Precise Asymptotics of Generalized Stochastic Order Statistics for Extreme Value Distributions

Transcription

1 Applied Mathematical Sciences, Vol. 5, 2011, no. 22, Precise Asymptotics of Generalied Stochastic Order Statistics for Extreme Value Distributions Rea Hashemi and Molood Abdollahi Department of Statistics, Rai University 67149, Kermanshah, Iran Abstract In this paper we mainly investigate the precise asymptotics of the generalied stochastic order statistics in the extreme value distributions based on the suitable counting process. In this work, we present the relations among the boundary function, weighted function, convergence rate and limit value of series, this will be enable us to introduce a unified approach to several kinds of ordered random variables. Keywords: Generalied stochastic order statistics, precise asymptotics, extreme value distributions 1 Introduction and Main Results Since Hsu and Robbins 1947 introduced the concept of complete convergence, there have been extensions in two directions. Let X, X k,k 1 be a sequence of i.i.d. random variables, S n = n k=1 X k,n 1, and ϕx and fx be the positive functions defined on [0,. One extension is to discuss the moment conditions, from which it follows that ϕnp S n ɛfn <, ɛ > 0, 1 n=1 where n=1 ϕn =. In this direction, one can refer to Hsu and Robbins 1947, Erdös 1949,1950 and Baum and Kat 1965, etc. They respectively studied the cases in which ϕn 1, fn =n and ϕn =n r/p 2, fn =n 1/p, where 0 <p<2, r p. Another extension departs from the convergence rate and limit value of n=1 ϕnp S n > ɛfn as ɛ a, a 0. A first result in this direction was Heyde 1975, who proved

2 1090 R. Hashemi and M. Abdollahi that lim ɛ 2 P S n ɛn =EX 2, 2 n=1 where EX = 0 and EX 2 <. For analogous results in the more general case, see Chen 1978, Gut and Spǎtaru 2000, Gut 2002, Spǎtaru Research in precise asymptotics has often focused on the part S n and its corresponding objects, but the samples maximum M n = max in X i, n 1 has relatively neglected. An important objective within the extreme value theory is the description of M n and its applications. On the other hand, M n is the special case of order statistics. Moreover, order statistics and record values play a crucial role in statistics and its applications; both modes describe random variables arranged in order of magnitude. In this work, we will study the precise asymptotics for generalied order statistics introduced by Kamps 1995, which permits a unified approach to several models of ordered random variables, e.g. ordinary order statistics, record values, sequential order statistics, progressive censoring, etc. We first introduce some concepts, notation and basic properties. Definition 1.1 The generalied inverse of the distribution function, df, F F t = infx R : F x t, 0 <t<1, is called the quantile function of the df F. The quantity x t = F t defines the t-quantile of F. Definition 1.2 Say that X or F belongs to the maximum domain of attraction of the extreme value distribution Gx, if there exist normaliing constants c 1n > 0, centering constants d n R, n 1, and df Gx, satisfying c 1 1n M d n d n Z G, denoted by X MDAGor F MDAG. By the famous Fisher-Tippett theorem see Embrechts et al, 1997, Theorem 3.2.7, extreme value distributions have only three types: Fréchet distributions, Gumbel distributions and Weibull distributions. It is well known that the standard Fréchet distribution, standard Gumbel distribution, and standard Weibull distributions have G 1,α, G 2,α, G 3,α forms respectively, where G 1,α x = exp x α Ix >0, for some α>0 G 2,α x = exp x α if x<0 1 if x>0 G 3,α x = exp exp x, x R. Definition 1.3 Let F be a df with right endpoint x F. Suppose there exists some <x F such that F has representation x 1 F x =c exp at dt, < x < x F,

3 Precise asymptotics 1091 where c is some positive constant, a. is a positive and absolutely continuous function with respect to the Lebesque measure with density a and lim x xf a x =0. Then F is called a Von Mises function, the function a. is the auxiliary function of F. Definition 1.4 Let n N, K>0, m 1,...,m n 1 R, M r = n 1 j=r m j, 1 r n 1, be parameters such that γ r = k + n r + M r 1 for all r 1,...,n 1, and let m =m 1,...,m n 1,ifn 2, m R arbitrary, if n =1. If the random variables Ur, n, m, k, r =1,...,n, possess a joint density function of the form n 1 n 1 f U1,n, em,k,...,un,n, em,k u 1,...,u n =k γ j 1 u i m i 1 u n k 1, j=1 on the cone 0 u 1... u n 1, then they are called uniform generalied order statistics. Let F be an arbitrary distribution function. The random variables Xr, n, m, k = F 1 Ur, n, m, k, r =1,...,n, are called generalied order statistics based on F in the particular case m 1 = m 2 =... = m n 1 = m variables are denoted by Ur, n, m, k and Xr, n, m, k, r =1,...,n. The parameters k, m j, j =1,...,n 1, determine the model of ordered random variables. For example, in the case k =1,m j = 1 one gets record values. A further example can be found in Kamps The paper of Nasri-Roudsari 1996 started to develop the extreme value theory of generalied order statistics. It was shown that well known results of extreme value theory for ordinary order statistics in the weak domain of attraction and normaliing constants carry over to generalied order statistics. In particular, the possible limit distribution of extreme generalied order statistics called generalied extreme value distribution were established Nasri-Roudsari, 1996, Theorem 3.3 and it was shown that an underlying distribution function of an i.i.d. sequence X 1,X 2,... belongs to the weak domain of attraction of an extreme value distribution Nasri-Roudsari, 1996, Corollary 3.7. As in Marohn 2002 and Nasri-Roudsari and Cramer 1999, we assume in the following that the underlying parameters m 1,...,m n 1 of the generalied order statistics are equal, i.e. m i = m> 1 such a condition seems to be restrictive; nevertheless, various interesting models are still included. For a discussion we refer to the Yan 2008, in his paper the precise asymptotics of generalied stochastic order statistics for the Frechet distribution which is one of the three possible distributions in the extreme value distributions is presented. In this paper we are mainly investigate the generaliation of the last mentioned paper to all of the extreme value distributions. Our main results are as follows: Let g 1 x and hx be positive and differentiable functions defined on [n 0,, which are both strictly increasing to, ϕx =g 1hxh x is monotone, in the case i=1

4 1092 R. Hashemi and M. Abdollahi ϕx be monotone nondecreasing, we assume lim n ϕn +1/ϕn = 1. In addition assume that the underlying distribution function F MDAG 2,α, α>0, and counting process Nt, t>0, satisfies t 1 P Nt λ>0, t > 0 ENt τ sup <, for fixed r 1 3 t>0 t τ where τ = k + r 1. And assume that gx, x n m+1 0, satisfy the following conditions: footnotesie ɛ >0, G 0 ɛ := λ ɛhn0 αm+1 1 g Γτ 1 ɛ 1 yλ 1 1/αm+1 y τ 1 e y dy 1 Γτ λ ɛhn 0 αm+1 g 1 ɛ 1 yλ 1 1/αm+1 y τ 1 e y dy if αm +1isodd if αm +1iseven 4 such that G 0 ɛ <, and lim G 0 ɛ =. 5 And for all Gɛ G 0 ɛ 1, ɛ 0, 8 >< >: Z 1 λ ɛg lim lim Gɛ 1 MG 0 ɛαm+1 M Z lim lim M for some θ>0, λ ɛg 1 1 MG 0 ɛαm+1 g 1 g 1 ɛ 1 yλ 1 1/αm+1 y τ 1 e y dy =0 ifαm +1is odd ɛ 1 yλ 1 1/αm+1 y τ 1 e y dy =0 ifαm +1is even, lim lim M Gɛɛ α θk+r 1m+1 y α θk+r 1m+1 dg 1 y = 0 7 g 1 1 G 0ɛM where g1 1 x, h 1 x are the inverse functions of g 1 x and h 1 x, M>1. Theorem 1.1 For the appropriate choice of functions ϕ and h as above, if the counting process Nt satisfies 3, G 0 satisfies 4 and 5 and G satisfies the equations 6 and 7, then we can write lim Gɛ ϕxp XNn r +1,Nn,m,k d n ɛhnc n =1, 8 where d n and c n can be chosen as in the following theorem2.1. 6

5 Precise asymptotics 1093 Now let G, satisfies the following conditions: ɛ >0, G 0ɛ := 1 Γτ λ exp m+1ɛhn0 0 g 1 ɛ 1 m +1 lnλ y y τ 1 e y dy <, 9 lim G 0ɛ =, 10 G ɛ G 0ɛ 1, ɛ 0, 11 λ exp m+1ɛg 1 lim lim 1 G 0ɛM ɛ 1 M ɛ ø G ɛ g 1 0 m +1 lnλ y y τ 1 e y dy =0, 12 lim lim M ɛ ø G ɛ h 1 og 1 1 G ɛm exp k +m + 1r 1 or lim lim M ɛ ø G ɛ ϕx h 1 og 1 1 G ɛm exp k +m + 1r 1 t k+m+1r 1 ϕx cɛhxc x + d x ɛhxc x+d x gt dx =0, at ɛhxc x+d x+tãɛhxc x+d x 1 du =0, au 13 where <x<x F, and g 1 1 x, h 1 x are the inverse functions of g 1 x and hx, and M > 1, c. and g. are measurable functions satisfying cx c > 0, gx 1asx x F, and ax is a positive, absolutely continuous function with respect to the Lebesque measure. Theorem 1.2 Let ϕ defined as above, and assume that the underlying distribution function F MDAG 3,α, α>0, and the counting process Nt, t>0, satisfies the conditions defined in 3. In addition choosing G as in the equation 13. Then we can write lim G ɛ XNn r +1,Nn,m,k d n ɛhnc n =1, 14 ϕxp where d n and c n as in the following proposition. Remark 1.1 We argue that the theorem of Yan, Wang and Cheng 2006 can be viewed as a particular case of theorems 1.1 and 1.2.

6 1094 R. Hashemi and M. Abdollahi 2 Some Lemmas and Theorems In this section we review some necessary theorems and lemmas. Theorem 2.1 Theorem in Embrechts The df F belongs to the maximum domain of attraction of G 2,α, α>0, if and only if x F < and F x F x 1 =x α Lx for some slowly varying function L. If F MDAG 2,α, then d c 1 n M n x F G 2,α, where the normaliing constants c n can be chosen as c n = x F F 1 n 1 and d n = x F and F MDAG 2,α x F <, F x F x 1 R α. Proposition 2.1 Proposition in Embrechts Suppose the df F is a von Misess function. Then F MDAG 3,α. A possible choice of norming constants is where a is the auxiliary of F. d n = F 1 n 1 and c n = ad n, 15 Theorem 2.2 Theorem in Embrechts The df F with right endpoint x f belongs to the maximum domain of attraction of G 3,α, α>0, if and only if there exists some <x F such that F has representation F x =cx exp x gt at dt, < x < x F, 16 where c and g are measurable functions satisfying cx c > 0, gx 1 as x x F, and ax is a positive, absolutely continuous function with respect to Lebesque measure with density a x having lim x xf a x =0. We can choose in this case, A possible choice for the function a is d n = F 1 n 1 and c n = ad n. 17 ax = xf x F t F x dt, x < x F, 18 motivated by von Misess functions, we call the function a in 17 an auxiliary function for F. For a r.v. X the function ax defined in 18 is nothing but the mean excess function ax =EX x X >x, x < x F ; 19 see also section 3.4 of Embrechts.

7 Precise asymptotics 1095 Theorem 2.3 Theorem in Embrechts The df F belongs to the maximum domain of attraction of G 3,α, α>0, if and only if there exists some positive function ã such that as x x F, F x + tãx F x = e t, t R. 20 A possible choice is ã = a as given in 18. lemma 2.1 Suppose that the counting process Nt, t>0, satisfies t 1 Nt ENt λ>0, t, sup τ t>0 <, for fixed r 1, and F MDAG. Then t τ there exist α n > 0, β n R, n 1 such that when n, Δ n,r = sup x P αn 1 XNn r +1,Nn,m,k β n >x H r,m,k,λ x 0. P 21 Here H r,m,k,λ x = 1 Γτ Γτ,λ log Gxm+1 and Γα, x denotes the incomplete gamma function which is defined by Γα, x = x tα 1 e t dt, α>0, x 0. lemma 2.2 Suppose that F MDAG 2,α, α>0. Then for ɛ >0, θ > 0, r 1, there exists 0 <c< such that P XNn r +1,Nn,m,k d n >ɛhnc n cɛhn α θk+r 1m+1. Proof. By Lemma 2.5 in Nasri-Roudsari 1996, for every i r, P Xi r +1,i,m,k d n ɛhnc n = 1 I M+1i r +1, k 1 1 F ɛhncn+dn + r 1, r =1,...,n m +1 k+m+1r 1i k c F ɛhncn + d n r +1 m+1 +r 1 c F ɛhncn + d n k+m+1r 1i k m+1 +r 1. 22

8 1096 R. Hashemi and M. Abdollahi By 21 and the basic renewal theorem, we have P XNn r +1,Nn,m,k d n ɛhnc n = 1 I M+1i r +1, k 1 1 F ɛhncn+dn m +1 + r 1 i=r k+m+1r 1 c F ɛhncn + d n i k m+1 +r i P Nn =i k+m+1r 1 c F ɛhncn + d n ENn k m+1 +r 1 k+m+1r 1 c F ɛhncn + d n n k m+1 +r i=r i=r i=r Since F MDAG 2,α, there exists a function L R 0, such that F x F x 1 = x α Lx. Suppose x F < and define F x =F x F x 1, x>0, then F R α, then F MDAG 2,α with normaliing constants c n = F 1 n 1 and d n =0. Then F x =x α Lx, x>0and F c n n 1, n, then F ɛhxc n F = ɛhxc n + d n. By 22, 23 and potter s theorem BinghamTheorem 1.5.6, for θ >0, we have P XNn r +1,Nn,m,k >ɛhxc n + d n ɛhnc α c n L k+r 1m+1 k+r 1m+1 ɛhnc n c α n Lc n 1 Lɛhnc = cɛ αk+r 1m+1 hn αk+r 1m+1 n k+r 1m+1 Lc n cɛ αk+r 1m+1 hn αk+r 1m+1 ɛhn k+r 1m+1 = c ɛhn α θk+r 1m lemma 2.3 Suppose that F MDAG 3,α, α>0. Then for ɛ >0, r 1, there exists 0 < c <, c n, d n and c. and g. such that c n = ad n and d n = F 1 n 1 and c., g. are measurable functions satisfying cx c>0, gx 1 as x x F, and ax is a positive, absolutely continuous function with respect to the Lebesque measure with density a x having limiting lim x xf a x =0,

9 Precise asymptotics 1097 x F, P XNn r +1,Nn,m,k d n ɛhnc n c c ɛhnc n + k+m+1r 1 d n ɛhnc n +d gt exp k +m + 1r 1 at dt, < t < x F. Proof. By Lemma 2.5 in Nasri-Roudsari 1996, for i r, P Xi r +1,i,m,k d n ɛhnc n i r +1, = 1 I m F ɛhnc n +d n k m +1 + r 1, r =1,...,n c F ɛhnc k+m+1r 1i n + d k n r +1 m+1 +r 1 c F ɛhnc k+m+1r 1i n + d k m+1 n +r By 21 and basic renewal theorem, we have P XNn r +1,Nn,m,k d n ɛhnc n = 1 I k m+1 i r +1, 1 1 F ɛhnc n +d n m +1 + r 1 i=r k+m+1r 1 c F ɛhnc n + d n i=r i k m+1 +r 1 P Nn =i c F ɛhnc k+m+1r 1ENn n + d n k m+1 +r 1 c F ɛhnc k+m+1r 1n n + d n k+m+1r 1 k+m+1r 1 = c cɛhnc n + d n exp k +m + 1r 1 ɛhnc n +d gt at dt. 26 lemma 2.4 Suppose F MDAG 3,α, α>0. Then for ɛ >0, r 1, there exists 0 <c<, and c n and d n and some positive function ã such that P XNn r +1,Nn,m,k d n ɛhnc n ɛhnc n +d +tãɛhnc n +d 1 c exp k +m + 1r 1 au du, t R, u < x F,

10 1098 R. Hashemi and M. Abdollahi where, a possible choice is ã = a as given in 18. The proof is similar to proof of Lemma 2.3, we omit it. 3 Proof of Results Proof of Theorem 1.1 is similar to proof of theorem 1.1 Jiago Yan, so we omit it. Proof of Theorem 1.1. When ϕx is non increasing, we have n n+1 ϕx H r,m,k,λ ɛhxdx = ϕn H r,m,k,λ ɛhn +1 n 0 ϕx H r,m,k,λ ɛhxdx n 0 ϕx H r,m,k,λ ɛhxdx 27 ϕx H r,m,k,λ ɛhxdx 28 n 0 ϕn H r,m,k,λ ɛhn. 29 By integration by parts and 27, 28, 29, we have lim G ɛ = lim G ɛ = lim G ɛ = lim G ɛ = lim G ɛ [ = lim G ɛ = lim G ɛ ϕn H r,m,k,λ ɛhn ϕx H r,m,k,λ ɛhxdx n=n 0 ɛhn 0 Hr,m,k,λ ɛhxdg 1 hx H r,m,k,λ tdg 1 ɛ 1 t [ Hr,m,k,λ tg 1 ɛ 1 t + ɛhn 0 [ lim H r,m,k,λ tg 1 ɛ 1 t+ lim t λt Γτ H r,m,k,λ tg 1 ɛ 1 t+ 1 Γτ = lim G ɛ lim H r,m,k,λ tg 1 ɛ 1 t+1. ] g 1 ɛ 1 tdh r,m,k,λ t ɛhn 0 λ exp m+1ɛhn0 0 λ exp m+1ɛhn0 0 ] g 1 ɛ 1 te τt e λe t dt ] g 1 ɛ 1 m +1 lnλ y yτ 1 e y dy H r,m,k,λ t = 1 τ λe tm+1 0 x τ 1 e x dx τ 1 Γτ λe tm+1 τe λe tm+1, t.

11 Precise asymptotics 1099 By 9, we have 1 k +m + 1r 1 g 1 ɛ 1 te tk+m+1r 1 e λe tm+1 = g 1 ɛ 1 te λe tm+1 e uk+m+1r 1 du t t g 1 ɛ 1 te λe tm+1 e uk+m+1r 1 du 0,t. Hence we get lim G ɛ ϕn H r,m,k,λ ɛhn = When ϕx is nondecreasing, by assumption of Theorem 1.2, we know for δ >0, there exists n 1 N, such that for n n 1, Then we have 1 + δ 1 ϕn ϕn δϕn. 31 n=n 1 +1 ϕn H r,m,k,λ ɛhn n δ ϕx H r,m,k,λ ɛhxdx n=n 1 ϕn H r,m,k,λ ɛhn. 32 Similarly, we get 30. Let bɛ =h 1 g1 1 G 0ɛM, ep > 0, M >1. By Lemma 2.3 in Wang and Yang 2003, for ɛ small enough, if ϕx is nondecreasing, then by 31 we have 4G 0 ɛm [bɛ] If ϕx is non increasing, then ϕn 1 + δ 1 4G 0 ɛm [bɛ]+1 [bɛ]+1 ϕn ϕn ϕn ϕn 0. 34

12 1100 R. Hashemi and M. Abdollahi By Lemma 2.1 and Lemma 2.1 of Jiago Yan 2008 and Toeplic s lemma, we have To prove 14, it suffices to show that lim Gɛ [bɛ]+1 lim Gɛ ϕnδ n,r =0. 35 n=[bɛ]+2 ϕn H r,m,k,λ ɛhn = lim Gɛ n=[bɛ]+2 ϕnp XNn r +1,Nn,m,k d n >ɛhnc n =0. 37 If ϕx is nondecreasing, then for every ɛ small enough, we have ϕx H r,m,k,λ ɛhxdx 1 + δ 1 bɛ n=[bɛ]+2 ϕn H r,m,k,λ ɛhn. 38 If ϕx is nondecreasing, then ϕx H r,m,k,λ ɛhxdx bɛ n=[bɛ]+2 ϕn H r,m,k,λ ɛhn. 39 By integration by parts, 38, 39, we get Gɛ bɛ ϕx Hr,m,k,λ ɛhxdx = Gɛ H r,m,k,λ ɛhxdg 1 hx = Gɛ = c cgɛ bɛ ɛhbɛ H r,m,k,λ tdg 1 ɛ 1 t g 1 ɛ 1 tdh r,m,k,λ t ɛhbɛ λ exp m+1ɛg 1 1 G 0 ɛm 0 g 1 ɛ 1 m +1 lnλ y yτ 1 e y dy. 40

13 Precise asymptotics 1101 Hence by 12, we get 36. By Lemma 2.3, 2.4, we prove 13. G ɛ ϕnp XNn r +1,Nn,m,k d n >ɛhnc n n>bɛ+1 cg ɛ n>bɛ+1 k+m+1r 1 ϕn cɛhnc n + d n ɛhnc n +d n gt exp k +m + 1r 1 at dt k+m+1r 1 cg ɛ ϕn cɛhnc n + d n bɛ ɛhxc x +d x gt exp k +m + 1r 1 at dt dx k+m+1r 1 cg ɛ cɛhxc x + d x h 1 og 1 G 0 ɛm ϕx exp k +m + 1r 1 ɛhxc x +d x gt at dt dx. By 13, we get 37. The proof of other part of Theorem 1.2 is similar above, we omit it. References N.H. Bingham, C.M. Goldie, J.L. Teúgels, Regular Variation, Cambridge University Press, Cambridge, L.E. Baum, M. Kat, Convergence Rates in the Law of Large Numbers, Trans. Amer. Math. Soc R. Chen, A Remark on the Tail Probability of a Distribution, J. Multivariate Anal P. Embrechts, C. Klüppelberg, T. Minkosch, Modeling Extremal Events for Insurance and Finance, Springer, Berlin, P. Erdös, On a Theorem of Hsu and Robbins, Ann. Math. Statist P. Erdös, Remark on my Paper on a Theorem of Hsu and Robbins, Ann. MAth. Statist J. Galambos, The Asymptotic Theory of Extreme Order Statistics. Second ed., Krieger Publishing Co., A. Gut, Precise Asymptotics for Record Times and the Associated Counting Process, Stochastic Process. Appl

14 1102 R. Hashemi and M. Abdollahi A. Gut, A. Spǎtaru, Precise Asymptotics in the Baum-Kat and Davis Law of Large Numbers, J. Math. Anal. Appl A. Gut, A. Spǎtaru, Precise Asymptotics in the Law of the Iterated Logarithm, Ann. Probab C.C. Heyde, A Supplement to the Strong Law of Large Numbers, J. Appl. Probab P.L. Hsu, H. Robbins, Complete Convergence and the Strong Law of Large Numbers, Proc. Natl. Acad. Sci. USA U. Kamps, A concept of Generalied Order Statistics, J. Statist. Plann. Inference F. Maron, Strong Domain of Attraction of Extreme Generalied Order Statistics, Extremes D. Nasri-Roudsari, Extreme Value Theory of Generalied Order Statistics, J. Statist. Plann. Inference D. Nasri-Roudsari, E. Cramer. On the Convergence Rates of Extreme Generalied Order Statistics, Extremes A. Spǎtaru, Precise Asymptotics in Spiter s Law of Large Numbers, J. Theoret. Probab Y.B. Wang, Y. Yang, A General Law of Precise Asymptotics for the Counting Process of Record Times, J. Math. Anal. Appl J.G. Yan, Y.B. Wang, F.Y. Cheng, Precise Asymptotics for Order Statistics of a Non-random Sample and a Random Sample, J. Systems Sci. Math. Sci Received: July, 2010