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Higher-order expansions for distributions of extremes from general error distribution
Journal of Inequalities and Applications volume 2014, Article number: 213 (2014)
Abstract
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.
1 Introduction
Let be a sequence of independent and identically distributed random variables with marginal cumulative distribution function (cdf) following the general error distribution ( for short), and let denote the partial maximum of . The probability density function (pdf) of is given by
where is the shape parameter, and denotes the Gamma function (Nelson [1]). Note that reduces to the standard normal distribution.
Recently, several contributions investigated asymptotic behaviors of normalized maxima from the . It is well known that the limiting distribution of extremes from the , i.e., the normal distribution, is a Gumbel extreme value distribution, see Leadbetter et al. [2] and Resnick [3]. Peng et al. [4] established the Mills type ratio of and proved that there exist normalizing constants and such that
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 [5] established the optimal uniform convergence rate as , i.e., the normal case; Peng et al. [6] extended the result to the case of . Both studies show that the optimal convergence rate of extremes from the is proportional to .
For more informative studies of extremes from the GED, Nair [7] considered higher-order expansions for distribution and moments of normalized maxima from the under optimal normalizing constants. Let denote the distribution function of the standard normal distribution , Nair [7] proved that
as , where the optimal normalizing constants and are given by
Here, and are, respectively, of the following form:
and
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 [8] for the log-normal distribution, and Liao et al. [9, 10] for skew distributions.
In order to derive the higher-order expansions of extremes from the , we cite some results from Peng et al. [4, 6]. The following Mills ratio of the is due to Peng et al. [4]:
which deduces the following distributional tail representation of :
for large , where
and
Noting that and , we may choose normalizing constants and satisfying the following equations:
Under these normalizing constants, we have
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).
Theorem 1 Let denote the cdf of with . Then:
-
(i)
For , with normalizing constants and given by (1.4), we have
(2.1)
as , where and are, respectively, given by
and
-
(ii)
For , with normalizing constants and , we have
(2.2)
as , where and are, respectively, given by
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 .
Lemma 1 Let and , respectively, denote the cdf and pdf of with ; for large , we have
with and given by (1.3).
Proof Using integration by parts we have
An application of L’Hospital’s rule shows that
Combining the latter with (1.2), (3.2), and (3.3), for large x we have
which is the desired result. □
Lemma 2 Let with normalizing constants and given by (1.4), then for we have
where and are given by Theorem 1.
Proof It is well known that as . By , we know that if and only if . The following fact holds by (1.2):
Let
It is easy to check that and
Hence,
and
By (3.1) we have
It follows from (3.5)-(3.8) that
where the last step is due to the dominated convergence theorem since
and
By arguments similar to (3.9), we have
The proof is complete. □
For , noting that the is the Laplace distribution with pdf given by
and the Laplace distributional tail can be written by
with . For the Laplace distribution, we have the following result.
Lemma 3 For , let with normalizing constants and . Then
where and are those given by Theorem 1.
Proof Noting that for , i.e., the Laplace distribution with pdf , we have
with normalizing constants and . So, by (1.4) and (3.13), we have
and
The proof is complete. □
Proof of Theorem 1 By (3.9) and (3.15), we have
as . For the case of , by Lemma 2 and (3.17), we have
as . Similarly, by Lemma 3 and (3.17), we get
as .
The proof is complete. □
References
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Acknowledgements
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).
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Authors’ contributions
PJ obtained the theorem and completed the proof. TL corrected and improved the final version. Both authors read and approved the final manuscript.
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Jia, P., Li, T. Higher-order expansions for distributions of extremes from general error distribution. J Inequal Appl 2014, 213 (2014). https://doi.org/10.1186/1029-242X-2014-213
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DOI: https://doi.org/10.1186/1029-242X-2014-213