Assessing bivariate tail non-exchangeable dependence

Assessing bivariate tail non-exchangeable dependence

STAPRO: 108556 Model 3G pp. 1–8 (col. fig: nil) Statistics and Probability Letters xxx (xxxx) xxx Contents lists available at ScienceDirect Stati...

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STAPRO: 108556

Model 3G

pp. 1–8 (col. fig: nil)

Statistics and Probability Letters xxx (xxxx) xxx

Contents lists available at ScienceDirect

Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro

Assessing bivariate tail non-exchangeable dependence ∗

Lei Hua , Alan Polansky, Paramahansa Pramanik Department of Statistics and Actuarial Science, Northern Illinois University, DeKalb, IL, 60115, United States of America

article

info

Article history: Received 28 August 2018 Received in revised form 9 July 2019 Accepted 14 July 2019 Available online xxxx Keywords: Conditional tail expectation Copula Tail dependence Tail behavior

a b s t r a c t Non-exchangeable dependence structures exist in the real world. In particular, when dependence patterns in joint distributional tails are important, such as in the fields of engineering, environmetrics and econometrics, one may need to detect the existence, and assess the strength of non-exchangeable dependence patterns in the tails. In this paper, we propose a sensible metric to quantify the degree of tail non-exchangeability of bivariate copulas. Based on the metric, we propose a practical method of assessing tail non-exchangeable dependence with uniform scores of bivariate data. An empirical example is used to demonstrate the usefulness of the proposed method. © 2019 Published by Elsevier B.V.

1. Introduction The goal of this study is to propose methods of assessing the degree of non-exchangeability in the tails of a bivariate copula. The study of tail non-exchangeability of bivariate copulas is particularly useful for understanding causality of extreme events and providing more tail-tailored statistical models for modeling tail non-exchangeable structures observed in the fields of risk management, quantitative finance, psychometrics, econometrics, environmetrics, etc. There have recently been quite a few papers focusing on the study of quantifying the degree of overall nonexchangeability. For example, Klement and Mesiar (2006) and Nelsen (2007) study the extremal cases of bivariate nonexchangeability using copulas, Durante et al. (2010) provide some axioms for measures of bivariate non-exchangeability, Durante and Mesiar (2010) study the cases for some specific bivariate copula families, Genest et al. (2012), Krupskii (2016) propose some statistical tests for bivariate non-exchangeability, and Harder and Stadtmüller (2014) extend the study of extremal non-exchangeability to multivariate cases. In this paper, we are interested in quantifying the degree of tail non-exchangeability for a bivariate copula. We first propose a sensible measure for quantifying the strength of tail non-exchangeability of copulas, and then derive asymptotic approximations for the tail behavior of the statistic to be proposed. There are some simple approaches that can be used to construct non-exchangeable copulas, such as the Marshall–Olkin copula, the generalized Clayton copula studied in Furman et al. (2015), the copulas constructed through comonotonic latent variables (Hua and Joe, 2017), and extreme value copulas constructed by non-exchangeable Pickands functions. In this paper, we consider Khoudraji’s device (Khoudraji, 1996) for generating non-exchangeable copulas. Let U1 and U2 each be a random variable uniformly distributed on [0, 1], denoted as Ui ∼ U (0, 1), i = 1, 2, and C be the copula of (U1 , U2 ). In order to have a sensible quantity for non-exchangeability in the tails of a bivariate copula, we can consider the difference between certain conditional tail quantities before and after U1 and U2 are switched. We propose to use the limiting property of η(t) = E[U1 |U2 > t ]/E[U2 |U1 > t ] as t → 1− to study the strength of tail non-exchangeability of the bivariate copula C . In Hua and Joe (2014) conditional tail expectations of the forms E[X1 |X2 > t ] and E[X1 |X2 = t ] ∗ Corresponding author. E-mail address: [email protected] (L. Hua). https://doi.org/10.1016/j.spl.2019.108556 0167-7152/© 2019 Published by Elsevier B.V.

Please cite this article as: L. Hua, A. Polansky and P. Pramanik, Assessing bivariate tail non-exchangeable dependence. Statistics and Probability Letters (2019) 108556, https://doi.org/10.1016/j.spl.2019.108556.

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are used to study the strength of tail dependence as t → ∞ between identical random variables, and in Bernard and Czado (2015), conditional quantiles are used to study the strength of tail dependence. We will focus on the case of E[U1 |U2 > t ] based on which a statistical test is more feasible. In what follows, we will first introduce the notation used in the paper. Then in Section 2, we propose a definition for tail non-exchangeability, and discuss some results for Khoudraji’s Device that can be used to generate non-exchangeability. Section 3 proposes an approach for testing the significance of tail non-exchangeability. Finally, Section 4 concludes the paper.

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1.1. Notation and symbols

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We define the survival copula of an ordinary copula C ∗ as ˆ C ∗ . In order to facilitate discussion, we use ˆ C ∗ as the copula before the non-exchangeable transformation in the sense of Khoudraji (1996). After the non-exchangeable transformation, the survival copula is denoted as ˆ C . Since survival copula itself is a copula function, throughout the paper we focus on discussing corresponding survival copulas directly.

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2. Tail non-exchangeability

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In dependence modeling one often uses copula functions to account for various dependence patterns appearing in the tail part of a joint distribution. This is particularly important when these patterns cannot be well modeled by the commonly-used multivariate models such as the multivariate Normal or Student-t distributions. Most of the commonly used bivariate copulas are of exchangeable structures, meaning that C (u, v ) ≡ C (v, u) for any (u, v ) ∈ [0, 1]2 . As copula modeling often plays an important role in accounting for dependence in the tails, one may be particularly interested in the non-exchangeable structure in the joint tails. Motivated by Hua and Joe (2014), where the tail behavior of E[X1 |X2 > t ] is studied for capturing the tail dependence strength between the bivariate random vector (X1 , X2 ), we introduce the following Definition 1. Let (U1 , U2 ) be a bivariate random vector with standard uniform marginals. Then the copula of the random vector (U1 , U2 ) is said to be tail exchangeable if the following condition holds: Condition I:

lim η(t) := lim

t →1−

t →1−

E[U1 |U2 > t ] E[U2 |U1 > t ]

= 1.

(1)

When the condition (1) does not hold, the copula C is said to be tail non-exchangeable. Remark 1. The departure of η(t) from 1 as t → 1− captures the degree It is easy to verify ∫ 1 of tail non-exchangeability. ∫1 that the above condition can be written as limt →1− η(t) = limt →1− [ 0 ˆ C (1 − x, 1 − t)dx]/[ 0 ˆ C (1 − t , 1 − x)dx] = 1, where ˆ C is the survival copula of C . So, clearly tail non-exchangeability is a copula property. Note that, the concept of tail non-exchangeability is different than the notion of tail asymmetry of copulas, and the latter means that the upper and lower tails are different. Definition 2. If lim inft →1− η(t) > 1, denoted as η(t) ≻ 1, t → 1− , then it is said that the copula C is tail non-exchangeable to the first element. Before we propose the methods for assessing the tail non-exchangeability in Section 3, let us look at some examples that are more analytically tractable to help us understand the behavior of η(t). There are many different approaches of constructing non-exchangeable copulas. In order to gain insights about how η(t) may behave from an analytical perspective, we consider applying Khoudraji’s device, a relatively more tractable transformation for getting non-exchangeable copulas; see, Khoudraji (1996), Genest et al. (1998, 2011, 2012). That is, with a given exchangeable (survival) copula ˆ C ∗, a non-exchangeable (survival) copula can be constructed as

ˆ C (u, v ) = u1−α1 v 1−α2 ˆ C ∗ (uα1 , v α2 ), (α1 , α2 ) ∈ [0, 1]2 .

(2)

Note that smaller values of α1 and α2 give more weight for the independence structure as opposed to the dependence suggested by ˆ C ∗ . After many numerical experiments with different copula families, we find that when C ∗ is an exchangeable bivariate copula, then α1 > α2 may imply that η(t) ≻ 1 as t → 1− , meaning that η(t) is eventually greater than 1 as t becomes sufficiently close to 1. We will show some examples later on, and we conjecture that such a pattern exists in many other copula families. As in general the conditional expectations do not have closed form solutions, Hua and Joe (2014) propose to use either the Laplace approximation or Watson’s lemma for asymptotic approximations, where exchangeable copulas are considered. In this paper, we derive a technical lemma that is helpful for approximating the tail behavior of η(t) as t → 1− . The following result gives a fairly useful tool in deriving an asymptotic approximation. Afterwards, we will be deriving approximations for some concrete parametric copula examples based on the lemma. Please cite this article as: L. Hua, A. Polansky and P. Pramanik, Assessing bivariate tail non-exchangeable dependence. Statistics and Probability Letters (2019) 108556, https://doi.org/10.1016/j.spl.2019.108556.

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Lemma 1. Let U1 , U2 ∼ U (0, 1) and their survival copula ˆ C be constructed as in (2) based on a given survival copula ˆ C ∗ . If limv→0+ ˆ C ∗ (u, v )/v = ξ (u) uniformly in u, then lim E[U1 |U2 > t ] = (1/α1 )

1



t →1−

u(2−2α1 )/α1 ξ (u)du.

(3)

1 2

3

0

Proof. For any given 0 < t < 1, letting y := − ln(1 − x), τ := − ln(1 − t),

E[U1 |U2 > t ] = (1 − t)−α2

1



(1 − x)1−α1 ˆ C ∗ ((1 − x)α1 , (1 − t)α2 )dx.

(4)

0





eα2 τ e−(2−α1 )yˆ C ∗ (e−α1 y , e−α2 τ ) dy.

=

(5)

0

Letting u = e−α1 y and v = e−α2 τ , lim E[U1 |U2 > t ] =

t →1−

4





e−(2−α1 )y lim [ˆ C ∗ (e−α1 y , v )/v]dy,

(6)

v→0+

0

5

which readily leads to the claim. □

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Remark 2. Based on the proof of Lemma 1, letting τ = − ln(1 − t) and further y = sτ , we can also write

7

E[U1 |U2 > t ] = τ





eτ g(s;τ ) ds,

(7)

8

0

where g(s; τ ) = α2 − s(2 − α1 ) + ln[ˆ C ∗ (e−α1 sτ , e−α2 τ )]/τ . Clearly, g(0; τ ) ≡ 0. When τ → ∞, the tail behavior of Eq. (7) will be dominated by g(s; τ ) on a neighborhood of s∗ that leads to the global maximum. Depending on whether s∗ = 0 or not, one may need either Watson’s lemma or Laplace’s method to derive a limiting approximation. We will give an example in Example 1 to briefly demonstrate the method. Remark 3. Since E[U1 |U2 > t ] is bounded from above by 1, Lemma 1 should be easy enough to help us derive the first order approximation for the tail behavior as t → 1− . Moreover, because of the convenience provided by the finite expectation, a similar technique may be applied for some other methods of constructing non-exchangeable bivariate copulas. Due to the limitation of the space, we now refrain ourselves from deriving results for the other non-exchangeability mechanism. Example 1 (Gumbel Copula). For Gumbel copula ˆ C ∗ (u, v ) = exp{−[(− ln u)θ + (− ln v )θ ]1/θ } with θ > 1, write w = − ln v . Then, ˆ C ∗ (u, v )/v = exp{w − ((− ln u)θ + w θ )1/θ } =: exp{ζ (w; u)}. It is clear that as w → ∞, i.e., as v → 1− , for an arbitrarily small ϵ > 0, ζ (w; u) → 0 pointwisely for any given u ∈ [ϵ, 1] which is compact. Moreover, ζ ′ (w; u) = 1 − ((− ln u)θ + wθ )(1−θ )/θ /[w1−θ ] ≥ 0 and thus ζ (w; u) is increasing in w. Therefore, exp{ζ (w; u)} → 1 uniformly in u ∈ [ϵ, 1] by Dini’s theorem. Then by Lemma 1 and letting ϵ → 0+ ,

E[U1 |U2 > t ] = (1/α1 )

ϵ

(∫

∫ 1) + ϵ

0

u(2−2α1 )/α1 [ˆ C ∗ (u, v )/v]du ∼ (1/α1 )

1



u(2−2α1 )/α1 du = 1/(2 − α1 ),

t → 1− .

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23

0

An alternative method based on Watson’s lemma can also be applied for this example as follows. Letting τ :=

− ln(1 − t),

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E[U1 |U2 > t ] = (τ /(2 − α1 ))





eτ g(s) ds,

(8)

26

0

)1/θ

with g(s) = α2 − s − α1θ (s/(2 − α1 ))θ + α2θ , which is decreasing in s and the maximum value is g(0) = 0. By Theorem 36 of Breitung (1994) and verifying the regularity conditions, one can derive that

(

E[U1 |U2 > t ] ∼ [τ /(2 − α1 )] · (τ

−1

) = 1/(2 − α1 ),

t→1 . −

Therefore, limt →1− η(t) = (2 − α2 )/(2 − α1 ), and if α1 > α2 then there is tail non-exchangeability to the first element. It is interesting to notice that such a first order approximation does not depend on the value of θ , the strength of dependence itself. Example 2 (Clayton Copula). For Clayton copula ˆ C ∗ (u, v ) = (u−δ + v −δ − 1)−1/δ with 0 < δ , write w = v δ . Then as + ∗ ˆ w → 0 , for an arbitrarily small ϵ > 0, C (u, v )/v = (u−δ w + 1 − w )−1/δ := ζ (w; u) → 1 pointwisely in u ∈ [ϵ, 1] which is compact. Moreover, ζ (w; a)′ ≤ 0 and thus ζ (w; a) is decreasing in w . By Dini’s theorem, ˆ C ∗ (u, v )/v → 1 uniformly + + in u ∈ [ϵ, 1] as v → 0 . Then by Lemma 1 and letting ϵ → 0 , E[U1 |U2 > t ] ∼ 1/(2 − α1 ), t → 1− . Therefore, limt →1− η(t) = (2 − α2 )/(2 − α1 ), which is the same as that for Gumbel copula. Please cite this article as: L. Hua, A. Polansky and P. Pramanik, Assessing bivariate tail non-exchangeable dependence. Statistics and Probability Letters (2019) 108556, https://doi.org/10.1016/j.spl.2019.108556.

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Example 3 (Frank Copula). For Frank copula ˆ C ∗ (u, v ) = (−1/δ ) ln(1 − [(1 − e−δ u )(1 − e−δv )]/[1 − e−δ ]) with 0 < δ , write a = (1 − e−δ u )/(1 − e−δ ) and thus a ∈ [0, 1] which is compact. Then it can be shown that ˆ C ∗ (u, v )/v = (−1/δ ) ln(1 − a(1 − e−δv ))/v := ζ (v; a) → a pointwisely in a as v → 0+ . Moreover, letting w = e−δv ∈ [e−δ , 1], ζ ′ (v; a) = (1/δ )[ln(1 − a(1 − w)) − ln(w)(aw)/(1 − a(1 − w))] =: g(w; a)/δ . Now, let us evaluate g(w; a). First, g(1; a) = 0. Second, g ′ (w; a) = − ln(w )a(1 − a)(1 − a + aw )−2 ≥ 0. Therefore, ζ ′ (v; a) = g(w; a)/δ ≤ 0. That is, ζ (v; a) is decreasing in v . By Dini’s theorem, as v → 0+ ,

ˆ C ∗ (u, v )/v → (1 − e−δ u )/(1 − e−δ ), uniformly in u.

7

Then by Lemma 1, lim E[U1 |U2 > t ] = (1/α1 )

1



t →1−

u(2−2α1 )/α1 (1 − e−δ u )/(1 − e−δ )du

0

= [α1 (1 − e−δ )]−1

[

α1 − δ (α1 −2)/α1 γ 2 − α1

(

2 − α1

α1



)]

,

(9)

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where γ (a, x) is the lower incomplete gamma function with x being the upper limit. The limit of E[U2 |U1 > t ] is the same as above except that the α1 needs to be replaced by α2 . Then limt →1− η(t) is just the ratio of these finite numbers. Numerical experiments suggest that Eq. (9) is increasing in α1 , and therefore, α1 > α2 may also imply that η(t) ≻ 1 as t → 1− .

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3. A test on tail non-exchangeable dependence

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In the previous section we have studied analytically the tail behavior of η(t) based on some parametric copulas. The η(t) function based on the conditional tail expectations was found to be useful in detecting tail non-exchangeability. In this section, we will develop a statistical test for tail non-exchangeability based on uniform scores of each univariate marginal. Suppose that we have observed a random sample (U11 , U21 ), . . . , (U1n , U2n ) from the random vector (U1 , U2 ) of uniform marginals with a copula C as the joint cdf. Then the empirical copula can be written as Cn (u1 , u2 ) = (1/n)

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n ∑

1{U1i ≤u1 ,U2i ≤u2 } .

i=1 20 21

Then a statistical test of bivariate tail non-exchangeability can be constructed by considering the estimator for η(t) = E[U1 |U2 > t ]/E[U2 |U1 > t ]. The empirical version of η(t) can be written as

ηˆ (t) =

22

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j∈G1 (t)

U1j

−1



j∈G2 (t)

U2j

|G2 (t)|

,

t ∈ [0, 1),

where G1 (t) := {j : U2j > t }, G2 (t) := {j : U1j > t }, and |Gi (t)| denotes the cardinality of the set Gi (t) for i = 1, 2. Under the null hypothesis of tail exchangeability ηˆ (t) should be close to 1 for t being close to 1. This property can be used as the basis for an approximate statistical test. Because the numerator and denominator of ηˆ (t) are essentially sample means, an asymptotic normality result can be easily established. Under the null hypothesis of tail exchangeability, we can derive the following result. Proposition 2. For a given t ∈ [0, 1) and Ui ∼ U (0, 1), i = 1, 2, if U1 and U2 are exchangeable whenever U1 > t and U2 > t, then d

n1/2 ηˆ (t) − 1 → Z ∼ N [0, ση2 (t)],

[

30

31

|G1 (t)|−1

]

n → ∞,

where

[ ] ση2 (t) = σ12 (t)/µ22 (t) + σ22 (t)µ21 (t)/µ42 (t) − 2σ12 (t)µ1 (t)/µ32 (t) (1 − t)2 , 2 |U2j > t)(1 − t) − µ21 (t)(1 − t)2 , σ22 (t) = E(U2j2 |U1j > t)(1 − t) − µ22 (t)(1 − t)2 , µ1 (t) = E(U1j |U2j > t), with σ12 (t) = E(U1j µ2 (t) = E(U2j |U1j > t), and σ12 (t) = E(U1j U2j |U1j > t , U2j > t)P(U1j > t , U2j > t) − µ1 (t)µ2 (t)(1 − t)2 .

Proof. Consider a set of bivariate column random vectors Z j = (B2j U1j , B1j U2j )⊤ for j = 1, . . . , n, where Bij = I(Uij > t) for i = 1, 2 and j = 1, . . . , n, with I(·) being an indicator function. This corresponds to a sequence of independent and identically distributed random vectors such that

µ(t) := E(Z i ) =

[ ] µ1 (t)(1 − t) , µ2 (t)(1 − t)

(10)

Please cite this article as: L. Hua, A. Polansky and P. Pramanik, Assessing bivariate tail non-exchangeable dependence. Statistics and Probability Letters (2019) 108556, https://doi.org/10.1016/j.spl.2019.108556.

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Fig. 1. Histograms of the 1000 simulated realizations of ηˆ (t) from Gumbel copula.

The covariance matrix of Z j is given by

[ Σ(t) =

1

σ σ12 (t) . σ12 (t) σ22 (t) 2 1 (t)

]

2

Under the null hypothesis of exchangeability when U1 > t and U2 > t, the Multivariate Central Limit Theorem then

3

d

implies that n1/2 (Z¯ n − µ(t)) → N (0, Σ(t)) as n → ∞. To obtain the weak convergence described in Proposition 2 we firstly apply the delta method (Polansky, 2011, Theorem 4.22) to obtain the result that

) ( ∑n B2j U1j d − 1 → N(0, ση2 (t)). n1/2 ∑ni=1 i=1

4 5

6

B1j U2j

p

The asymptotic normality of ηˆ (t) then follows from Slutsky’s Theorem and the fact that |G1 (t)|/|G2 (t)| → 1 as n → ∞. □ 1/2

[ ] ηˆ (t) − 1 /σˆ η (t) to a normal

In order to use Proposition 2 to develop a statistical test we can fix t and compare n rejection region where σˆ η (t) is an estimate of ση (t) based either on its empirical version or an alternative method such as the bootstrap. In the case of a bootstrap based estimator, parametric and nonparametric bootstrap methods are easily adapted. For the parametric bootstrap, resamples from the joint distribution under the assumption of exchangeability ∗ ∗ ∗ ∗ ) using a fitted copula. For the nonparametric bootstrap , U2n ), . . . , (U1n , U21 can be generated by first simulating (U11 ∗ + (1 − B)U2j∗ and one can generate a resample under the hypothesis of exchangeability by simulating U˜ 1j = BU1j

∗ U˜ 2j = (1 − B)U1j + BU2j∗ , for j = 1, 2, . . . , n where (U1j∗ , U2j∗ ) is a simulated observation from the empirical distribution of (U11 , U21 ), . . . , (U1n , U2n ), and B is a Bernoulli random variable with success probability equal to 1/2 that is independent ∗ , U2j∗ ). of (U1j As the methods in this paper focus on tail exchangeability, t should be chosen to be fairly close to 1. However, one must balance the choice of t against the amount of data available beyond t, as this affects the standard error of ηˆ (t). Overall, the necessity of the choice of t in the method is unappealing as the results of the test may change based on the value of t. As an alternative we suggest aggregating the test over a sequence of values of t, using the maximum difference from one as the test statistic. In particular, let ti ’s be a grid of values such that 0 < p = t0 < t1 < · · · < tg = q < 1. The test statistic for testing the null hypothesis of tail exchangeability is then given by

∆ = max |ηˆ (ti ) − 1|. i=0,...,g

The rejection region for this test statistic can be computed using the bootstrap. The weak convergence described in Proposition 2 may require large sample sizes to be effective especially when t is chosen to be very close to 1. This is due to the fact that the effective sample is reduced because only data in the upper-tail is used to compute ηˆ 1 (t). Further, the fact that ηˆ 1 (t) is a ratio implies that skewness may remain a part of the sampling distribution until sample sizes become very large. A small computer based empirical study confirms these results. For this study we generated 1000 samples of size n = 100, 500, 750 and 1000 from Gumbel copula with parameter δ = 4. We computed ηˆ (t) on each simulated sample where t was taken to be 0.95. Histograms and normal quantile plots for each sample size are given in Figs. 1 and 2 respectively. The plots demonstrate both that the convergence to a normal distribution does take place, but also that the convergence may be slow. This bolsters the idea of using the bootstrap in practical situations when the sample size is not very large. In order to study the power of the proposed test, we simulate tail non-exchangeable copulas using Khoudraji’s device. Table 1 summarizes the results for these simulated data. We find that the proposed test is fairly powerful when the sample size is large enough, with the exception that when both the strength of dependence and the degree of tail nonexchangeability are too weak, say δ = 1.5 and α1 = 0.9, it becomes much harder to detect such a subtle degree of tail non-exchangeability. Regarding the effect of dependence strength, the test is generally more powerful when the dependence is relatively stronger. Please cite this article as: L. Hua, A. Polansky and P. Pramanik, Assessing bivariate tail non-exchangeable dependence. Statistics and Probability Letters (2019) 108556, https://doi.org/10.1016/j.spl.2019.108556.

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Fig. 2. Normal quantile plots of the 1000 simulated realizations of ηˆ (t) from Gumbel copula.

Fig. 3. Scatter plots of the observed Uranium vs Lithium and Titanium data, and their uniform scores. Table 1 Statistical test power for different strength of dependence (δ ), degrees of tail non-exchangeability (α1 ), and sample sizes (S). The Gumbel copula is simulated with larger δ indicating stronger dependence, and α1 and α2 ≡ 1 are the parameters of Khoudraji’s device with smaller α1 indicating stronger tail non-exchangeability. ∆ is the test statistic. The p-values are each derived based on 500 simulated random samples.

δ

α1

S



p-value

δ

α1

S



p-value

δ

α1

S



1.5 2.0 1.5 2.0 1.5 2.0

0.3 0.3 0.6 0.6 0.9 0.9

200 200 200 200 200 200

0.421 0.374 0.148 0.216 0.100 0.049

0.082 0.046 0.464 0.070 0.364 0.678

1.5 2.0 1.5 2.0 1.5 2.0

0.3 0.3 0.6 0.6 0.9 0.9

500 500 500 500 500 500

0.179 0.225 0.202 0.196 0.076 0.074

0.150 0.036 0.058 0.022 0.704 0.430

1.5 2.0 1.5 2.0 1.5 2.0

0.3 0.3 0.6 0.6 0.9 0.9

1000 1000 2000 2000 2000 2000

0.268 0.292 0.124 0.131 0.054 0.073

p-value 0.004

<0.001 0.010

<0.001 0.364 0.010

16

In order to demonstrate the application of this test we consider the data of Cook and Johnson (1986) which consists of the observed log-concentration of seven chemical elements in 655 water samples collected near Grand Junction, Colorado. We will concentrate on the joint distributions of Uranium and Lithium, and Uranium and Titanium, respectively. The original data and their uniform scores are plotted in Fig. 3, and the uniform scores are computed as (R˜ 1j , R˜ 1j ) for j = 1, . . . , n where R˜ ij = [Ranki (Uij ) − 0.5]/n, where Ranki (Uij ) is the rank of Uij relative to Ui1 , . . . , Uin for i = 1, 2 and j = 1, . . . , n. The tail exchangeability was tested for each case using the bootstrap methodology outlined above. Figs. 4 and 5 show the results of those calculations. The left panel of each figure indicates the value of ηˆ (t) for values of t corresponding to the range between 0.75 and 0.95. The gray lines are the values of ηˆ (t) computed on 500 nonparametric bootstrap resamples. The right panel in each figure shows a histogram of the values of the test statistic ∆ computed on each of the 500 resamples. The vertical line indicated the observed value of ∆ from the sample. The bootstrap estimates of the p-values for testing the null hypothesis of tail exchangeability are given in Table 2. It is clear from the estimated p-values that there is substantial evidence against tail exchangeability for the joint distributions of Uranium and Titanium while there is little evidence against tail exchangeability for the joint distributions of Uranium and Lithium. Moreover, the left panel of Fig. 5 suggests that there exists significant tail non-exchangeability to the first element between Uranium and Titanium, as ηˆ (t) tends to be larger than 1 when t becomes sufficiently close to 1.

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4. Concluding remark

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Our primary goal throughout the paper is to assess bivariate tail non-exchangeable dependence. We first propose the function η(t) = E[U1 |U2 > t ]/E[U2 |U1 > t ] to measure the strength of tail non-exchangeability. Asymptotic Please cite this article as: L. Hua, A. Polansky and P. Pramanik, Assessing bivariate tail non-exchangeable dependence. Statistics and Probability Letters (2019) 108556, https://doi.org/10.1016/j.spl.2019.108556.

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Fig. 4. Results of the bootstrap test of the null hypothesis of tail exchangeability for the joint distribution of Uranium and Lithium.

Fig. 5. Results of the bootstrap test of the null hypothesis of tail exchangeability for the joint distribution of Uranium and Titanium.

Table 2 Bootstrap estimates of the p-values for testing the null hypothesis of tail exchangeability for the chemical concentration data. Chemicals



p-value

Chemicals



p-value

Uranium vs Lithium

0.0922

0.700

Uranium vs Titanium

0.5300

0.002

approximations are derived for non-exchangeable bivariate copulas that are constructed by Khoudraji’s device. Then a statistical test based on an empirical version of η(t) is proposed to formally test tail non-exchangeability. As empirically one cannot show limiting properties, we consider a series of upper sets of data on which η(t) is calculated. We have proved asymptotic normality of the proposed test statistic under mild conditions as the sample size goes to infinity. A real data is used to illustrate the usefulness of the proposed method. When α1 > α2 for Khoudraji’s device, we find a consistent pattern among different copulas that η(t) ≻ 1; that is, there is tail non-exchangeability to the first element. We conjecture that such a pattern may exist in many other bivariate parametric copula families. In addition to the study reported in the paper, we have also tried transforming each univariate marginal to different distributions such as exponential and Pareto to conduct a similar test. We have noticed that the choice of univariate marginals does not affect the tail behavior of η(t) very much and thus the conclusion of the proposed test for the chemical concentration data. However, the choice of using uniform scores as the pseudo data for conducting the test has a faster convergence speed in the sense of Proposition 2. Therefore, we suggest using the uniform scores as discussed in the paper. Nevertheless, the power of such a test based on transforming each marginal to different distributions remains an open question. The influence from different patterns of tail dependence on the tail behavior of η(t) deserves more research. The Frank copula example does suggest that the strength of tail dependence may affect the tail behavior of η(t). It would be interesting to study the cases of tail non-exchangeability when some generic tail dependence patterns such as usual tail dependence, intermediate tail dependence, tail quadrant independence and tail negative dependence appear; we refer to Hua and Joe (2011, 2013) for details about those different tail dependence patterns. Please cite this article as: L. Hua, A. Polansky and P. Pramanik, Assessing bivariate tail non-exchangeable dependence. Statistics and Probability Letters (2019) 108556, https://doi.org/10.1016/j.spl.2019.108556.

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The approach proposed in this paper only works for bivariate tail non-exchangeability. For tail non-exchangeability of multivariate cases, one can employ the approach for all the pairwise bivariate marginals. However, bivariate exchangeability does not imply multivariate exchangeability. For example, for a trivariate Gaussian copula with the three correlation coefficients not being the same, although all bivariate marginal copulas are exchangeable, the trivariate copula itself is not exchangeable. Therefore, further research is required for the latter case.

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Acknowledgments

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The authors thank the reviewers for their helpful suggestions and comments that lead to a better presentation of the paper; any remaining errors are the authors own. The main revision of the paper was conducted when the first author was visiting the Department of Statistics at University of Illinois at Urbana–Champaign, and the support is acknowledged. References Bernard, C., Czado, C., 2015. Conditional quantiles and tail dependence. J. Multivariate Anal. 138, 104–126, High-Dimensional Dependence and Copulas. Breitung, K., 1994. Asymptotic Approximations for Probability Integrals. Springer. Cook, R., Johnson, M.E., 1986. Generalized Burr–Pareto-logistic distributions with applications to a uranium exploration data set. Technometrics 28 (2), 123–131. Durante, F., Klement, E.P., Sempi, C., Úbeda-Flores, M., 2010. Measures of non-exchangeability for bivariate random vectors. Statist. Papers 51 (3), 687–699. Durante, F., Mesiar, R., 2010. L∞ -Measure of non-exchangeability for bivariate extreme value and archimax copulas. J. Math. Anal. Appl. 369 (2), 610–615. Furman, E., Su, J., Zitikis, R., 2015. Paths and indices of maximal tail dependence. Astin Bull. 45, 661–678. Genest, C., Ghoudi, K., Rivest, L.-P., 1998. ‘‘Understanding relationships using copulas,’’ by Edward Frees and Emiliano-Valdez, January 1998. N. Am. Actuar. J. 2 (3), 143–149. Genest, C., Kojadinovic, I., Nešlehová, J., Yan, J., et al., 2011. A goodness-of-fit test for bivariate extreme-value copulas. Bernoulli 17 (1), 253–275. Genest, C., Nešlehová, J., Quessy, J.-F., 2012. Tests of symmetry for bivariate copulas. Ann. Inst. Statist. Math. 64 (4), 811–834. Harder, M., Stadtmüller, U., 2014. Maximal non-exchangeability in dimension d. J. Multivariate Anal. 124, 31–41. Hua, L., Joe, H., 2011. Tail order and intermediate tail dependence of multivariate copulas. J. Multivariate Anal. 102, 1454–1471. Hua, L., Joe, H., 2013. Intermediate tail dependence: a review and some new results. In: Li, H., Li, X. (Eds.), Stochastic Orders in Reliability and Risk: In Honor of Professor Moshe Shaked. In: Lecture Notes in Statistics, Springer, pp. 291–311. Hua, L., Joe, H., 2014. Strength of tail dependence based on conditional tail expectation. J. Multivariate Anal. 123, 143–159. Hua, L., Joe, H., 2017. Multivariate dependence modeling based on comonotonic factors. J. Multivariate Anal. 155, 317–333. Khoudraji, A., 1996. Contributions à l’étude des copules et à la modélisation de valeurs extrêmes bivariées (Ph.D. thesis). Université Laval Québec, Canada. Klement, E.P., Mesiar, R., 2006. How non-symmetric can a copula be? Comment. Math. Univ. Carolin. 47 (1), 141–148. Krupskii, P., 2016. Copula-based measures of reflection and permutation asymmetry and statistical tests. Statist. Papers 1–23. Nelsen, R.B., 2007. Extremes of nonexchangeability. Statist. Papers 48 (2), 329–336. Polansky, A.M., 2011. Introduction to Statistical Limit Theory. CRC Press, Bocca Raton, Fl.

Please cite this article as: L. Hua, A. Polansky and P. Pramanik, Assessing bivariate tail non-exchangeable dependence. Statistics and Probability Letters (2019) 108556, https://doi.org/10.1016/j.spl.2019.108556.