Tail properties and asymptotic distribution for maximum of LGMD

Transcription

1 Tail properties asymptotic distribution for maximum of LGMD a Jianwen Huang, b Shouquan Chen a School of Mathematics Computational Science, Zunyi Normal College, Zunyi, 56300, China b School of Mathematics Statistics, Southwest University, Chongqing, 40075, China Abstract. We introduce logarithmic generalized Maxwell distribution which is an extension of the generalized Maxwell distribution. Some interesting properties of this distribution are studied the asymptotic distribution of the partial maximum of an independent identically distributed sequence from the logarithmic generalized Maxwell distribution is gained. Keywords. Extreme value distribution; Logarithmic generalized Maxwell distribution; Mills ratio; Partial Maximum; Tail Properties. Mathematics Subject Classification: Primary 6E0, 60E05; Secondary 60F5, 60G5. Introduction The generalized Maxwell distribution GMD for short, a generalization of ordinary Maxwell distribution, is proposed by Vodă 009. The GMD has a variety of applications in statistics, physics chemistry. Its most common application is in statistical mechanics. Some recent examples of this have: constructing fractional rheological constitutive equations Schiessel et al., 995; be friction model suitable for quick simulation control Farid et al., 005; forecasting the temporal change of opening angle in multiple time scales electroscalar wave Zhang et al., 008; Arbab Satti, 009; project of the time related to behavior of viscoelastic materials Monsia, 0. The probability density function pdf the cumulative distribution function cdf of the GMD with the parameter k > 0 are respectively, k g k x = k/ σ +/k Γ + k/ xk xk σ G k x = x g k t dt for x > 0, where σ is a positive constant Γ is the Gamma function. Corresponding author. address: hjw @6.com J. Huang.

2 Mills 96 gave a well-known inequality Mills ratio result for the stard normal cdf Φx with pdf φx as follows: for x > 0 as x. x + x φx < Φ x < x φx,. Φ x φx x,. Peng et al. 009 extended the Mills results to the case of the general error distribution: λ v v x v + for v > x > 0, for v > 0 v λv x v < T v x < λv v t v x v x v,.3 T v x t v x λv v x v,.4 [ ] / as x, where λ = /v Γ/v Γ3/v, T v x is the cdf with pdf t v x. Huang Chen 04 investigated the similar results of GMD, i.e., σ k x k < G k x < σ g k x k x k for k > /, σ > 0 x > 0, for k > 0, σ + k xk,.5 G k x g k x σ k x k,.6 as x. The above-mentioned inequalities such as.,.3,.5 Mills type-ratios such as.,.4,.6 play an important role in considering some tail behavior extremes of economic financial data. In this paper, we define the logarithmic generalized Maxwell distribution denoted by LGMD, which is a natural extension of the generalized Maxwell distribution. One motivation of considering LGMD is to obtain more efficient results as parameter estimators when rom models were supposed with the LGMD error terms instead of normal ones. Meanwhile, the LGMD can be ected to be a better model for certain modern areas. The present paper is to derive the Mills-type inequality, Mills-type ratio, the tail distributional representation for the LGMD. As an important application, the asymptotic distribution of the partial maximum of independent identically distributed i.i.d. with common LGMD is investigated. First we provide the definition of LGMD. Definition.. Let X denote a rom variable which obeys the GMD. Set Y = X. Then we call that Y obeys the LGMD, denoted by Y LGMDk with parameter k > 0.

3 Easily check that the pdf of Y LGMDk is f k x = kx k/ σ +/k Γ + k/ log log xk xk σ for x > 0, where parameter k > 0, σ is a positive constant. Let F k x denote the cdf of Y, i.e., F k x = x 0 f k t dt for x > 0. Note that the LGMD reduces to the logarithmic Maxwell distribution when k =. The rest of the article is organized as follows. In Sec., we derive some results concerning Millstype ratios tail behavior of LGMD. Sec. 3, we consider the it distribution of the partial maximum of i.i.d rom variables following the LGMD the suitable norming constants needed. The result is also extended to the case of a finite mixture of LGMDs. Mills Ratio Tail Properties of LGMD In this section, we derive some results including Mills inequality, Mills ratio of LGMD. For LGMD GMD, note that G k log x = F k x G k log x x g k log x = F kx. f k x So, by Lemma. Theorem. in Huang Chen 04, we have the following two results. Theorem.. Let F k f k respectively denote the cdf pdf of LGMD with parameter k > /. We have the inequality below for all x >, σ k xlog x k < F kx < σ σ xlog x k + f k x k k log xk,. where σ is a positive constant. Corollary.. For fixed k > 0, as x, we have F k x f k x σ k xlog x k.. Remark.. Since the LGMDk are reduced to the logarithmic Maxwell distribution as k =, so by Theorem. Corollary., we derive the inequality Mills ratio of the logarithmic Maxwell distribution, i.e., σ xlog x f x < F x < σ xlog x + σ log x f x, for x > as x. F x f x σ x log x, 3

4 Remark.. For k > /, Corollary. gives F k DΛ, i.e., there exist norming constants α n > 0 β n R which make sure Fk nα nx + β n converges to x. Since d/dxf k x = k f k x x log x + k log xk, σ by Corollary., we have F k x d/dxf k x f k x f k x as x. Hence, it follows by Proposition.8 in Resnick 987 that F k DΛ. The choice of norming constants α n β n is discussed by Theorem 3.. Finner et al. 008 investigated the asymptotic behavior of the ratio of the Student s t normal distributions as the degrees of freedom u = ux satisfies x 4 x u = β [0,..3 The main motivation of the work is to consider the false discovery rate in multiple testing problems with large numbers of hypotheses extremely small critical values for the smallest ordered p value; in detail, see Finner et al In this section, we study the asymptotic behavior of the ratio of pdfs the ratio of the tails of the LGMD the logarithmic Maxwell distribution. Firstly, we consider the case of k. Secondly, we consider the case of x for fixed k. Theorem.. For k > 0, let x = xk be such that for some γ R. Then Proof. Note that k+/ σ +/k Γ+k/ kσ 3 π / f x k f k x = k = γ log x log log x f x γ k f k x = σ.4 F x γ k F k x = σ..5 as k, so log x k log log k x k x σ σ log x k σ log x k = log x = k σ k log log x log x γ = k σ log x γ = σ. The condition of the theorem deduces log x k as k. According to Corollary., Remark..4,.5 can be deduced. 4

5 Theorem.3. For fixed k, we have f x f k log x /k = k+/ Γ + k/ log x /k π / kσ /k x.6 log x /k F x x F k log x /k = k+/ Γ + k/ π / σ /k..7 Proof. Note that.6 follows from fundamental calculation. By Corollary., Remark..6, we have so.7 follows. log x /k F x x F k log x /k = x kx f x log x /k f k log x /k = k+/ Γ + k/ π / σ /k, 3 Asymptotic Distribution of the Maximum By applying Corollary., we could establish the distributional tail representation for the LGMD. Theorem 3.. Under the conditions of Theorem., we have x gt F k x = cx ft dt for large enough x, where cx = k/ σ /k Γ + k/ /σ + θ x ft = σ k tlog t k, gt = σ k log t k, where θ x 0 as x. Proof. For large enough x, by Corollary., we have F k x = σ k log x k xf k x + θ x = log k/ σ /k Γ + k/ log x = σ k/ σ /k Γ + k/ 5 e log xk σ x e + θ x klog t k σ t dt + θ x t log t

6 = k/ σ /k Γ + k/ x gt = cx ft dt, e σ where θ x 0 as x. The desired result follows. x k σ log t k e k σ tlog t k dt + θ x Remark 3.. As t gt =, ft > 0 on [, is absolutely continuous function t f t = 0 in Theorem 3., an application of Theorem 3. Corollary.7 in Resnick 987 shows F k DΛ, where DΛ denotes the domain of attraction Λx = x. In this we consider the asymptotic distribution of the normalized maximum of a sequence of i.i.d. rom variables following LGMD. Remark. Theorem 3. showed that the distribution of partial maximum converges to Λx. So, the following work is to find the suitable norming constants. Theorem 3.. Let {X n, n } be an i.i.d. sequence from the LGMD with k > /. Let M n = max{x k, k n}. We have where P M n α n x + β n = x, n α n = k σ /k σ /k log n /k + σ/k log n /k log log n k log k log Γ + k/ /k k /k σ /k log n /k + log + σ/k log n /k log log n k /k k log k log Γ + k/ k β n = /k σ /k log n /k + σ/k log n /k log log n k /k k log k log Γ + k/. Proof. Since F k DΛ, there must be norming constants a n > 0 b n R which make sure that n P M n b n /a n x = x. By Proposition. in Resnick 987 Theorem 3., the norming constants can be chosen that a n b n satisfy the equations: b n = / F k n a n = fb n. Note that F k x is continuous, then F k b n = n. By Corollary., we have nk σ log b n k b n f k b n, as n, i.e., n k σ k Γ + k log b n log b n k σ, as n, so log n k log k log σ log Γ + k + log log b n log b n k σ 0, 3. as n, which deduces log b n k σ log n, 6

7 as n, thus k log log b n log log σ log log n 0, as n, hence log log b n = log + log σ + log log n + o. k Putting the above equality into 3., we have which deduces that log b n k = σ log n + k log log n k log log Γ + k + o, k log b n = k σ k log n k therefore b n = k σ k log n k + σ k log n k + o log n k + log log n k log k log Γ + k k log n k k =β n + o log n k k σ k log n k, + o log n, log log n k log k log Γ + k where β n = /k σ /k log n /k + σ/k log n /k /k k log log n k log k log Γ + k/. Hence, we have α n = fβ n σ /k σ /k log n /k + σ/k log n /k log log n k log k log Γ + k/ = /k k k /k σ /k log n /k + log + σ/k log n /k k. log log n k /k k log k log Γ + k/ It is easy to check that n α n /a n = n b n β n /α n = 0. Hence, by Theorem..3 in Leadbetter et al. 983, the result follows. Remark 3.. Theorem 3. shows that the it distribution of the normalized maximum from the logarithmic Maxwell distribution is the Gumbel extreme value distribution with norming constants σ / σlog n / + α n = / σlog n / + log σ 3/ log n log log n logπ / / / + σ 3/ log n / log log n logπ / / β n = / σlog n / σ + 3/ log n / log log n logπ / /. 7

8 At the end of this section, we extend the result of Theorem 3. to the case of a finite mixture of LGMDs. Finite mixture distributions or models have been widely applied in various areas like Chemistry Roeder, 994 image video databases Yang Ahuja, 998. Recently, some extreme statistical scholars have also studied them. Mladenović 999 have considered extreme values of the sequences of independent rom variables with common mixed distributions containing normal, Cauchy uniform distributions. Peng et al. 00 have investigated the it distribution its corresponding uniform convergence rate for a finite mixed of onential distribution. If the distribution function df F of a rom variable ξ have F x = p F x + p F x + + p r F r x, we say that ξ obeys a finite mixture distribution F, where F i, i r denote different dfs of the mixture components. The weight coefficients have the condition that p i > 0, i =,,, r r j= p j =. Next, we consider the extreme value distribution from a finite mixture with component dfs F ki obeying LGMDk i, where the parameter k i > for i r k i k j for i j. Denote the df of the finite mixture by for x > 0. F x = p F k x + p F k x + + p r F kr x 3. Theorem 3.3. Let {Z n, n } be a sequence of i.i.d. rom variables following the common df F given by 3.. Let M n = max{z, Z,, Z n }. Then P Mn β n x = x n holds with the norming constants α n α n = σ/k /k σ /k log n /k /k klog n /k β n = /k σ /k log n /k + σ/k log n /k /k k log log n + k log p k log k log Γ + k/, where σ = max{σ,, σ r }, p = p i + +p ij, where i s {i, σ i = σ k = k i }, s j r k = min{k,, k r }. Proof. By 3., we have By Theorem., we have r i= p i σ i k i F x = r p i F ki x. i= log x k i xf ki x < F x 8

9 < r i= p i σ i k i log x k i x + σ i log x k i f ki x k i for all x >, according to the definition of f k, which implies where B k x = k i k p log x k σ k Γ + k p log x < k σ k Γ + k log xk σ + A k x < F x + σ k log xk A k x = k p i σ k Γ + k log x k k i k k i pσ k i Γ + k i σ k p i σ k Γ + k k i pσ k i Γ + k i + σ i k i log x k i + σ k log xk log xk σ + B k x, 3.3 log x k σ log xki σi log xki σi as x since k = min{k, k,, k r }. Combining together, for large enough x, we obtain F x p F k x 3.6 as x, where F k denotes the cdf of the LGMDk, σ p are defined by Theorem 3.3. By Proposition.9 in Resnick 987, we can derive F DΛ. The norming constants can be obtained by Theorem The proof is complete. Funding This work was supported by the National Natural Science Foundation of China Grant no. 775 the SWU grant for Statistics Ph.D. References [] Arbab, A. I., Satti, Z. A On the generalized Maxwell equations their prediction of electroscalar wave. Prog. Phys. : 8-3. [] Farid, A., Vincent, L., Jan, S The generalized Maxwell-Slip model: a novel model for friction simulation compensation. IEEE T. Automat. Contr. 50: [3] Finner, H., Dickhaus, T., Roters, M Asymptotic tail properties of Students t-distribution. Commun. Stat. Theor. Methods 37: [4] Finner, H., Dickhaus, T., Roters, M Dependency false discovery rate: asymptotics. Ann. Statist. 35: [5] Huang, J., Chen, S. 04. Tail behavior of the generalized Maxwell distribution. Commun. Stat. Theor. Methods Accepted 9

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