Convergence rate of extremes from Maxwell sample
© Liu and Liu; licensee Springer. 2013
Received: 30 May 2013
Accepted: 1 October 2013
Published: 7 November 2013
For the partial maximum from a sequence of independent and identically distributed random variables with Maxwell distribution, we establish the uniform convergence rate of its distribution to the extreme value distribution.
MSC:62E20, 60E05, 60F15, 60G15.
One interesting problem in extreme value theory is to consider the convergence rate of some extremes. For the uniform convergence rate of extremes under the second-order regular variation conditions, see Falk , Balkema and de Haan , de Haan and Resnick  and Cheng and Jiang . For the extreme value distributions and their associated uniform convergence rates for given distributions, see Hall and Wellner , Hall , Peng et al. , Lin and Peng  and Lin et al. .
The MD and the convergence rate of extremes from Maxwell sample have been widely used in the field of physics. We establish the uniform convergence rate of its distribution to the extreme value distribution and give an improved proof for the pointwise convergence rate of MD.
This paper is organized as follows. Section 2 gives some auxiliary results. In Section 3, we present the main result. Related proofs are given in Section 4.
2 Auxiliary results
To establish the uniform convergence of to its extreme value distribution , we need some auxiliary results. The first result is the decomposition of , which is the following result.
For simplicity, throughout this paper, let C be a generic positive constant whose value may change from line to line, and let , (, ) be absolute positive constants.
where , , . So, , , implying . For large n, we have the following result.
which is the desired result. □
3 Main results
In this section we present the pointwise convergence rate and the uniform convergence rate of to its extreme value distribution under different normalizing constants. The first result is the pointwise convergence of extremes under the normalizing constants given by (1.2).
for large n, where , are defined in (1.2).
Recently Liu and Fu  proved the result, we present an improved proof for the pointwise convergence rate in Section 4.
The following is the uniform convergence rate of extremes under the appropriate normalizing constants and given by (1.3) and (1.4), which shows that the optimal convergence rate is proportional to .
where and are defined by (1.3) and (1.4), respectively.
where . By similar arguments, we can obtain (4.2).
The proof is complete. □
Together with (4.12), we establish (4.7).
Hence (4.8) is proved.
which is (4.9).
This is (4.10). The proof is complete. □
The research was supported by the National Natural Science Foundation of China (11171275) and the Fundamental Research Funds for the Central Universities (XDJK2012C045).
- Falk A: Rate of uniform convergence of extreme order statistics. Ann. Inst. Stat. Math. 1986, 38(2):245–262.MathSciNetView ArticleMATHGoogle Scholar
- Balkema AA, de Haan L: A convergence rate in extreme-value theory. J. Appl. Probab. 1990, 27: 577–585. 10.2307/3214542MathSciNetView ArticleMATHGoogle Scholar
- de Haan L, Resnick SI: Second order regular variation and rates of convergence in extreme value theory. Ann. Probab. 1996, 24: 97–124.MathSciNetView ArticleMATHGoogle Scholar
- Cheng S, Jiang C: The Edgeworth expansion for distributions of extreme values. Sci. China Ser. A 2001, 4: 427–437.MathSciNetView ArticleMATHGoogle Scholar
- Hall WJ, Wellner JA: The rate of convergence in law of the maximum of an exponential sample. Stat. Neerl. 1979, 33: 151–154. 10.1111/j.1467-9574.1979.tb00671.xMathSciNetView ArticleMATHGoogle Scholar
- Hall P: On the rate of convergence of normal extremes. J. Appl. Probab. 1979, 16: 433–439. 10.2307/3212912MathSciNetView ArticleMATHGoogle Scholar
- Peng Z, Nadarajah S, Lin F: Convergence rate of extremes for the general error distribution. J. Appl. Probab. 2010, 47: 668–679. 10.1239/jap/1285335402MathSciNetView ArticleMATHGoogle Scholar
- Lin F, Peng Z: Tail behavior and extremes of short-tailed symmetric distribution. Commun. Stat., Theory Methods 2010, 39: 2811–2817. 10.1080/03610920903132087MathSciNetView ArticleMATHGoogle Scholar
- Lin F, Zhang X, Peng Z, Jiang Y: On the rate of convergence of STSD extremes. Commun. Stat., Theory Methods 2011, 40: 1795–1806. 10.1080/03610921003695726MathSciNetView ArticleMATHGoogle Scholar
- Liu B, Fu Y: The pointwise rate of extremes for Maxwell distribution. J. Southwest Univ., Nat. Sci. 2013, 5: 86–99.Google Scholar
- Resnick SI: Extreme Values, Regular Variation and Point Processes. Springer, New York; 1987.View ArticleMATHGoogle Scholar
- Leadbetter MR, Lindgren G, Rootzen H: Extremes and Related Properties of Random Sequences and Processes. Springer, New York; 1983.View ArticleMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.