Mathematical Problems in Engineering

Volume 2012 (2012), Article ID 280869, 14 pages

doi:10.1155/2012/280869

Research Article

Tail Dependence for Regularly Varying Time Series

Ai-Ju Shi1,2 and Jin-Guan Lin¹

¹Department of Mathematics, Southeast University, Nanjing 210096, China

²College of Science, Nanjing University of Posts and Telecommunications, Nanjing 210003, China

Received 9 January 2012; Accepted 14 March 2012

Academic Editor: Ming Li

Copyright © 2012 Ai-Ju Shi and Jin-Guan Lin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We use tail dependence functions to study tail dependence for regularly varying (RV) time series. First, tail dependence functions about RV time series are deduced through the intensity measure. Then, the relation between the tail dependence function and the intensity measure is established: they are biuniquely determined. Finally, we obtain the expressions of the tail dependence parameters based on the expectation of the RV components of the time series. These expressions are coincided with those obtained by the conditional probability. Some simulation examples are demonstrated to verify the results we established in this paper.

1. Introduction

Copula is a useful tool for handling multivariate distributions with given univariate margins. A copula is a distribution function, defined on the unit cube , with uniform one-dimensional margins . For any , ; the survival copula is , the joint survival function of copula is . Given a copula , let

then is a multivariate distribution with univariate margins . On the other hand, given a distribution with margins , there exists a copula such that (1.1) holds. And copula is unique if are all continuous (Sklar [1], Nelsen [2]).

In generally, copula forms a natural way to describe the dependence between series when making abstraction of their marginal distributions. Overviews of the probabilistic and statistical properties of copula are to be found in [1–6].

Tail dependence plays an important role among dependence measures due to its ability to describe dependence among extreme values (Frahm et al. [7], Resnick [8, 9], and Nikoloulopoulos et al. [10]) which is introduced by Joe [4]. The issue of tail dependence is mainly for heavy tailed phenomena, heavy tailed phenomenon in fractal time series. It is extensively studied and applied in insurance, risk management, traffic management and engineering management, and so forth. [11–27].

Researchers find various multivariate distributions with heavy tails to describe the extremal or tail dependence, see, Pisarenko and Rodkin [13], Hult and Lindskog [28], and Fang et al. [29]. Many interesting tail quantities have been derived via standard methods: coefficients of tail dependence [30–37] and tail dependence copulas (Charpentier and Segers [38]).

In this paper, we are interested in the tail behavior of the time series which have the form:

where the scale variable is independent of random vector . And is multivariate regularly varying with distribution function having copula .

This distribution is a generalized class, including, for example, multivariate Pareto and multivariate elliptical distribution as special ones. Especially, the multivariate distribution is included in it. As an example, we will justify the results through multivariate copula.

In order to analyze the tail dependence behavior of (1.2), we first study the tail dependence functions via intensity measure. Then using the relation between tail dependence parameter and the tail dependence functions, we explore the explicit representations of the tail dependence parameters.

The outline of this paper is as follows. After some preliminaries about multivariate regularly varying series and dependence functions in Section 2, detailed results for the tail dependence functions are discussed in Section 3, the expressions of tail dependence parameters for RV time series are demonstrated in Section 4, and multivariate distribution is demonstrated as an example in Section 5.

Throughout, is a random vector with joint distribution function and copula . Minima and maxima will be denoted by and , respectively. The Cartesian product is denoted by for any .

2. Preliminaries

Definition 2.1. The -dimensional random vector is said to be regularly varying with index if there exists a random vector with values in a.s., where denotes the unit sphere in with respect to the norm , such that, for all ,

as . The symbol stands for vague convergence on ; vague convergence of measures is treated in detail in Kallenberg [39]. The distribution of is referred to as the spectral measure of . For further information on multivariate regular variation we refer to Resnick [8, 9].

In fact, (2.1) is equivalent to the following expression

where is an intensity measure or Radon measure on and {} is a sequence an of nonnegative numbers.

From the Definition 2.1, we can see that the regularly varying distribution is connected with intensity measure . The following lemma yields the explicit relation between them which can be found in [8].

Lemma 2.2. Let random vector be regularly varying with index and distribution function , then it is equivalent to the following.

(1)There exists an intensity measure on , such that for every Borel set bounded away from the origin that satisfies ,

with the homogeneous condition .(2)There exists an intensity measure on , such that

for all continuous points of . According to Lemma 2.2, one notices that for any nonnegative multivariate regularly varying random vector , its nondegenerate univariate margins have regularly varying right tails and with the same index of also, that is,

where is a slowly varying function.

Lemma 2.3 (Breiman [40]). Let and be two independent nonnegative random variables, be regularly varying with index . If there exists a , such that , then

The multivariate version of the Lemma belongs to Basrak et al. [41]. It is said that, if is regularly varying in the sense of (2.2), is a random matrix, independent of , with for some , then

where denotes vague convergence on .

Definition 2.4 (Kluppelberg et al. [42]). Let be the distribution function of random vector with continuous margins and copula . For any , the lower dependence function is defined as

and the upper dependence function is defined as

The upper exponent function is defined as

where .

From the definition, we can verify the elementary properties listed in Proposition 2.5 of the tail dependence function. We denote and are the upper tail and lower dependence parameters of , respectively, where is a nonempty subset of . is the margin of with component indexes in .

Proposition 2.5. For any ,

where is the margin copula of .

For any nonempty ,

Proof. According to the definition of , we get

similarly,

Note that

combined with (2.9), the first part is determined. The second part can be verified similarly.

We can obtained the proof only paying attention to .

From the proposition, the upper tail dependence function of copula is the lower one of its survival copula . And in most fractal time series, from the point of view of either theory or applications, people only need to understand the right tail of the data, so we focus on the upper tail function and coefficient in the following.

We first study the upper tail dependence function of multivariate regularly varying time series in (1.2) using the intensity measure.

3. The Upper Tail Dependence Function for RV Time Series

Theorem 3.1. Let be RV time series with regularly varying index , distribution function , copula , and the stochastic representation as (1.2). If the margins are tail equivalent as , then the upper tail dependence function can be written as

and the upper exponent function can be written as

Proof. For any ,

Since every margin is regularly varying with the same index , we obtain that

where is slowing varying function. So for any , as ,

where as . So the equation becomes

in other words,

Now we let , then

so, .

As , , so we get that

And since the margins are equivalent, that is, as . We have as (Resnick [8]). So for sufficient small , , and , combining (3.3) and (2.3), we obtain that

In order to calculate , we recall the inclusion-exclusion formula, it says that

is valid for any finite set and arbitrary events , where .

Using this formula, (2.10) becomes

By using the same method of (3.3), the following equation holds:

Corollary 3.2. Under the same conditions as Theorem 3.1, the following result holds

Proof. By (2.4), one can see that . So we can get the result immediately by letting all in (3.2).

According to Theorem 3.1 and Corollary 3.2, we can represent the intensity measure through the tail dependence function as the following Corollary.

Corollary 3.3. Under the same conditions as Theorem 3.1, one has

4. The Upper Tail Dependence Parameters for Regularly Varying Time Series

According to Proposition 2.5 and Theorem 3.1, we can express the tail dependence parameters by their tail dependence functions. In this section, we will deduce the upper tail dependence parameters of time series with multivariate varying distribution in (1.2) by this method. Hereafter, we let be the intensity measure of with copula . Where is regularly varying at with index , with survival function , and is a slowly varying function. So for any nonnegative vector , we have

by inserting and into the representation, then,

Similarly, we have,

Consequently, we get the main result as follows.

Theorem 4.1. Let be regularly varying time series with the same regularly varying index and the stochastic representation given in (1.2), the margins are tail equivalent as . If there exists a holds for , then the upper tail dependence parameter of is

Proof. We first calculate the tail dependence function of . In the following, let and be the copula of and , respectively. Denote

where

Note that is strictly increasing transformation of , and the tail dependence function and the parameter are all copula properties. Hence and have the same tail dependence functions. By Lemma 2.3, one can see that the marginal variables of vector are tail equivalent and regularly varying with the same index as as . Denote the intensity measures of and by and , respectively. According to (2.7),

Now by (4.6), we see that,

combining this with (4.3), for any nonnegative , we obtain the intensity measure given by

Hence, we have

Substituting this measure into (3.1), we get the upper tail dependence function of vector as follows:

Since and have the same tail dependence functions, we have

By (2) in Proposition 2.5, we obtain the upper tail dependence parameters of vector .

5. Examples

Let in (1.2) be , where is a matrix with , and is a semidefinite matrix, is uniformly distributed on the unit sphere (with respect to Euclidean distance) in . We know that conforms to an elliptical contoured distribution (Fang et al. [43]). The tail dependence of the elliptical contoured distribution has been discussed in Schmidt [33]. Here we select the distribution to display our results in Theorem 4.1 as a special case.

If , then has the stochastic representation ([43]):

where and are independent, .

Let . Then and is regularly varying with index at . So the vector is regularly varying according to Schmidt [33].

For the upper tail dependence that only relies on the tail behavior of the random vector, we can focus, without loss of generality, on the random vector with zero mean vector. Furthermore, since the strictly increasing transformation of does not change the copula, has the same copula as , where and . Thus . It is evident that becomes the correlation matrix of the random vector. Consequently, we may assume that the covariance matrix is the correlation matrix. In this situation, all have the same margins as . So are all equal for any . Under these assumptions, using (4.4), we get the upper tail dependence parameter of as

This is coincided to the one obtained in Shi and Lin [34].

6. Simulations

In Section 4, we obtain the expressions of the tail dependence indexes about RV time series in (1.2). In Section 5, we display our result in the multivariate distribution as example. In this Section, we will illustrate these results by some Monte Carlo simulated numerical examples. Given that be generated from the multivariate normal distribution , then the upper tail dependence indices of can be estimated by

We estimate the upper tail dependence parameter of 3-dimensional distribution under autoregressive of order 1 (AR(1)), exchangeable(EX), Toeplitz(TOEP), and unstructured(UN) correlation structure, respectively. For each correlation matrix, we first generate 80,000 pseudorandom vectors, then use (5.2) to estimate tail dependence parameter for different . Specifically, we do the following simulations.

Let and , respectively. The corresponding upper tail dependence parameters are denoted by and . and are under AR(1) and EX correlation structure, respectively, the simulated values of , about different are computed and plotted in Figure 1. and are under TOEP and UN correlation structure, the corresponding results are demonstrated in Figure 2.

(a)

(b)

Figure 1: The estimation of , under AR(1) (the left one) and EX (the right one) correlation structure.

(a)

(b)

Figure 2: The estimation of , under TOEP (the left one) and UN (the right one) correlation structure.

From the two figures, in spite of the correlation structure, decreased and approached 0 quickly as increased to , which is the tail dependence index for multivariate normal copula.

Many researchers try to discuss the monotonicity of the tail dependence parameter about the regular varying index. Embrechts et al. [11] proved that the tail dependence of the bivariate distribution is decreasing about the regular varying index , and demonstrated that the tail dependence parameter is decreasing in by numerical results. But From the right graph in Figure 2., these conclusions are not always correct when .

7. Conclusion

In the paper, we mainly study tail dependence of RV time series in (1.2). We use tail dependence function and intensity measure to express tail dependence parameters. Using tail dependence function, we do not need to consider the explicit representation of the copula. We first discuss the tail dependence function of the RV time series due to the propositions of the regularly varying function, connecting the biuniquely determined property between the tail dependence function and the intensity measure. Then we calculate the explicit formula of the upper tail dependence parameter about the RV time series under some conditions. In fact, we can obtain the extreme upper tail dependence index (Shi and Lin [34]) very similarly to Theorem 4.1, for concise, we omit it here.

Copula of continuous variables is invariant under strictly increasing transformation (Nelsen [2]). In order to obtain the tail dependence function of random vector , we shift to solve that of in (4.5), which is just a strictly increasing transformation of .

At last, we select the distribution as a special case to display our result, they are coincided to the one given in [34]. The monotonicity of the tail dependence parameters about the regular varying index is still an open problem. Under what constraints the tail dependence parameters will be deceasing in the variation index? We are still interested in the problem. We will discuss it in the following work in details. In engineering application, when we confront fractal time series and seasonal data, we can model the tail dependence property via the tail dependence function if the data is consistent with the constraint conditions in our work.

Acknowledgment

The authors are very grateful to the referees for their comments and suggestions on the earlier version of the paper, which led to a much improved paper. This project is supported by MSRFSEU 3207011102, NSFC 11171065, and NSFJS BK2011058.

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