On the distributional expansions of powered extremes from Maxwell distribution

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1 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. address: wjj@swu.edu.c J. Wag.

2 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 σ = σ = σ = 5 σ = pdf 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..

3 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

4 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

5 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 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

6 + σ4 b 4 8 t x 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 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

7 + σ4 x b 4 t t + t 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 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

8 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 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

9 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 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 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

10 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 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

11 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.

12 E x E x E x Table : Absolute errors betwee actual values ad their asymptotics of the cdf at x = 0.7 with σ =

13 G x G x G x Table : Absolute errors betwee actual values ad their asymptotics of the pdf at x = 0.7 with σ =

14 actual values first-order asymptotics secod-order asymptotics third-order asymptotics actual values first-order asymptotics secod-order asymptotics third-order asymptotics a σ = 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 actual values first-order asymptotics secod-order asymptotics third-order asymptotics 0.86 actual values first-order asymptotics secod-order asymptotics third-order asymptotics a σ = 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

15 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 t x + 8 t x σ + x + t x e x b + 8 t x 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

16 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 Refereces Smith, LR: Uiform rates of covergece i extreme-value theory. Adv. Appl. Probab. 4, Leadbetter, MR, Lidgre, G, Rootzé, H: Extremes ad Related Properties of Radom Sequeces ad Processes. Spriger, New York 98 Galambos, J 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., Hall, P., 979. O the rate of covergece of ormal extremes. J. Appl. Probab. 6, Hall, P., 980. Estimatig probabilities for ormal extremes. Adv. Appl. Probab., Nair, K. A. 98. Asymptotic distributio ad momets of ormal extremes. Aals of Probability, 9, Liao, X., & Peg, Z. 0. Covergece rates of limit distributio of maxima of logormal samples. Joural of Mathematical Aalysis ad Applicatios, 95, Li, F., Zhag, X., Peg, Z., & Jiag, Y. 0. O the rate of covergece of stsd extremes. Commuicatios i Statistics - Theory ad Methods, 400, 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, Du, L., & Che, S. 06. Asymptotic properties for distributios ad desities of extremes from geeralized gamma distributio. Joural of the Korea Statistical Society, 45, Che, S., & Du, L. 07. Asymptotic expasios of desity of ormalized extremes from logarithmic geeral error distributio. Commuicatios i Statistics, 467, 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

17 5 Liao, X, Peg, Z, Nadarajah, S: Asymptotic expasios of the momets of skew-ormal extremes. Stat. Probab. Lett. 8, Jia, P, Liao, X, Peg, Z: Asymptotic expasios of the momets of extremes from geeral error distributio. J. Math. Aal. Appl. 4, Madl, F Statistical Physics d Editio. New Jersey: Joh Wiley & Sos. 8 Shim, J. W., & Gatigol, R. 0. How to obtai higher-order multivariate hermite expasio of maxwellcboltzma distributio by usig taylor expasio? Zeitschrift Fr Agewadte Mathematik Ud Physik, 64, Tomer, S.K. ad Pawar, M.S. 05. Estimatio procedures for Maxwell distributio uder type-i progressive hybrid cesorig scheme. Joural of Statistical Computatio ad Simulatio, 85, Shim, J. W. 07. Parametric lattice boltzma method. Joural of Computatioal Physics, 8. Liu, C., Liu, B. 0. Covergece rate of extremes from Maxwell sample. J. Iequal. Appl. 0: 477. Available at: Huag, J., Che, S. Tail behavior of the geeralized Maxwell distributio. Commuicatios i Statistics-Theory ad Methods. 06, 454: Dar, A.A., A. Ahmed ad J.A. Reshi. Bayesia aalysis of Maxwell-Boltzma distributio uder differet loss fuctios ad prior distributios. Pak. J. Statist. 07, 6, Huag, J., Wag, J., Luo, G., & He, J. Tail properties ad approximate distributio ad expasio for extreme of LGMD. Joural of Iequalities & Applicatios, 07b, 07:-6. 5 Huag, J., Wag, J., Luo, G. Pu, H. Higher order expasio for momets of extreme for geeralized Maxwell distributio, Commuicatios i Statistics-Theory ad Methods, 08, 474: Huag J.W., Wag J.J. O asymptotic of extremes from geeralized Maxwell distributio. Bulleti of the Korea Mathematical Society, 08a, 55: Huag J.W., Wag J.J. Higher order asymptotic behaviour of partial maxima of radom sample from geeralized Maxwell distributio uder power ormalizatio. Applied Mathematics-A Joural of Chiese Uiversities, 08b,: Resick, S.I Extreme Value, Regular Variatio, ad Poit Processes. New York: Spriger. 7