Higher-order expansions for distributions of extremes from general error distribution
© Jia and Li; licensee Springer. 2014
Received: 2 January 2014
Accepted: 14 May 2014
Published: 27 May 2014
In this short note, with optimal normalizing constants, the higher-order expansion for a distribution of normalized partial maximum from the general error distribution is derived, by which one deduces the associate convergence rate of the distribution of the extreme to the Gumbel extreme value distribution.
MSC:62E20, 60E05, 60F15, 60G15.
where is the shape parameter, and denotes the Gamma function (Nelson ). Note that reduces to the standard normal distribution.
i.e., is in the domain of attraction of Λ, which we denote by . For the uniform convergence rate of normalized maxima from the , Hall  established the optimal uniform convergence rate as , i.e., the normal case; Peng et al.  extended the result to the case of . Both studies show that the optimal convergence rate of extremes from the is proportional to .
In this short note, the aim is to establish a higher-order expansion for the distribution of normalized maxima from the for . For some recent related work on uniform convergence rates and higher-order expansions of extremes for given distributions, see Liao and Peng  for the log-normal distribution, and Liao et al. [9, 10] for skew distributions.
This paper is organized as follows. Section 2 provides the main results. Some auxiliary results and the proofs of the main results are given in Section 3.
2 Main result
In this section, we provide asymptotic expansions of a distribution for the partial maximum of the GED with normalizing constants and given by (1.4).
- (i)For , with normalizing constants and given by (1.4), we have(2.1)
- (ii)For , with normalizing constants and , we have(2.2)
Remark 1 The main result coincides with (1.1) as the GED reduces to the standard normal distribution .
Remark 2 From (1.2) and (1.4), it is easy to check that . Hence, for , Theorem 1(i) shows that the convergence rate of to its ultimate extreme value distribution is proportional to , while for the case of , Theorem 1(ii) shows that the convergence rate is proportional to .
3 The proofs
In order to prove the main results, we need some auxiliary lemmas. The first lemma deals with a decomposition of the distributional tail representation of .
with and given by (1.3).
which is the desired result. □
where and are given by Theorem 1.
The proof is complete. □
with . For the Laplace distribution, we have the following result.
where and are those given by Theorem 1.
The proof is complete. □
The proof is complete. □
This work was supported by the National Natural Science Foundation of China (11171275), the Natural Science Foundation Project of CQ (cstc2012jjA00029) and the Fundamental Research Funds for the Central Universities (XDJK2013C021, XDJK2014D020).
- Nelson DB: Conditional heteroskedasticity in asset returns: a new approach. Econometrica 1991, 59: 347–370. 10.2307/2938260MathSciNetView ArticleMATHGoogle Scholar
- Leadbetter MR, Lindgren G, Rootzén H: Extremes and Related Properties of Random Sequences and Processes. Springer, New York; 1983.View ArticleMATHGoogle Scholar
- Resnick SI: Extreme Values, Regular Variation and Point Processes. Springer, New York; 1987.View ArticleMATHGoogle Scholar
- Peng Z, Tong B, Nadarajah S: Tail behavior of the general error distribution. Econometrica 2009, 38: 1884–1892.MathSciNetMATHGoogle 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
- Nair KA: Asymptotic distribution and moments of normal extremes. Stat. Probab. Lett. 1981, 9: 150–153.MathSciNetMATHGoogle Scholar
- Liao X, Peng Z: Convergence rate of limit distribution of maxima of lognormal samples. J. Math. Anal. Appl. 2012, 395: 643–653. 10.1016/j.jmaa.2012.05.077MathSciNetView ArticleMATHGoogle Scholar
- Liao X, Peng Z, Nadarajah S: Tail properties and asymptotic expansions for the maximum of logarithmic skew-normal distribution. J. Appl. Probab. 2013, 50: 900–907. 10.1239/jap/1378401246MathSciNetView ArticleMATHGoogle Scholar
- Liao X, Peng Z, Nadarajah S, Wang X: Rates of convergence of extremes from skew normal sample. Stat. Probab. Lett. 2014, 84: 40–47.MathSciNetView ArticleMATHGoogle Scholar
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