Tail properties and approximate distribution and expansion for extreme of LGMD.

We introduce logarithmic generalized Maxwell distribution motivated by Vodă (Math. Rep. 11:171-179, 2009), which is an extension of the generalized Maxwell distribution. Some interesting properties of this distribution are studied and the asymptotic distribution of the partial maximum of an i.i.d. sequence from the logarithmic generalized Maxwell distribution is gained. The expansion of the limit distribution from the normalized maxima is established under the optimal norming constants, which shows the rate of convergence of the distribution for normalized maximum to the extreme limit.

Introduction

The generalized Maxwell distribution (GMD for short), a generalization of ordinary Maxwell (or classical Maxwell) distribution, was proposed by Vodă [1]. With the rapid development of economy and science and technology, some of the existing distribution functions cannot meet the needs of research. For example, for some skewed data, it is appropriate to describe and fit them only by using some logarithmic models. Therefore, the recent development of some new distribution functions and the study of logarithmic case of the distribution functions have become hot issues in the statistical field. For more details, please refer to [2-8]. In this paper, we define the logarithmic generalized Maxwell distribution (for brevity LGMD), which is a natural prolongation of the generalized Maxwell distribution. In addition to the previously mentioned, one motivation of thinking of LGMD is to obtain more efficient results as parameter estimators when random models were supposed with the LGMD error terms instead of normal ones. Other aspects, like compressive sensing, we hope the LGMD could be used to model impulsive noise [9].

The GMD has a variety of applications in statistics, physics, and chemistry. The probability density function (pdf) and the cumulative distribution function (cdf) of the GMD with the parameter are respectively, and for , where σ is a positive constant and is the gamma function.

Mills [10] gave a well-known inequality and Mills’ ratio conclusion for the standard Gauss cdf with respect to its pdf as follows: for , and as .

Peng et al. [11] extended the Mills results to the case of the general error distribution: for and , and for as , where , and is the general error cdf with pdf . Huang and Chen [12] investigated similar results of GMD, viz., for , and , and for , as . The above-mentioned Mill type inequalities such as (1.1), (1.3), and (1.5) and Mills’ type ratios such as (1.2), (1.4), and (1.6) play an important role in considering some tail behavior and extremes of economic and financial data.

The present paper is to derive the Mills’ inequality, Mills’ ratio, and the distributional tail expression for the LGMD. As an important application, the asymptotic distribution of the partial maximum of i.i.d. variables with common LGMD is investigated. As another significant application, with appropriate normalized constants, the distributional expansion of the normalized maxima from LGMD is obtained. Moreover, we indicate that rate of convergence of the distribution of normalized maxima to corresponding extreme value limit is of the order of .

First of all, we provide the definition of LGMD.

Definition 1.1

Set X stand for a random variable which obeys the GMD. Set . Then Y is termed obeying the LGMD, denoted by with parameter .

Easily check that the pdf is for , where parameter , and σ is a positive constant. Suppose that for . Observe that the LGMD decreases to the logarithmic Maxwell distribution when .

The rest of the article is organized as follows. In Section 2, we derive some interesting results including Mills-type ratios and tail behaviors of LGMD. In Section 3, we discuss the asymptotic distribution of normalized maxima of i.i.d. random variables following the LGMD and the suitable norming constants. We generalize the result to the case of a finite blending of LGMDs. In Section 4, we establish the asymptotic expansion of the distribution of the normalized maximum from LGMD under optimal choice of norming constants. As a byproduct, we obtain the convergence speed of the distribution of the normalized partial maxima to its limit.

Mills’ ratio and tail properties of LGMD

In this part, we obtain some significant results including Mills’ inequality, Mills’ ratio of LGMD.

As to LGMD and GMD, observe that and Hence, by Lemma 2.2 and Theorem 2.1 in Huang and Chen [12], the two results below follow.

Theorem 2.1

Suppose that and respectively represent the cdf and pdf of LGMD with parameter . We have the inequality below, for all , where σ is a positive constant.

Corollary 2.1

For fixed , as , we have

Remark 2.1

Since the are reduced to the logarithmic Maxwell distribution as , so by Theorem 2.1 and Corollary 2.1, we derive the Mill’s inequality and Mills’ ratio of the logarithmic Maxwell distribution, viz., for , and as .

Remark 2.2

For , Corollary 2.1 gives , i.e., there are norming constants and which ensure converges to , as . Since by Corollary 2.1, we obtain as . Hence, applying Proposition 1.18 in Resnick [13], we obtain . As to how to choose the norming constants and will be explored by Theorem 3.2.

Finner et al. [14] investigated the asymptotic property of the ratio of the Student t and Gauss distributions as the degrees of freedom satisfies The main motivation of the work is to consider the false discovery rate in multiple testing problems with large numbers of hypotheses and extremely small critical values for the smallest ordered p value; for details, see Finner et al. [15]. In the following, we investigate the asymptotic property of the ratio of pdfs and the ratio of the tails of the LGMD and the logarithmic Maxwell distribution. Firstly, we think over the situation of . Secondly, we think over the situation of for fixed k.

Theorem 2.2

For , let be such that for some . We obtain and

Proof

Observe that as , therefore By (2.4), it is easy to check that as . Again applying (2.4), we have Combining (2.7), Corollary 2.1, Remark 2.1, and (2.5), representation (2.6) can be derived. □

Theorem 2.3

For fixed k, we have and

It is easy to verify (2.8) by fundamental calculation. By Corollary 2.1, Remark 2.1, and (2.8), we have Hence (2.9) follows. □

Limiting distribution of the maxima

By applying Corollary 2.1, we could establish the distributional tail representation for the LGMD.

Theorem 3.1

Under the conditions of Theorem  2.1, we have for large enough x, where and where as .

For large enough x, by Corollary 2.1, we have where as . The desired result follows. □

Remark 3.1

As , on is absolutely continuous function and in Theorem 3.1, an application of Theorem 3.1 and Corollary 1.7 in Resnick [13] shows , and the norming constants and can be chosen by such that where denotes the domain of attraction .

Here we establish the asymptotic distribution of the normalized maximum of a sequence of i.i.d. random variables following LGMD. Remark 2.2 and Theorem 3.1 showed that the distribution of partial maximum converges to . So, the following task is to look for the associated suitable norming constants.

Theorem 3.2

Suppose that be an i.i.d. sequence from the LGMD with . Let . We have where and

Since , there must be norming constants and which make sure that . By Proposition 1.1 in Resnick [13] and Theorem 3.1, we can make choice of the norming constants and satisfying the equations: and . Note that is continuous, then . By Corollary 2.1, we have as , viz., as , and so as , from which one deduces as , thus as , hence Putting the equality above into (3.3), we have from which one induces that therefore where Hence, we have It is easy to check that and . Hence, by Theorem 1.2.3 in Leadbetter et al. [16], the proof is complete. □

Remark 3.2

Theorem 3.2 shows that the limit distribution of the normalized maximum from the logarithmic Maxwell distribution is the extreme value distribution with norming constants and

At the end of this section, we generalize the result of Theorem 3.2 to the situation of a finite blending of LGMDs.

Finite mixture distributions (or models) have been widely applied in various areas such as Chemistry [17] and image and video databases [18]. Specifically, related extreme statistical scholars have studied them. Mladenović [19] have considered extreme values of the sequences of independent random variables with common mixed distributions containing normal, Cauchy and uniform distributions. Peng et al. [20] have investigated the limit distribution and its corresponding uniform rate of convergence for a finite mixed of exponential distribution.

If the distribution function (df) F of a random variable ξ have we say that ξ obeys a finite mixed distribution F, where , stand for different dfs of the mixture components. The weight coefficients satisfy the condition that , and .

Next, we think of the extreme value distribution from a finite blending with constituent dfs obeying , where the parameter for and for . Denote the cumulative df of the finite blending by for .

Theorem 3.3

Suppose that be a sequence of i.i.d. random variables following the common df F given by (3.4). Set . Now holds with the norming constants and where , and , , , and .

By (3.4), we have By Theorem 2.1, we have for all , according to the definition of , which implies where and as since . Combining (3.5)-(3.7) with (2.2), for large enough x, we obtain as , where represents the cdf of the , and σ and p are defined by Theorem 3.3. By Proposition 1.19 in Resnick [13], we can derive . The norming constants can be obtained by Theorem 3.2 and (3.8). The desired result follows. □

Asymptotic expansion of maximum

In this section, we establish an high-order expansion of the distribution of the extreme from the LGMD sample.

Theorem 4.1

For the norming constants and given by (3.1), we have where and

Corollary 4.1

Under the condition of Theorem  4.1, we have for large n, where is given by Theorem 4.1.

The result directly follows from Theorem 4.1. The detailed proof is omitted. □

In order to prove Theorem 4.1, we need several lemmas. The following lemma shows a decomposition of the distributional tail representation of the LGMD.

Lemma 4.1

Let denote the cdf of the LMGD. For large x, we have with and given by Theorem  3.1.

By integration by parts, we have Using L’Hospital’s rules yields One easily checks that with and determined by Theorem 3.1. Combining with (4.1)-(4.3), we complete the proof. □

Lemma 4.2

Set with the norming constants and given by (3.1), then

By (3.2), we have as , with the norming constants and given by (3.1). It is not difficult to verify that and For large n we have and By (4.4)-(4.6), the desired result follows. □

Lemma 4.3

Set to denote the minimum of and with norming constants and given by (3.1). Then with , given by Theorem  4.1.

For any positive integers m and , by Corollary 2.1 and the fact that , we have For any and , we have and as . Here set By Lemmas 4.1, 4.2, (4.9), and (4.10), we have By (4.8)-(4.11), Lemma 4.2, and the dominated convergence theorem, we have For all , for large n, which implies and as .

By (4.13), (4.14), and Lemma 4.2, we have with . The proof is completed. □

Proof of Theorem 4.1

By Lemma 4.3, we have as . Once again by Lemma 4.3, we have where is provided by Theorem 4.1. The proof is completed. □

Conclusion

Motivated by Vodă [1], we put forward the logarithmic generalized Maxwell distribution. We discuss tail properties and the limit distribution of the distribution. We extend the results to the case of a finite mixture distribution. With the optimal norming constants, we establish the high-order expansion of the distribution of maxima from logarithmic generalized Maxwell distribution, by which we derive the convergence rate of the distribution of maximum to the associate extreme limit.