2,δ (x, F 5 PDA(Q (0, 5,δ (x and F j PDA(Q (0, t,δ (x, t 3, 4, 6}, j =, 3, 4. Then, n ( +l m+ (P 2( log x δ + P 5 x m+ t m+ +l 2 e t dt, 0 < x <, ( +l m+ ( P 5 x m+ t m+ +l 2 e t dt, x >.

9 Elsaah, Vishaarma and Tan / ProbStat Forum, Volume, July 208, Pages (V Let F(x = 4 P if i (x, here 4 P i =, 0 < P i <, F PDA(Q (0, PDA(Q (0, t,δ (x, t 3, 4, 6}, j = 2, 3, 4. Then for x > 0, e get n ( +l m+ ( P x m+ t 5,δ (x and F j Proof. The proof is obvious from Theorems.2 and 2.6 by the same technique as that in the proof of Corollary 2.0. Remar 2.2. If Ξ (0, ( +l z m+ t m+ +l 2 e t dt, then n l+:n (B n(x e z. Therefore, the discussion in Corollaries 2.0 and 2. can be immediately given for the ordinary order statistics Mixture extreme value theory via different normalizations Theorem 2.3. For any symmetric generalized order statistics from a mixture of ζ distributions F i (x ith mixture distribution function F(x = ζ P ζ if i (x, 0 < P i <, P i =, let m >, > 0, l be a fixed integer l n and L(K i (x and R(K i (x be the left and the right end points of a nondegenerate distribution function K i (x, respectively. Suppose F i (x are normalized by different strongly monotone continuous sequences B i,n (x} n= such that, for i ζ, the folloing conditions hold (C u F n i (B i,n(x K i (x, 0 < K i (x <, (C u 2 n[f i (B n (x F i (B i,n (x] η i (x, 0 < η i (x <, for some strongly monotone continuous sequence B n (x} n=. Then, hen max i ζ L(K i (x < x < max i ζ R(K i (x, e have n ( m+ + l ( t ζ [log(k i(x P i +P iη i(x] m+ Proof. Since F(x = ζ P if i (x, e get F(B n /(m+(x = P i F i (B i,n /(m+(x + [ P i Fi (B n /(m+(x F i (B i,n /(m+(x ] ζ = P ζ F ζ (B ζ,n /(m+(x + P i F i (B i,n /(m+(x + [ P i Fi (B n /(m+(x F i (B i,n /(m+(x ]. (4 Since P ζ = ζ P i, from (4 e have ζ [ F(B n /(m+(x = F ζ (B ζ,n /(m+(x P i Fζ (B ζ,n /(m+(x F i (B i,n /(m+(x ] + [ P i Fi (B n /(m+(x F i (B i,n /(m+(x ]. (5

10 Elsaah, Vishaarma and Tan / ProbStat Forum, Volume, July 208, Pages Thus, from (5 e get n /(m+ ( F(B n /(m+(x = n /(m+ ( F ζ (B ζ,n /(m+(x P i n /(m+ [ F i (B n /(m+(x F i (B i,n /(m+(x ] P i [n /(m+ ( F i (B i,n /(m+(x ζ + ] n /(m+ ( F ζ (B ζ,n /(m+(x. (6 Suppose that (C u is satisfied. Then, from Lemma 2. and Remar 2.5 for i =, 2,..., ζ e have n( F i (B n (x log K i (x. Therefore, (C u and (C2 u become and n /(m+ ( F i (B n /(m+(x log K i (x (7 n /(m+ [F i (B n /(m+(x F i (B i,n /(m+(x] η i (x, (8 respectively. Then, from (6, (7 and (8 e get n /(m+ ( F(B n /(m+(x log K ζ (x It then follos, since ζ P i = P ζ, that n /(m+ ( F(B n /(m+(x ζ + P i η i (x P i ( log K i (x + log K ζ (x. (9 P i (log K i (x + η i (x. (20 Therefore, from (20 e get ( m + + n ( F(B n /(m+(x m+ n [( F(B n /(m+(x] m+ [ ] m+ n /(m+ ( F(B n /(m+(x [ P i (log K i (x + η i (x From (2 and (2 and by taing F = ( F(B n /(m+(x m+, N = m+ + n and R = m+ + l in Lemma 2.2, the proof can be completed. Theorem 2.4. For any symmetric generalized order statistics from a mixture of ζ distributions F i (x ith mixture distribution function F(x = ζ P ζ if i (x, 0 < P i <, P i =, let m >, > 0, l be a fixed integer l n and L(E i (x and R(E i (x be the left and the right end points of a non-degenerate distribution function E i (x, respectively. Suppose F i (x are normalized by different strongly monotone continuous sequences A i,n (x} n= such that, for i ζ, the folloing conditions hold (C l ( F i (A i,n (x n n E i (x, 0 < E i (x <, ]m+. (2

11 Elsaah, Vishaarma and Tan / ProbStat Forum, Volume, July 208, Pages (C l 2 n[f i (A n (x F i (A i,n (x] µ i (x, 0 < µ i (x <, for some strongly monotone continuous sequence A n (x} n=. Then, hen max i ζ L(E i (x < x < max i ζ R(E i (x, e have (A n(m+ (x (l ζ [log(e i(x P i P iµ i(x] tl e t dt. Proof. The proof is similar to the proof of Theorems 2.7 and 2.3, except the obvious changes. Corollary 2.5. From Theorems 2.3 and 2.4, e have the folloing special cases (I For ordinary order statistics, i.e., m = 0 and =, e get (i Ξ (0, n l+:n (B n(x (ii Ξ (0, (A n(x ζ (l (l (iii Ξ (0, n:n (B n (x = F n (B n (x (iv Ξ (0, :n (A n(x [log(k i(x P i P iη i(x] tl e t dt. [log(ei(x P i P iµ i(x] tl e t dt. ζ [K i(x exp(η i (x] Pi. ζ [E i(x exp(µ i (x] Pi. (II If the observations come from the same distribution, i.e., K i (x = K(x, η i (x = η(x, E i (x = E(x and µ i (x = µ(x for i ζ, then e have (i ( (ii (A n(m+ (x and Ξ (0, (A n(x (iii Ξ (0, n:n (B n (x K(x exp(η(x. (iv :n (A n(m+(x and Ξ (0, :n (A n(x +l ( m+ (log K(x+η(x t m+ m+ (l log E(x µ(x tl e t dt. E(x exp(µ(x. 3. Applications In this section some illustrative examples of the most practically important distributions are obtained, hich lend further support to our theoretical results. Example 3.. (logistic, Exponential( and Gumbel distributions. Let X, X 2,..., X n be a random sample dran from a mixed population hose distribution function F(x = P F (x + P 2 F 2 (x + P 3 F 3 (x, here P + P 2 + P 3 =, 0 < P i <, i =, 2, 3, F (x = +e, < x <, F x 2 (x = e x, x > 0 and F 3 (x = exp( e x, < x <. One can choose the common linear normalization, B n (x = x+log n. Then B n /(m+ = x+ m+ log n. It is not difficult to sho that n( F i(x+log n e x, or, equivalently, Fi n (x + log n n exp( e x, i =, 2, 3. Then, e have n l+:n (x + log n m + n ( m+ + l and Ξ (0, n:n (x + log n exp( e x. e x(m+ t m+ +l 2 e t dt Example 3.2. (Poer( and Exponential( distributions. Let X, X 2,..., X n be a random sample dran from a mixed population hose distribution function F(x = PF (x + ( PF 2 (x, here 0 < P <, F (x = x, 0 < x < and F 2 (x = e x, x > 0. One can choose the common linear normalization, A n (x = x n. Then, e get A x n(m+(x = n(m+. It is not difficult to sho that nf i( x n x, i =, 2.

12 Elsaah, Vishaarma and Tan / ProbStat Forum, Volume, July 208, Pages ( or, equivalently, ( F i ( x n n n e x, i =, 2. Then, e have x n(m+ and Ξ (0, ( x n (l x tl e t dt and Ξ (0, ( x :n n n e x. Example 3.3. Let X, X 2,..., X n be a random sample dran from a mixed population hose distribution function F(x = PF (x + ( PF 2 (x, here 0 < P <, F (x = exp[ ( log x α ], 0 < x 0, x 0,, x >, exp[ (log( x and F 2 (x = α ], x, One can choose the common poer normalization, B, x >. n (x = x n α S(x. Then B n /(m+(x = x n α(m+ S(x. It is not difficult to sho that ( ] n [F i x n /α S(x n F i (x, i =, 2. Then, e have n l+:n ( x n α(m+ S(x ( m+ + l ( and Ξ (0, n:n x n α S(x ( m+ (log([f (x] P [F 2(x] P m+ t m+ +l 2 e t dt [F (x] P [F 2 (x] P. Example 3.4. (Pareto and Frechet distributions. Let X, X 2,..., X n be a random sample dran from a mixed population hose distribution function F(x = PF (x + ( PF 2 (x, here 0 < P < 0, x <, exp[ x, F (x = x α and F, x, α > 0 2 (x = α ], x 0, α > 0 One can choose the 0, x < 0. common poer normalization, B n (x = (nx /α. Then, e get B n /(m+(x = (n m+ x /α. It is not difficult to sho that n( F i ((nx /α n x, i =, 2. Then, e get ( n l+:n (n m+ x /α n ( m+ + l ( and Ξ (0, n:n (nx /α = n e /x. x (m+ t m+ +l 2 e t dt Similarly, one can choose the common linear normalization, B n (x = n /α x. Then, B n /(m+(x = n /α(m+ x. It is not difficult to sho that n( F i (n /α x x α, i =, 2. Then, ( n l+:n n /α(m+ x n ( m+ + l ( and Ξ n:n (0, n /α x = n e x α. x α(m+ t m+ +l 2 e t dt Remar 3.. It is orth mentioning that at this stage e are unable to find a suitable example here the normalizing constants are different to support the assumptions (C u 2 and (C l 2 of Theorems 2.3 and 2.4, respectively. This ill be investigated in our future or. Acnoledgments The authors greatly appreciate the corrections and helpful suggestions by the referee that significantly improved the paper. Elsaah also than Prof. Kai-Tai Fang for his guidance and ind support during this or.

13 References Elsaah, Vishaarma and Tan / ProbStat Forum, Volume, July 208, Pages [] Alaady, M.A, Elsaah, A.M, Hu, J. and Qin, H., 206. Asymptotic behavior of non-identical multivariate mixture. ProbStat Forum, 09, July 206, [2] AL-Hussaini, E.K. and El-Adll, E.M., Asymptotic distribution of normalized maximum under finite mixture models. Statist. Probab. Lett. 70, [3] Balema, A.A. and de Haan, L., 972. On R. von Mises condition for the domain of attraction of exp( e x. Ann. Math. Stat. 43, [4] Baraat, H.M., 997. Asymptotic properties of bivariate random extremes. J. Statist. Plann. Inference 6, [5] Baraat, H.M and Nigm, E.M, Extreme order statistics under poer normalization and random sample size. Kuait J. Sci. Eng. 29(, [6] Elsaah, A.M., 204. Generalized Order Statistics Under Finite Mixture Models: Asymptotic Theory and Applications. Lap Lambert Academic Publishing, Riga. [7] Everitt, B.S. and Hand, D.J., 98. Finite Mixture Distribution. Chapman and Hall, London. [8] Gnedeno, B.V., 943. Sur la distribution limite du terme maximum dune serie al eatoire. Ann. Math. 44, [9] Ga, W., Goo, H., Choi, Y.H. and Ahn, J.Y., 206. Extreme value theory in mixture distributions and a statistical method to control the possible bias. J. Korean Statist. Soc. 45, [0] Kamps, U., 995. A Concept of Generalized Order Statistics. Teubner, Stuttgart. Update. 3, Wiley, Ne Yor. [] Leadbetter, M.R., Lindgren, G. and Rootzen, H., 983, Extremes and related properties of random sequences and processes. Springer, Ne Yor. [2] Lindsay, B.G., 995. Mixture Models: Theory, Geometry and Applications. The Institute of Mathematical Statistics, Hayard, CA. [3] Maclachlan, G.J. and Basford, K.E., 988. Mixture Models: Applications to Clustering. Marcel Deer, Ne Yor. [4] Mohan, N.R. and Ravi, S., 992. Max domains of attraction of univariate and multivariate p-max stable las. Theory Probab. Appl. 37, [5] Nasri-Roudsari, D., 996. Extreme value theory of generalized order statistics. J. Statist. Plann. Inference 55, [6] Nasri-Roudsari, D., 999. Limit distributions of generalized order statistics under poer normalization. Commun. Statist.- Theory Meth. 28(6, [7] Nasri-Roudsari, D. and Cramer, E., 999. On the convergence rates of extreme generalized order statistics. Extremes 2, [8] Pantcheva, E., 985. Limit theorems for extreme order statistics under nonlinear normalization. Lecture Notes in Mathematics, vol. 55, pp [9] Smirnov, N. V. 949, 952. Limit distributions for the terms of a variational series. Original Russian in Trudy Math. Inst. Stelov., 25 (949, -6. [20] Sreehari, M. and Ravi, S., 200. On extremes of mixture of distributions. Metria 7, [2] Titterington, D.M., Smith, A.F.M. and Maov, U.E., 985. Statistical Analysis of Finite Mixture Distributions, Wiley. Ne Yor.