O the distributioal expasios of powered extremes from Maxwell distributio Jiawe Huag a, Xilig Liu b, Jiaju Wag a, Zhogqua Ta c, Jigyao Hou a, Hao Pu d a School of Mathematics ad Statistics, Southwest Uiversity, Chogqig, 40075, Chia b School of Mathematics ad Iformatio, Chia West Normal Uiversity, Nachog, 6700, Chia c Departmet of Statistics, Jiaxig Uiversity, Jiaxig, 4000, Chia d School of Mathematics, Zuyi Normal College, Zuyi, 5600, Chia Abstract. I this paper, asymptotic expasios of the distributios ad desities of powered extremes for Maxwell samples are cosidered. The results show that the covergece speeds of ormalized partial maxima relies o the powered idex. Additioally, compared with previous result, the covergece rate of the distributio of powered extreme from Maxwell samples is faster tha that of its extreme. Fially, umerical aalysis is coducted to illustrate our fidigs. Keywords. Asymptotic expasio; desity; Maxwell distributio; powered extreme. AMS Classificatio: Primary 6E0, 60E05; Secodary 60F5, 60G5. Itroductio I extreme value theory, researchers recetly focus o ivestigatig the quality of covergece of ormalized maxx k, k := M of a sample. For the covergece rate of ormalized M, geeral cases were discussed by Smith, Leadbetter et al., Galambos ad de Haa ad Resick 4, ad specific cases were cosidered by Hall 5, 6, Nair 7, Liao ad Peg 8, Li et al. 9, 0, Du ad Che,, ad Huag et al.. Hall 6 derived the asymptotics of distributio of ormalized M t, the powered extremes for give power idex t > 0. Zhou ad Lig 4 improved Hall results ad proved that the covergece speed of distributios ad desities of extremes depeds o the power idex. Nair 7 established the asymptotic expasios of ormalized maximum from ormal samples. Liao et al. 5 ad Jia et al. 6 geeralized Nair s work to skew-ormal distributio ad geeral error distributio, respectively. Sice the Maxwell distributio was proposed by James Clerk Maxwell 7, a variety of applicatios of it i physics i particular i statistical mechaics have bee foud; see Shim ad Gatigol 8, Tomer ad Pawar 9 ad Shim 0 ad some statisticias ad reliability egieers have ivestigated the statistical properties of it as well, see, 7. The aim of this paper is to ivestigate the distributioal tail represetatio of X t with X followig Maxwell distributio ad the limitig distributio of ormalized M t, ad obtai asymptotic expasios of distributio ad desity of powered maximum from Maxwell distributio. Correspodig author. E-mail address: wjj@swu.edu.c J. Wag.
Let X, be a sequece of idepedet idetically distributed i.i.d. radom variables with margial cumulative distributio fuctio cdf F obeyig the Maxwell distributio abbreviated as F MD, ad as before let M = maxx i, i deote the partial maximum of X,. The probability desity fuctio pdf of the MD is defied by x fx = π σ exp x σ, x > 0,. where σ > 0 is the scale parameter. Figure presets the graph of pdf of Maxwell distributio. It shows that with the scale parameter icreasig, the tail of pdf of MD becomes much heavier. 0.6 0.5 σ = σ = σ = 5 σ = 8 0.4 pdf 0. 0. 0. 0 0 5 0 5 0 x Figure : Probability desity fuctio of Maxwell distributio Liu ad Liu showed that F DΛ, i.e., the max-domai of attractio of Gumbel extreme value distributio ad the ormalizig costats a ad b ca be give by a = σ b. ad π σ b exp b σ =. such that lim PM a x + b = Λx = exp exp x..4 The paper is costructed as follows. Sectio presets auxiliary lemmas with proofs. The mai results are give i Sectio. Numerical studies preseted i Sectio 4 compare the precisio of the true values with its approximatios. Sectio 5 provides the proofs of mai results. Auxiliary results To prove the mai results, the followig auxiliary lemmas are eeded. Lemma.. Let F x ad fx respectively represet the cdf ad the pdf of MD with σ > 0, respectively. For large x, we have F x = σ x fx + σ x σ 4 x 4 + σ 6 x 6 + Ox 8..
The proof of Lemma. is derived by itegratio by parts. The followig lemma gives the distributioal tail represetatio of X t with X MD. Lemma.. Suppose that 0 < t. Let F t x deote the cdf of X t with X MD. The for large x, we get x g t u F t x = C t x exp f t u du,. where ad C t x σ π exp σ as x, g t x = σ x /t as x, f t x = σ tx t with f tx 0 as x.. Proof. Combiig with., we get F t x = σ fx t + σ x t σ 4 x 4 t + σ 6 x 6 t + Ox 8 t x t = σ π exp x t x = C t x exp with f t x = σ tx t, g t x = σ x /t ad C t x σ σ + t log x + σ x t σ 4 x 4 t + σ 6 x 6 t + Ox 8 t g t u f t u du + σ x t σ 4 x 4 t + σ 6 x 6 t + Ox 8 t π exp σ as x..4 Applyig the result of Lemma. ad Corollary.7 8, the followig result holds. Propositio.. Uder the coditios of Lemma., we have F t x DΛ, where DΛ is the domai of Λx = exp exp x. The, our aim is to select the suitable ormalizig costats which esure that the distributio of maximum teds to its extreme value limit. A combiatio of. ad.4, we obtai that d = b t. It follows from. that c = f t d = σ tb t t = σ tb t..5 The followig work is to fid the special ormalizig costats c ad d for the case of powered idex t =. Similarly, it is ecessary to establish the distributioal tail represetatio of X with X MD. Lemma.. Assume that t =. Let F x stad for the cdf of X with X MD. The for large x, we get x g u F x = C x exp f u du,.6
where ad C x σ π exp σ g x = + σ4 as x, x f x = σ + σ x as x, with f x 0 as x..7 Proof. Similar to the case of t, we get F x = σ fx x + σ x σ 4 x + σ 6 x + Ox 4 = σ fx + σ x σ 4 x + σ x + σ 6 x + σ x + Ox 4 x = σ π exp = σ π exp x σ + log x + log exp σ + σ σ 4 x + 4σ 6 x + Ox 4 x x g u f u du σ 4 x + 4σ 6 x + Ox 4 with g x = + σ 4 t ad f x = σ + σ t, where the third equality follows from the fact that + x a = + ax + aa /x + Ox for all a R, as x 0..8 Similar to the case of t, we have the followig result: Propositio.. Uder the assumptios of Lemma., we get F x DΛ, where DΛ is the domai of Λx = exp exp x. Now we discuss how to fid the costats c, d. Aalogous to the case of t, we may make choice of d = b ad c = f d = σ + σ b. Ispired by c, ow chage Let d = b + σ 4 b, c = f d = σ + σ b σ 6 b 6 + Ob 0 σ + σ b..9 T x, t = F c x + d /t F c x + d /t. The followig lemmas preset the expasios of the two terms of desities of M t d /c. Lemma.4. For ormalizig costats c ad d determied by.5 ad 0 < t, we have T x, t = Λx A t, xe x b + A t, xe x A t, x e x b 4 + Ob 6.0 4
as, where ad A t, x = σ 4 t x 4 A t, x = σ + x + 8 t x. + 6 t 5 tx x x.. Proof. Let δ x, t = c x + d /t. Oe ca easily see that c x + d > 0 for large ad fixed x R. By., for large, we have b σ log. The, by.5, we have δx, a t = b a + aσ x b + aa tσ4 x b 4 + aa ta tσ6 x 6b 6,. where it follows from the fact that + x a = + ax + aa /x + aa a /6x + Ox 4, for a R, as x 0. The, we get σ fδ x, t a = δ x, t σ π b exp b σ + σ x b b = σ fb e x b tσ x b + σ x + tσ4 x b 4 + tσ4 x b 4 b + σ x c = e x + σ x + σ4 x b 4 b b + tσ4 x b 4 t tσ4 x b 4 + t x + t tσ6 x 6b 6 + t tσ6 x + Ob 6 b 6 + t σ 4 x 4 8b 4 + Ob 6 8 t x + t t 5 tx + + Ob 6 6.4 where a follows from. with a = ad, b is from the fact that e x = + x + x / + Ox, as x 0 ad c is due to.. Furthermore, we get + σ δ a = + σ b = + σ b x, t σ 4 δ 4 x, t + Oδ 6 x, t σ x b + Ob 4 σ4 b 4 + Ob + Ob 6 σ4 b 4 + x + Ob 6,.5 where a is from. with a = ad 4. By Lemma., we get F δ x, t = σ fδ x, t δ x, t a = e x + σ b + σ δ x, t σ 4 δ + x + t x 5 x, t + Oδ 6 x, t
+ σ4 b 4 8 t x 4 + 6 t 5 tx x x + Ob 6 =: e x + A t, xb where a is due to.4 ad.5. Accordigly, + A t, xb 4 + Ob 6,.6 F δ x, t = exp log F δ x, t a = Λx exp A t, xe x b A t, xe x b 4 + Ob 6 b = Λx A t, xe x b + A t, xe x A t, x e x b 4 ad e F δ x, t x = + Ob 6,.7 + Ob = ob η, η 6,.8 where a is from the fact that log x = x + Ox, as x 0, ad b follows from that Taylor s expasio of e x. The desired result follows by.7 ad.8. Lemma.5. For the ormalizig costats c ad d determied by.5 ad 0 < t, we have d dx F c x + d /t = e x + σ x as. + σ4 x b 4 Proof. It is ot hard to check that Therefore, we get d dx F c x + d /t a = σ b b t tx t t + t 6 5 6 t x + 8 t x + Ob 6,.9 d dx F c x + d /t = t c c x + d /t fc x + d /t. π exp b σ b = fb σ e x b tσ x + tσ x b c = e x + σ x b + σ x b b + t tσ4 x b 4 + tσ4 x b 4 + tσ x b t tσ4 x b 4 t tx + Ob 6 + t tσ6 x b 6 + t tσ4 x b 4 + t σ 4 x 4 8b 4 + Ob 6 + Ob 6 6
+ σ4 x b 4 t t + t 6 5 6 t x + 8 t x + Ob 6, where a follows from. with a = t,.5 ad.4 for the expasio of fδ x, t with δ x, t = c x + d /t, b is from the fact that e x = + x + x / + Ox, as x 0 ad c is due to.. The proof is complete. Lemma.6. For the ormalizig costats c ad d determied by.9 ad t =, we have T x, t = Λx B t, xe x b 4 B t, xe x b 6,.0 as, where B t, x = σ 4 x + x +. ad B t, x = σ 6 4 x + x x + 7.. Proof. The proof of the case of t = is similar to the case of 0 < t. Note that c = σ + σ b, d = b + σ 4 b for t =. So, we get The, we have β a = b a δ x, = c x + d / = b + σ b x + σ 4 x + b 4 / =: β. + aσ x b + aσ4 b 4 Further, we get σ fβ a b = β π σ exp + σ x b b σ + x a x a aσ6 x b 6 + x 4 a 6 x.. + σ4 b 4 σ e x b + x x σ6 x b 6 σ6 + x σ + x b + σ4 + x b 4 b = e x σ b σ4 b 4 x x 6b 6 + x x + σ6 4x x x 7 6 b 6,.4 where a is from. with a = ad ad e x = + x + x / + Ox, as x 0, ad b is due to.. Besides, applyig. with a =, 4 ad 6, we get + σ β σ 4 β 4 = + σ b + σ 6 β 6 σ x b σ4 b 4 + Oβ 8 + x x + Ob 6 7
σ 4 b 4 4σ x b + Ob 4 = + σ b σ4 b 4 Combiig with Lemma.,.4 ad.5, we get + σ 6 b 6 + Ob x + + σ6 b 6 4x x +..5 F β = e x σ4 b 4 x + x + + σ6 4 b 4 x + x x + 7 =: e x + B t, xb 4 + B t, xb 6..6 The remaider proof is the same as the case of 0 < t. We omit it. The proof is complete. Lemma.7. For the ormalizig costats c ad d determied by.9 ad t =, we have d dx F c x + d /t = e x σ4 b 4 x x + σ6 4 b 6 x x x +, as. Proof. By.4 ad after observig that c = σ + σ b, we get + σ σ + σ6 4x d dx F β = e x b = e x σ4 b 4 The proof is complete. b x x σ4 b 4 + σ6 x x b 6 b 6 4 x x x + x x 7 6..7 As we metioed i the itroductio, Liu ad Liu obtaied the poitwise covergece rate of distributio of partial maximum to its limitig distributio. Their mai results are stated as follows. Theorem.. Suppose that X, is a sequece of i.i.d. radom variables with cdf MD. The, for large, where â = Mai result F â x + ˆb x log log Λx Λxe,.8 6 log σ log / ad ˆb = σ log / + σ log log + σ log π log /..9 I this sectio, we establish the higher-order expasios of the cdf ad the pdf of powered maximum from MD sample. 8
Theorem.. i For 0 < t ad the ormalizig costats c ad d give by.5, we have P M t c x + d = Λx e x A t, xb + e x e x A t, x A t, x b 4 + Ob 6,. where A t, x = σ + x + t x. ad A t, x = σ 4 8 t x 4 + 6 t 5 tx x x.. ii For t = ad the ormalizig costats c ad d give by.9, we have P M t c x + d = Λx e x B t, xb 4 e x B t, xb 6,.4 where B t, x = σ 4 x + x +.5 ad B t, x = σ 6 4 x + x x + 7..6 Remark.. From Theorem., oe ca easily see that the covergece rates of powered maximum of cdf for MD are proportioal to / log ad /log for power idex 0 < t ad t =, respectively, sice /b σ log by.. Remark.. From Theorems. ad. ii, we ca observe that the covergece speed of powered extreme of cdf for MD is better tha that of extreme of cdf. I the followig we provide the higher-order expasios of the pdf of powered maximum. Theorem.. i For 0 < t ad the ormalizig costats c ad d give by.5, we have where d dx P M t c x + d = Λ x + P t, xb + P t, xb 4 + Ob 6,.7 t x P t, x = σ + x + e x + t x t x ad P t, x =σ 4 t x + x + e x 9
5t x 4 5 t 8 + t x 8 6 t 0 5 t 6 t 6 x + t + x e x x + t t x. ii For t = ad the ormalizig costats c ad d give by.9, we have where ad d dx P M t c x + d = Λ x + Q t, xb 4 + Q t, xb 6,.8 Q t, x = σ 4 x + x + e x x + x + Q t, x = σ 6 4 x + x x + 7 e x 4 x + x + x. Remark.. From Theorem., it is ot difficult to observe that the covergece speeds of powered extreme of pdf for MD are the same order of / log ad /log for power idex 0 < t ad t =, respectively, because of /b σ log by.. Remark.4. For t =, the ormalizig costats c ad d are ot give by.9, but we choose them as follows: the we derive c = σ σ b ad d = b σ 4 b,.9 P M t c x + d =Λx e x σ x + + e x σ 4 e x σ 6 4 b 6 b + x + x + x + 4 b 4 e x x + e x x + e x x + x x b 4 x + x +.0 ad d dx P M t c x + d = Λ x σ b e x x + x + σ4 b 4 + x x + + σ6 b 6 e x x + 5x + 5x + e x 4xx + e x 4x + x + x + e x + x x 7 6.. Obviously, the covergece rates of the cdf ad the pdf of powered extreme give by.4 ad.8, which are proportioal to /log, are faster tha that give by.0 ad.. Cosequetly, the ormalizig costats c ad d determied by.9 are optimal. 0
4 Numerical aalysis I this sectio, we coduct umerical studies to illustrate the accurateess of higher-order expasios for the cdf ad the pdf of M t. Let T i x ad S i x, i =,,, respectively represet the first-order, the secod-order ad the third-order approximatios of the cdf ad the pdf of M t. Sice the aalysis of the case of t is similar to that of t =, we oly cosider the situatio of t =. By Theorems. ad., we obtai ad T x = Λx, T x = Λx e x B t, xb 4, T x = Λx e x B t, xb 4 e x B t, xb 6, S x = Λx exp x, S x = Λx exp x + Q t, xb 4, S x = Λx exp x + Q t, xb 4 + Q t, xb 6. Easily observe that the secod-order approximatio ad the third-order relate to the sample size. I order to compare the precisio of true values with its approximatios, let E i x = F c x + d T i x ad G i x = c F c x + d f c x + d S i x c x + d respectively stad for the absolute errors of the cdf ad the pdf, where i =,,. We utilize MATLAB to compute the approximatios ad the true values of the cdf ad the pdf of M. First, we estimate the absolute errors of the cdf of M at x = 0.7, where the sample size varies from 5 to 000 with step size 5. For give x = 0.7, umerical aalysis results of E i x are recorded i Table 4. The table demostrates that the precisio of all three kids of approximatios of the cdf ca be refied as the sample size icreases. To order to idicate the precisio of all approximatios more ituitive with the chage of the sample size, the actual values ad its approximatio of the cdf of M are plotted versus the values of with x =.5. Figure evideces that the larger, the better all asymptotics. Secodly, we estimate the absolute errors of the pdf of M at x = 0.7, where the value of the sample size rages from 75 to 5000 with step legth 75. Table 4 lists the umerical aalysis results of G i x, where i =,,. Table 4 reveals that the precisio of all three kids of approximatios of the pdf ehaces as the sample size grows. To clear the precisio of all approximatios more ituitive with, the actual ad its approximatios of the pdf of M are plotted versus the values of with x =.5. Figure idicates that as the sample size becomes larger, all approximatios become better.
E x E x E x 5 0.0690569 0.008774565 0.0075947 50 0.0457459 0.0086906808 0.0078589009 75 0.04677 0.0084469 0.007800784 00 0.0958 0.008865886 0.0076896494 5 0.086684 0.0080447997 0.007579459 50 0.04684 0.0078976489 0.007478856 75 0.05009 0.0077754585 0.00788468 00 0.0088746 0.007665944 0.00707058 5 0.00664804 0.0075705 0.0074689 50 0.0047474 0.00748677 0.007665087 75 0.000764 0.00740697 0.007059645 00 0.0058089 0.00744685 0.00704874 5 0.000998 0.00778407 0.0069964575 50 0.00989766 0.007978455 0.006947848 75 0.0097868048 0.0076578 0.00690464 400 0.0096840969 0.0075658 0.006859889 45 0.0095898478 0.007068889 0.0068975869 450 0.00950867 0.0070409995 0.006789964 475 0.0094088789 0.00698480 0.0067464458 500 0.009446449 0.0069468 0.006778 55 0.00970706 0.006905958 0.0066798748 550 0.009056689 0.0068705584 0.0066490777 575 0.009409 0.0068689469 0.0066966674 600 0.00908746 0.00680478587 0.00659565 65 0.00904547 0.00677405 0.0065645566 650 0.00897056 0.0067447575 0.006586679 675 0.0089885 0.0067655745 0.006577957 700 0.0088687746 0.0066895057 0.00648985 75 0.0088408 0.0066648955 0.006466754 750 0.0087760598 0.00668467 0.00644444946 775 0.0087597 0.00664808 0.0064987 800 0.008690794 0.006590968 0.00640079 85 0.0086505088 0.00656849 0.006895685 850 0.0086777 0.006546665 0.0066486 875 0.008574495 0.0065556 0.00644995 900 0.00858648 0.0065050869 0.00655 95 0.0085056 0.006485447 0.00607744 950 0.00846987 0.006465988 0.00689998 975 0.008479 0.006447809 0.006704687 000 0.0084056099 0.00649098 0.006566887 Table : Absolute errors betwee actual values ad their asymptotics of the cdf at x = 0.7 with σ =
G x G x G x 75 0.00856746 0.005859446 0.00554667797 750 0.0070098 0.005405507 0.004946905 5 0.006558405 0.004797558 0.0046054464 500 0.0060457 0.00457905959 0.00440749 875 0.00596478 0.00445796 0.00467709 50 0.005776798 0.00499798 0.0045766 65 0.005648995 0.0040856 0.0040608 000 0.0054990795 0.004887475 0.00985069 75 0.0059866 0.004048464 0.0099986 750 0.005989067 0.0098706069 0.008605898 45 0.0057070 0.00989 0.008750 4500 0.005445 0.008845007 0.007676576 4875 0.00507859 0.0084058 0.00769095 550 0.005089706 0.008007480 0.0068978076 565 0.004964549 0.00764409 0.00655747 6000 0.004944645 0.00705877 0.0064765 675 0.00486868 0.00699856 0.005950557 6750 0.00485485 0.00670886 0.0056798 75 0.0047850008 0.0064886 0.005448964 7500 0.0047474097 0.0067709 0.005860668 7875 0.0047085 0.0059744 0.00496075 850 0.0046787885 0.0057066 0.004748504 865 0.004647084 0.00549648 0.0045474 9000 0.00467484 0.00597666 0.0045595 975 0.0045895666 0.00509969 0.0047987 9750 0.004569575 0.00494765 0.0040004057 05 0.004564847 0.004786 0.0084654 0500 0.00459 0.00457005 0.00675949 0875 0.0044884509 0.004408694 0.0058574 50 0.00446588547 0.00457844 0.00778509 65 0.0044446057 0.00404988 0.007497 000 0.00444759 0.0096706 0.00086 75 0.00440475 0.0087 0.0097797 750 0.004840096 0.006906445 0.00846556 5 0.004656084 0.005654 0.007509 500 0.0044766007 0.00479647 0.00607899 875 0.0040094 0.00788 0.0049456 450 0.00450 0.0004766 0.0084795 465 0.004976807 0.0008879 0.00777608 5000 0.00485447 0.0097959 0.00748 Table : Absolute errors betwee actual values ad their asymptotics of the pdf at x = 0.7 with σ =
0.8 0.8 0.798 0.798 0.796 0.794 actual values first-order asymptotics secod-order asymptotics third-order asymptotics 0.796 0.794 actual values first-order asymptotics secod-order asymptotics third-order asymptotics 0.79 0 000 4000 6000 8000 0000 a σ = 0.79 0 000 4000 6000 8000 0000 b σ = 0.5 Figure : Actual values ad its asymptotics of the cdf of M with x =.5. The actual values draw i black, the first-order approximatios draw i blue, the secod-order approximatios draw i red ad the third-order approximatio draw i gree. 0.86 actual values first-order asymptotics secod-order asymptotics third-order asymptotics 0.86 actual values first-order asymptotics secod-order asymptotics third-order asymptotics 0.84 0.84 0.8 0.8 0.8 0.8 0.78 0 000 4000 6000 8000 0000 a σ = 0.78 0 000 4000 6000 8000 0000 b σ = 0.5 Figure : Actual values ad its asymptotics of the pdf of M with x =.5. The actual values draw i black, the first-order approximatios draw i blue, the secod-order approximatios draw i red ad the third-order approximatio draw i gree. 5 Proof of mai result Proof of Theorem.. By some fudametal calculatios, we get P M t c x + d = F c x + d /t F c x + d /t. 5. First, we cosider the case of 0 < t. By.6 ad similar discussios as for.7 ad.8, we get F δ x, t = Λx A t, xe x b + A t, xe x A t, x e x b 4 + Ob 6, 5. where A t, x ad A t, x are determied by. ad., ad e F δ x, t x = + Ob = ob η, η 6. 5. 4
A combiatio of 5. ad 5. implies that. holds. For the case of t =, by similar argumets as for 0 < t, the desired result follows. The proof is complete. Proof of Theorem.. Oe ca easily check that d d dx P M t c x + d = dx F c x + d /t F c x + d /t + F c x + d /t. 5.4 For 0 < t, combiig with Lemmas.4 ad.5, we get d Λ x dx P M t c x + d = + σ x b t tx + σ4 x b 4 t t + t 6 5 6 t x + 8 t x σ + x + t x e x b + 8 t x 4 + 6 t 5 tx x x σ 4 e x b 4 = σ t x b + x + e x + t x t x + σ4 t x b 4 + x + e x 5t x 4 5 t 8 6 t 0 x + t + x e x + t x 8 =P t, xb which deduces.7. 5 t 6 t 6 + P t, xb 4 + Ob 6, x + + Ob 6 + x + t x e x t t x + Ob 6 + Ob 6 The followig is for the case of t =. By 5.4 ad Lemmas.6 ad.7, we gai d Λ x dx P M t c x + d = σ4 b 4 x x + σ6 4 b 6 x x x + + σ4 e x b 4 x + x + σ6 e x 4 b 6 x + x x + 7 = σ4 b 4 x + x + e x x + x + σ6 b 6 =Q t, xb 4 4 x + x x + 7 e x 4 x + x + x + Q t, xb 6, 5
which proves.8. The proof of Theorem. is fiished. Ackowledgmets Fudig This work was supported by Natural Sciece Foudatio of Chia grat umber 66705, grat umber 6700 ad Fudametal Research Fuds for the Cetral Uiversities grat umber XDJK05A007, Youth Sciece ad techology talet developmet project No.Qia jiao he KY zi 08, Sciece ad techology Foudatio of Guizhou provice grat umber Qia ke he Ji Chu 066, Guizhou provice atural sciece foudatio i Chia grat umber Qia Jiao He KY 0655. Refereces Smith, LR: Uiform rates of covergece i extreme-value theory. Adv. Appl. Probab. 4, 600-6 98 Leadbetter, MR, Lidgre, G, Rootzé, H: Extremes ad Related Properties of Radom Sequeces ad Processes. Spriger, New York 98 Galambos, J. 987. The asympotic theory of extreme order statistics. Secod Editio New York, Wiley. 4 de Haa, L, Resick, SI: Secod-order regular variatio ad rates of covergece i extreme-value theory. A. Probab., 97-4 996 5 Hall, P., 979. O the rate of covergece of ormal extremes. J. Appl. Probab. 6, 4-49. 6 Hall, P., 980. Estimatig probabilities for ormal extremes. Adv. Appl. Probab., 49-500. 7 Nair, K. A. 98. Asymptotic distributio ad momets of ormal extremes. Aals of Probability, 9, 50-5. 8 Liao, X., & Peg, Z. 0. Covergece rates of limit distributio of maxima of logormal samples. Joural of Mathematical Aalysis ad Applicatios, 95, 64-65. 9 Li, F., Zhag, X., Peg, Z., & Jiag, Y. 0. O the rate of covergece of stsd extremes. Commuicatios i Statistics - Theory ad Methods, 400, 795-806. 0 Li, F., Peg, Z., & Yu, K. 06. Covergece rate of extremes for the geeralized short-tailed symmetric distributio. Bulleti of the Korea Mathematical Society, 55, 549-566. Du, L., & Che, S. 06. Asymptotic properties for distributios ad desities of extremes from geeralized gamma distributio. Joural of the Korea Statistical Society, 45, 88-98. Che, S., & Du, L. 07. Asymptotic expasios of desity of ormalized extremes from logarithmic geeral error distributio. Commuicatios i Statistics, 467, 459-478. Huag, J., Wag, J., & Luo, G. 07a. O the rate of covergece of maxima for the geeralized maxwell distributio. Statistics A Joural of Theoretical & Applied Statistics, -. 4 Zhou, W., Lig, C. 06. Higher-order expasios of powered extremes of ormal samples. Statistics ad Probability Letters,, -7. 6
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