On a generalization of Archimedean copula family

Accepted Manuscript On a generalization of Archimedean copula family Jiehua Xie, Feng Lin, Jingping Yang PII: DOI: Reference:

S0167-7152(17)30051-2 http://dx.doi.org/10.1016/j.spl.2017.02.001 STAPRO 7842

To appear in:

Statistics and Probability Letters

Received date: 26 July 2016 Revised date: 4 December 2016 Accepted date: 1 February 2017 Please cite this article as: Xie, J., Lin, F., Yang, J., On a generalization of Archimedean copula family. Statistics and Probability Letters (2017), http://dx.doi.org/10.1016/j.spl.2017.02.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

On a generalization of Archimedean copula family Jiehua Xiea , Feng Lina , Jingping Yangb,∗ a Department

b LMEQF,

of Financial Mathematics, Peking University, Beijing 100871, P.R.China Department of Financial Mathematics, Peking University, Beijing 100871, P.R.China

Abstract This paper introduces a new family of multivariate copula functions defined by two generators, which is a multi-dimensional extension of the bivariate copula presented in Durante et al. (2007a). The copula family is also a generalization of Archimedean copula family to allow for tail dependence. The probabilistic structure of the copula function is given. Some properties of the copula function are discussed, such as multivariate tail dependence and uniqueness. Keywords: Generalization of Archimedean copula, Probabilistic structure, Multivariate tail dependence

1. Introduction Copula method is a fundamental tool in dependence modeling (Nelsen, 2006) and it has been applied in many areas, such as finance, insurance and econometrics (McNeil et al., 2005; Denuit et al., 2005). An n-dimensional copula is a multivariate distribution function with uniform [0,1] marginal distributions. By Sklar’s Theorem (Nelsen, 2006) it is known that for n-dimensional distribution F with marginal distributions Fi , i = 1, . . . , n, there exists a copula C such that F(x1 , . . . , xn ) = C(F1 (x1 ), . . . , Fn (xn )), xi ∈ (−∞, +∞), i = 1, . . . , n. If the marginal distributions Fi , i = 1, . . . , n are continuous, the copula function C is unique. Archimedean copulas enjoy considerable popularity in various fields. An n-dimensional Archimedean copula is expressed as Cnφ (u1 , u2 , . . . , un ) = φ [−1] (φ (u1 ) + · · · + φ (un )), u1 , . . . , un ∈ [0, 1],

(1)

where the generator φ is a continuous and strictly decreasing convex function φ : [0, 1] → [0, +∞] satisfying that φ (1) = 0. The pseudo-inverse φ [−1] : [0, +∞] → [0, 1] is defined by φ [−1] (t) = φ −1 (min{φ (0),t}), where φ −1 is the inverse function of φ . φ

McNeil and Neˇslehov´a (2009) proved that Cn is an n-dimensional copula if and only if φ [−1] is n-monotone on [0, ∞), i.e., φ [−1] k

n−2

has derivatives up to the order n − 2 satisfying (−1)k dtd k φ [−1] (t) ≥ 0 on (0, ∞) for 1 ≤ k ≤ n − 2, and that (−1)n−2 dtd n−2 φ [−1] (t) is convex and nonincreasing on (0, ∞). Note that Archimedean copula is fully characterized by its generator φ (Alsina et al., 2006). Now Archimedean copulas have been widely used in actuarial science (Frees and Valdez, 1998; Albrecher et al., 2011), quantitative finance (McNeil et al., 2005; Sch¨onbucher, 2003) and biostatistics (Chaieb et al., 2006; Lakhal et al., 2008), due ∗ Corresponding

author Email address: [email protected] (Jingping Yang)

Preprint submitted to Statistics and Probability Letters

February 6, 2017

to their analytically tractable forms and good properties, as well as to the easy procedures for choosing an Archimedean copula to fit a data set (Wang and Wells, 2000). Most of the Archimedean copulas have few parameters, which make the copulas less flexible. Thus it is interesting to consider the generalization of Archimedean copula family. Liebscher (2008) discussed the problem by focusing on asymmetry. Durante et al. (2007a) presented a two-dimensional copula function defined by Cφ ,ψ (u1 , u2 ) = φ [−1] (φ (u1 ∧ u2 ) + ψ(u1 ∨ u2 )),

(2)

where u1 ∧ u2 = min{u1 , u2 }, u1 ∨ u2 = max{u1 , u2 }, ψ is a continuous decreasing function from [0, 1] to [0, +∞], ψ(1) = 0 and φ is a continuous and strictly decreasing convex function from [0, 1] to [0, +∞], and the conditions on φ and ψ were given for Cφ ,ψ (u1 , u2 ) being a copula function. Durante (2014) gave the stochastic mechanism to sample random variables from copulas of type (2). Note that Cφ ,ψ (u1 , u2 ) is a generalization of two-dimensional Archimedean copula with two generators φ and ψ, and Cφ ,φ (u1 , u2 ) is a bivariate Archimedean copula with generator φ . One interesting problem is to consider the extension of the bivariate copula Cφ ,ψ (u1 , u2 ) to multivariate case (n ≥ 3). Unfortunately, as pointed out by Durante et al. (2007a), their method can not be applied to high dimensional case. In this paper, by applying the idea of Marshall-Olkin distribution (Marshall and Olkin, 1967), we will consider the extension of bivariate copula Cφ ,ψ (u1 , u2 ) in Durante et al. (2007a). Note that the idea of Marshall-Olkin distribution has been applied in sampling random variables from copulas of type (2)(Durante, 2014) and new constructions of multivariate copulas (Mai et al., 2016). We will present one multivariate copula family with two generators to extend the copula function Cφ ,ψ (u1 , u2 ) to multivariate case. Our multivariate copula family is also a generalization of Archimedean copula family with two generators. The probabilistic structure of our multivariate copula will be given by applying Marshall-Olkin ideas, and some properties of the copula function are provided. Furthermore, the multivariate tail dependence coefficients of the copula function will be given. As an extension of the multivariate Archimedean copula, our copula family is more flexible than Archimedean copula. The rest of the paper is organized as follows. In Section 2, we introduce our multivariate copula family and the probabilistic structure for the copula function is provided. In section 3, the multivariate tail dependence coefficients of the copula family are derived. Section 4 focuses on comparison of the copulas in our copula family, where the identity between two copula functions is discussed. Conclusions are drawn in Section 5.

2. Generalization of Archimedean copula family In this section, we present one family of copula functions with two different generators to extend Archimedean copula and the bivariate copula in Durante et al. (2007a), and some examples are provided.

2

2.1. Some preparations First we give some notations. For fixed n ≥ 2, let Φn be the class of Archimedean generator φ , where φ is continuous and strictly decreasing convex function from [0, 1] to [0, +∞] with φ (1) = 0, and its pseudo-inverse function is n-monotone on [0, ∞). Let Φ∞ be the class of Archimedean generator φ whose pseudo-inverse φ [−1] is completely monotonic function, k

i.e., (−1)k dtd k φ [−1] (t) ≥ 0,t ∈ (0, ∞) for all k = 1, 2, . . .. Thus Φ∞ stands for the class of generators which can generate one Archimedean copula in any dimension n ≥ 2. See McNeil and Neˇslehov´a (2009) for more details. It is easy to see that Φ2 ⊇ Φ3 ⊇ · · · ⊇ Φ∞ . Denote by Ψ the class of all functions ψ : [0, 1] → [0, +∞] that are continuous, decreasing and satisfy ψ(1) = 0. Lemma 2.1. Suppose that (φ , ψ) ∈ Φn+1 × Ψ and ψ − φ is increasing in [0,1]. Denote FX (x) = φ [−1] (ψ(x)), x ∈ [0, 1],

FX ∗ (x) = φ [−1] (φ (x) − ψ(x)), x ∈ [0, 1],

(3)

and FX (x) = FX ∗ (x) = 0, x < 0. Then FX (x), x ∈ [0, 1] and FX ∗ (x), x ∈ [0, 1] are distribution functions. Proof. Notice that ψ(x) ≤ φ (x), x ∈ [0, 1]. In fact, since ψ(x) − φ (x), x ∈ [0, 1] is increasing, then for all x ∈ [0, 1), ψ(x) − φ (x) ≤ ψ(1) − φ (1) = 0, thus one can get ψ(x) ≤ φ (x), x ∈ [0, 1]. Moreover, it is easy to verify that limx→0− FX (x) = limx→0− FX ∗ (x) = 0, FX (1) = FX ∗ (1) = 1. From the definitions of Φn+1 and Ψ, one knows that φ and ψ are continuous on [0, 1] and φ [−1] is continuous on [0, ∞). Hence, FX (x) = φ [−1] (ψ(x)) is continuous on (0, 1]. Next we will prove that FX (x) is right-continuous at x = 0: (a) First we consider the case ψ(0) < +∞. Notice that FX (0) = φ [−1] (ψ(0)) in this case. Since φ [−1] is continuous on [0, ∞) and ψ is continuous on [0, 1], then limx→0+ FX (x) = limx→0+ φ [−1] (ψ(x)) = φ [−1] (ψ(0)) = FX (0). Thus FX (x) is right-continuous at x = 0. (b) When ψ(0) = +∞, by the increasing property of ψ − φ we can get φ (0) = +∞, thus φ is strictly decreasing on [0,1]

with φ [−1] = φ −1 and FX (0) = φ −1 (ψ(0)) = 0. Since limx→0+ FX (x) = limx→0+ φ −1 (ψ(x)) = 0 = FX (0), then FX (x) is also right-continuous at x = 0. Similarly, from ψ(x) ≤ φ (x), x ∈ [0, 1] we get that FX ∗ (x) = φ [−1] (φ (x)−ψ(x)) is also continuous on (0, 1]. Now we prove

that FX ∗ (x) is right-continuous at x = 0. In the case limx→0+ (φ (x) − ψ(x)) < +∞, we have FX ∗ (0) = φ [−1] (φ (0) − ψ(0)) and lim FX ∗ (x) = lim φ [−1] (φ (x) − ψ(x)) = φ [−1] (φ (0) − ψ(0)) = FX ∗ (0);

x→0+

x→0+

In the case limx→0+ (φ (x) − ψ(x)) = +∞, we have φ (0) = +∞, FX ∗ (0) = 0 and lim FX ∗ (x) = lim φ −1 (φ (x) − ψ(x)) = 0 = FX ∗ (0).

x→0+

x→0+

Combining the above two limits we conclude that FX ∗ (x) is right-continuous at x = 0. On the other hand, since φ [−1] is decreasing on [0, ∞] and ψ is decreasing on [0,1], then FX (x) = φ [−1] (ψ(x)) is increasing on [0, 1]. From that ψ − φ is increasing in [0,1], we know that FX ∗ (x) = φ [−1] (φ (x) − ψ(x)) is increasing on [0, 1]. Summarizing the above results, we conclude that FX (x), x ∈ [0, 1] and FX ∗ (x), x ∈ [0, 1] are distribution functions. 3

Remark 2.1. Notice that functions FX (x) and FX ∗ (x) are right-continuous, not necessarily continuous at x = 0. In fact, let ψ(t) = 1 − t and φ ∈ Φn+1 /Φ∞ such that ψ − φ is strictly increasing in [0,1]. In this case, FX (x) and FX ∗ (x) are right-

continuous at x = 0. Furthermore, 1 = ψ(0) < φ (0) < ∞ and φ −1 (1) > 0. Then, FX (0) = φ [−1] (1) = φ −1 (1) > 0 and FX ∗ (0) = φ [−1] (φ (0) − 1) = φ −1 (φ (0) − 1) > φ −1 (φ (0)) = 0. From the definitions of functions FX (x) and FX ∗ (x), we know that FX (x) = FX ∗ (x) = 0, x < 0. Therefore, FX (x) and FX ∗ (x) are not continuous at x = 0 in this case. 2.2. Definition of the copula function φ ,ψ

Given a pair of generators (φ , ψ) ∈ Φn+1 × Ψ, the function Cn

: [0, 1]n → [0, 1] is defined by


Cnφ ,ψ (u1 , u2 , . . . , un ) = φ [−1] φ (u(1) ) + ψ(u(2) ) + · · · + ψ(u(n) ) , u1 , . . . , un ∈ [0, 1],

(4)

where u(i) , i = 1, 2, . . . , n are the order statistics of ui , i = 1, 2, . . . , n, i.e., u(1) ≤ · · · ≤ u(n) . In the following, by applying the φ ,ψ

idea from Marshall-Olkin distributions, we will provide the conditions on generators φ and ψ such that Cn

is a multivariate

copula. Furthermore, we will also give the probabilistic structure of this multivariate copula function. Theorem 2.1. Suppose that (φ , ψ) ∈ Φn+1 × Ψ and the function ψ − φ is increasing in [0,1]. φ ,ψ

(a). The n-dimensional function Cn (u1 , . . . , un ) in (4) is a copula function; (b). Let X = (X1 , . . . , Xn , X ∗ ) be a random vector from the (n + 1)-dimensional Archimedean copula with generator φ , where Xi and X ∗ have marginal distributions FX (·) and FX ∗ (·) in (3) respectively. Denote Zi = max{Xi , X ∗ }, i = 1, . . . , n. Then Zi , φ ,ψ

i = 1, . . . , n are uniform [0, 1] random variables and (Z1 , . . . , Zn ) has multivariate copula Cn . Proof. Consider the joint distribution function of random vector (Z1 , . . . , Zn ). For ui > 0, i = 1, . . . , n, P(Z1 ≤ u1 , . . . , Zn ≤ un ) = P (X1 ≤ u1 , . . . , Xn ≤ un , X ∗ ≤ min{u1 , . . . , un }) = φ [−1] (φ (FX (u1 )) + · · · + φ (FX (un )) + φ (FX ∗ (min{u1 , . . . , un })))


= φ [−1] φ (FX (u(1) )) + φ (FX ∗ (u(1) )) + φ (FX (u(2) )) + · · · + φ (FX (u(n) )) ,

(5)

where u(1) ≤ u(2) ≤ · · · ≤ u(n) are the order statistics of ui , i = 1, . . . , n.

For fixed i, i = 1, . . . , n, setting ui = u and u j = 1, j = 1, . . . , n, j 6= i in (5), we can get the marginal distribution function FZi (u) = φ [−1] (φ (FX (u)) + φ (FX ∗ (u))) .

(6)

Notice that, from the definition of pseudo-inverse φ [−1] , we have φ [−1] (φ (t)) = t and φ (φ [−1] (t)) = min{φ (0),t}. Since (φ , ψ) ∈ Φn+1 × Ψ and the function ψ − φ is increasing in [0,1], we get φ (u) − ψ(u) ≤ φ (0) − ψ(0) ≤ φ (0) for all u ∈ [0, 1] such that φ (FX ∗ (u)) = φ (φ [−1] (φ (u) − ψ(u))) = min{φ (0), φ (u) − ψ(u)} = φ (u) − ψ(u), u ∈ [0, 1]. 4

(7)

Also, from the facts that function ψ is decreasing on [0, 1] and ψ(t) ≤ φ (t) for all t ∈ [0, 1], we know that ψ(u) ≤ ψ(0) ≤ φ (0) for all u ∈ [0, 1] such that φ (FX (u)) = φ (φ [−1] (ψ(u))) = min{φ (0), ψ(u)} = ψ(u), u ∈ [0, 1].

(8)

Substituting (7) and (8) into (6), we obtain FZi (u) = u, for u ∈ [0, 1]. Hence Zi , i = 1, . . . , n are uniform [0, 1] random variables. Also, substituting (7) and (8) into (5), we have

φ ,ψ

i.e., Cn


P(Z1 ≤ u1 , . . . , Zn ≤ un ) = φ [−1] φ (u(1) ) + ψ(u(2) ) + · · · + ψ(u(n) ) , ui ∈ [0, 1], i = 1, 2, . . . , n,

(9)

φ ,ψ

is a multivariate cumulative distribution function, and (Z1 , . . . , Zn ) has multivariate copula Cn . φ ,ψ

Remark 2.2. (1) Theorem 2.1 gives the probability structure of the copula Cn . It also provides a methodology for generating random vectors from the copula, which is a generalization of the sampling method presented in Durante (2014). (2) Durante et al. (2007a) obtained the corresponding result of Theorem 2.1(a) for n = 2. On the other hand, the copula φ ,ψ

function Cn

φ ,φ

is also a generalization of multivariate Archimedean copula. In fact, for ψ ≡ φ ∈ Φn+1 , Cn

is an n-dimensional

Archimedean copula with generator φ . From the following examples we can see that the copula functions of type (4) include some well-known copula functions. Example 2.1. For an increasing function f (t) : [0, 1] → [0, 1] satisfying f (1) = 1 and that the function f (t)/t,t ∈ (0, 1] is decreasing, let φ (t) = − lnt and ψ(t) = − ln f (t). Then one can obtain n

Cnφ ,ψ (u1 , u2 , . . . , un ) = u(1) ∏ f (u(i) ), u1 , . . . , un ∈ [0, 1], i=2

coinciding with the multivariate copula for aggregation process introduced by Durante et al. (2007b) and being also a special case of exchangeable multivariate copula family presented by Mai et al. (2016). Note that the stochastic mechanism of copula φ ,ψ

Cn

is similar to the exchangeable multivariate copula construction presented in Mai et al. (2016). In fact, both methods are

inspired by the Marshall-Olkin mechanism (Marshall and Olkin, 1967) of shock models. In particular, let f (t) = t α , α ∈ [0, 1], one obtains the resulting copula  α Cnφ ,ψ (u1 , u2 , . . . , un ) = u(1) u(2) · · · u(n) , u1 , . . . , un ∈ [0, 1].

Multivariate distributions of this kind were first presented by Marshall-Olkin (1967), and the above copula can also be considered as a multivariate version of the Cuadras-Augé family of copulas (Cuadras and Augé, 1981). Example 2.2. Choosing φ ∈ Φn+1 and ψ(t) = αφ (t), α ∈ [0, 1], we have

φ ,ψ


Cnφ ,ψ (u) = φ [−1] φ (u(1) ) + αφ (u(2) ) + · · · + αφ (u(n) ) , u1 , . . . , un ∈ [0, 1].

(10) φ ,ψ

If α = 0, then Cn (u) = u(1) , which is the multivariate Fréchet-Hoeffding upper bound copula. If α = 1, Cn (u) is an n-dimensional Archimedean copula with generator φ . 5

φ ,ψ

A distinctive property of copula Cn

which distinguishes it from the Archimedean copula family is its singular component

on the diagonal. Next, suppose that U1 , . . . ,Un are uniform [0, 1] random variables whose joint distribution function is the φ ,ψ

copula Cn . We will calculate the value of P(U1 = · · · = Un ). φ ,ψ

Proposition 2.1. (The singular component of Cn ) Let (φ , ψ) ∈ Φ∞ × Ψ, ψ − φ be increasing in [0,1] and U1 , . . . ,Un be φ ,ψ

uniform [0, 1] random variables whose joint distribution function is the copula Cn . If ψ(u) is differentiable in (0,1] with ψ(0) = +∞ and limu→0+ (φ (u) − ψ(u)) = +∞, then it holds that ˆ 1 0 ψ (u) P(U1 = · · · = Un ) = 1 − n du. 0 −1 0 φ (φ (φ (u) + (n − 1)ψ(u)))

(11)

Proof. Suppose that X ∗ and Xi , i = 1, . . . , n have distributions FX ∗ (·) and FX (·) in (3) respectively, and X = (X1 , . . . , Xn , X ∗ ) is a random vector from the (n + 1)-dimensional Archimedean copula with generator φ . Denote Xmax = max{X1 , . . . , Xn }. It is obvious that P(X ∗ ≤ u, Xmax ≤ v) = φ [−1] (φ (FX ∗ (u)) + nφ (FX (v))), u ∈ [0, 1], v ∈ [0, 1]. Note that if φ ∈ Φ∞ , then φ (0) = +∞

and the pseudo-inverse coincides with the inverse, i.e., φ [−1] = φ −1 . Hence, using (3), we get

P(X ∗ ≤ u, Xmax ≤ v) = φ −1 (φ (u) − ψ(u) + nψ(v)), u ∈ [0, 1], v ∈ [0, 1].

(12)

Applying the stochastic representation in Theorem 2.1(b), the sequence max{Xi , X ∗ }, i = 1, . . . , n are uniform [0, 1] φ ,ψ

random variables with joint distribution Cn . Hence, we can define Ui = max{Xi , X ∗ }, i = 1, . . . , n. From the proof of Lemma 2.1, we know that FX (x) and FX ∗ (x) are continuous on (0,1] and right-continuous at x = 0. Then by the assumptions (φ , ψ) ∈ Φ∞ × Ψ, ψ(0) = +∞ and limu→0+ (φ (u) − ψ(u)) = +∞, we have FX (0) = limx→0+ φ −1 (ψ(x)) = 0 and FX ∗ (0) = limx→0+ φ −1 (φ (x) − ψ(x)) = 0, thus FX (x) and FX ∗ (x) are continuous at x = 0. In summary, Xi , i = 1, . . . , n and X ∗ are continuous random variables and it holds that P(U1 = · · · = Un ) = P(X ∗ > Xmax ).

(13)

Then applying (12) and (13), we can get (11). As an example, we discuss the singular component of copula (10). Let φ ∈ Φ∞ . Then the assumptions of Proposition 2.1 are satisfied. Hence, using formula (11) we get P(U1 = · · · = Un ) = (1 − α)/(1 + (n − 1)α). Thus it is easy to see that copula (10) has a singular component on the diagonal when α ∈ [0, 1). Furthermore, the value of P(U1 = · · · = Un ) is independent of the selected function φ . 3. Tail dependence Copulas with different tail dependence are usually required for constructing stochastic models for examining extreme events (MeNeil et al., 2005). Several ideas were proposed to study the tail dependence (Nelsen, 2006), and these concepts are useful to describe the dependence between risks in actuarial and financial risk management (McNeil et al., 2005; Fernándezφ ,ψ

Sánchez et al., 2016). In this section, we will show the tail dependence of the copula function Cn . 6

First we give the notion of multivariate tail dependence coefficients introduced by Li (2009). Definition 3.1. (Li, 2009). For Xi ∼ Fi , i = 1, 2, . . . , n, with joint copula Cn , the multivariate upper and lower tail dependence I |Ihc

coefficients λUh

I |I c

and λLh h , respectively, are given by I |Ihc

= lim P (Fi (Xi ) > u, ∀i ∈ Ih |Fj (X j ) > u, ∀ j ∈ Ihc ) ,

Ih |Ihc

= lim P (Fi (Xi ) ≤ u, ∀i ∈ Ih |Fj (X j ) ≤ u, ∀ j ∈ Ihc ) ,

λUh

u→1−

λL

u→0+

for all 0/ 6= Ih ⊂ I = {1, 2, . . . , n} and Ihc = I \ Ih with respectively cardinal h ≥ 1 and n − h ≥ 1. From the above definition we can see that I |Ihc

λUh

= lim

u→1−

Cˆn (1 − u, . . . , 1 − u) , Ic Cˆnh (1 − u, . . . , 1 − u)

I |Ihc

λLh

= lim

Ic

u→0+

Cn (u, . . . , u) Ic

Cnh (u, . . . , u)

,

(14)

Ic

where Cˆn denotes the survival copula of the copula function Cn , and Cˆnh (Cnh ) denotes the marginal copula of Cˆn (Cn ) for all Ic

I c (1)

components being one except those in Ihc , i.e., Cˆnh (u, . . . , u) = Cˆn ((u) Ih {1}|{2}

IA (x) = 0. Note that if n = 2, λU

{1}|{2}

and λL

I c (n)

, . . . , (u) Ih

), with IA (x) = 1 if x ∈ A, otherwise

are the upper and lower tail dependence coefficients, respectively. φ ,ψ

and

Applying the principle of inclusion and exclusion, the survival copula of Cn can be expressed as   n  n  Cˆnφ ,ψ (1 − u, . . . , 1 − u) = 1 + ∑   (−1)k φ [−1] (φ (u) + (k − 1)ψ(u)), k=1 k Ihc ,φ ,ψ

Cˆn I c ,φ ,ψ

here Cˆnh

 n − h   k [−1] (φ (u) + (k − 1)ψ(u)), (1 − u, . . . , 1 − u) = 1 + ∑   (−1) φ k=1 k n−h

I c (1)

φ ,ψ

(1 − u, . . . , 1 − u) = Cˆn ((1 − u) Ih



I c (1)

, . . . , (1 − u) Ih

(15)

(16)

).

Since limu→1− (φ (u) + (k − 1)ψ(u)) = 0 = φ (1) < φ (0), k = 1, . . . , n, one can replace φ [−1] by φ −1 during the calculation I |I c

of λUh h . By applying equation (14), one can get the multivariate upper dependence coefficient   n   1 + ∑nk=1   (−1)k φ −1 (φ (u) + (k − 1)ψ(u)) k I |I c {1,2,...,h}|{h+1,...,n}   λUh h = λU = lim u→1−  n−h  1 + ∑n−h  (−1)k φ −1 (φ (u) + (k − 1)ψ(u)) k=1  k

(17)

if the limit exists.

φ ,ψ

Similarly, for copula function Cn I |Ihc

λLh

we can derive its multivariate lower tail dependence coefficient {1,2,...,h}|{h+1,...,n}

= λL

φ [−1] (φ (u) + (n − 1)ψ(u)) u→0+ φ [−1] (φ (u) + (n − h − 1)ψ(u))

= lim

if the limit exists. 7

(18)

Another dependent coefficient is extremal dependent coefficient introduced by Frahm (2006). We recall that the notion of the upper (lower) extremal dependent coefficient λUEDC (λLEDC ) is defined as the probability that all univariate margins are large (small) given that at least one of them is large (small). Definition 3.2. (Frahm, 2006). For Xi ∼ Fi , i = 1, 2, . . . , n, with joint copula Cn , the upper and lower extremal dependent coefficients λUEDC and λLEDC , respectively, are given by λUEDC = lim P u→1−

λLEDC = lim P u→0+



 min {Fi (Xi )} > u| max {Fi (Xi )} > u ,

1≤i≤n



1≤i≤n

 max {Fi (Xi )} ≤ u| min {Fi (Xi )} ≤ u . 1≤i≤n

1≤i≤n

From these definitions, it is easy to see that λUEDC and λLEDC can be simplified as Cn (u, . . . , u) Cˆn (1 − u, . . . , 1 − u) , λLEDC = lim . u→0+ 1 − Cˆn (1 − u, . . . , 1 − u) u→1− 1 −Cn (u, . . . , u)

λUEDC = lim

φ ,ψ

Substituting the survival copula (15) of Cn

into the equation above, we get upper and lower extremal dependent coefficients φ ,ψ

for the generalization Archimedean copula Cn , that is  λUEDC = lim

u→1−

 n  1 + ∑nk=1   (−1)k φ −1 (φ (u) + (k − 1)ψ(u)) k 1 − φ −1 (φ (u) + (n − 1)ψ(u))

λLEDC = lim

u→0+

if the limits exist.





(19)

,

φ [−1] (φ (u) + (n − 1)ψ(u)) 

(20)

 n  ∑nk=1   (−1)k−1 φ [−1] (φ (u) + (k − 1)ψ(u)) k 0

0

0

Theorem 3.1. Suppose that (φ , ψ) ∈ Φn+1 × Ψ, ψ − φ is increasing in [0, 1], φ (1−) and ψ (1−) exist and φ (1−) 6= 0. For the copula function 0

ψ (1−) 0 φ (1−)

φ ,ψ Cn

defined in (4), we have

{1,2,...,h}|{h+1,...,n}

6= 1, then λU

{1,2,...,n−1}|{n} λU

= 1−

0

ψ (1−) 0 φ (1−)

0

ψ (1−)

and λUEDC =

1− 0 φ (1−)

0 ψ (1−)

1+(n−1) 0 φ (1−)

. Moreover, if

= 1 for h = 1, . . . , n − 2.

Proof. From (17) we have  n   1 + ∑nk=1   (−1)k φ −1 (φ (u) + (k − 1)ψ(u)) k 

{1,2,...,n−1}|{n}

λU

= lim

u→1−

1 − φ −1 (φ (u))

Applying L’Hôpital rule to the equation above, we get  
0 n φ φ −1 (φ (u))  n  {1,2,...,n−1}|{n} k+1 λU = lim ∑   (−1) 0 φ (φ −1 (φ (u) + (k − 1)ψ(u))) u→1− k=1 k 8

.

0

ψ (u) 1 + (k − 1) 0 φ (u)

!

.

0

0

0

Since ψ(1) = φ (1) = 0, φ (1−) and ψ (1−) exist and φ (1−) 6= 0, then   ! 0 n ψ (1−)  n  {1,2,...,n−1}|{n} k+1 λU = ∑   (−1) 1 + (k − 1) 0 φ (1−) k=1 k     ! 0 0 n n ψ (1−)  n  ψ (1−)  n  = 1− 0 (−1)k+1  (−1)k+1   k + 0 ∑ ∑ φ (1−) k=1 φ (1−) k=1 k k = 1−



where the formulas ∑nk=1 (−1)k+1 

n k

0

ψ (1−) , 0 φ (1−)





n

 = 1 and ∑nk=1 (−1)k+1 

k

(21)



 k = 0 are used in the last equality.

Similarly, from (17), for h = 1, . . . , n − 2 we have          n   1 + ∑nk=1   (−1)k φ −1 (φ (u) + (k − 1)ψ(u)) / 1 − φ −1 (φ (u))     k {1,2,...,h}|{h+1,...,n}     . λU = lim u→1−     n − h   1 + ∑n−h  (−1)k φ −1 (φ (u) + (k − 1)ψ(u)) / {1 − φ −1 (φ (u))} k=1      k {1,2,...,n−1}|{n}

Using the same method of calculating λU 0

equation above are equal to 1 − ψ0 (1−) . Thus if φ (1−)

0

, it yields that the limits of the numerator and the denominator in the

ψ (1−) 0 φ (1−)

{1,2,...,h}|{h+1,...,n}

6= 1, then λU

= 1 for h = 1, . . . , n − 2. λUEDC can be

calculated similarly and we omit its proof. The theorem is proved. {1,2,...,n−1}|{n}

Letting ψ ≡ φ in Theorem 3.1, we derive that λU

= 0 for n ≥ 2, which was obtained by Fernández-Sánchez

et al. (2016). Hence, it shows that Archimedean copula does not have upper tail dependence given that the first derivative φ ,ψ

0

of generator φ (1−) 6= 0 is finite. However, the copula function Cn

of type (4) may have upper tail dependence when

0

φ (1−) 6= 0 is finite, and the multivariate tail dependence coefficients depend on the value of 0

0

ψ (1−) . 0 φ (1−)

0

Denote θ1 := − lims→1− ((1 − s)φ (s)/φ (s)) if the limit exists. In the case φ (1−) 6= 0, then θ1 = 1 and from Charpentier and Sergers (2009) and Bernardino and Rullière (2016) we know that the upper tail of a multivariate Archimedean copula {1,2,...,h}|{h+1,...,n}

with generator φ exhibits asymptotic independence and λU

= 0 for h = 1, . . . , n − 2. In the generalization

of Archimedean copula family (4), by choosing a suitable generator ψ, the multivariate upper tail dependence coefficient {1,2,...,h}|{h+1,...,n}

λU

0

can be 1 for h = 1, . . . , n − 2 even if φ (1−) 6= 0. {1,2,...,h}|{h+1,...,n}

Remark 3.1. (1) Under some assumptions, the upper tail dependence coefficient λU φ ,ψ

Cn

of the copula function

does not depend on the value of n. φ ,ψ

0

(2) For one Archimedean generator φ with φ (1−) 6= 0, by choosing a suitable generator ψ, the multivariate copula Cn {1,2,...,n−1}|{n}

have arbitrary multivariate upper tail dependence coefficients λU

more flexible for practical application than Archimedean copula. 9

may

φ ,ψ

and λUEDC . Hence, the copula function Cn

is

Clayton copula, Frank copula and Gumbel copula are three important classes of copula functions in Archimedean family. In the following, we will give the multivariate tail dependent coefficients for copula function (10) with these three types of generators as an illustration. Example 3.1. Following Example 2.2, we choose the generator φ as the generators of Clayton copula, Frank copula and Gumbel copula, respectively. Note that the generators of the above three Archimedean copulas can be expressed as φ (t) =  −θt  −1 t −θ − 1, θ > 0, φ (t) = − ln ee−θ −1 , θ > 0, and φ (t) = (− lnt)θ , θ ≥ 1. Moreover, the generator ψ(t) = αφ (t), where    n  α ∈ [0, 1]. Then the tail dependent coefficients are listed in the following tables, here χn (α, β ) = ∑nk=1   (−1)k−1 (1 + k (k − 1)α)β . Note that the multivariate tail dependent coefficients λU

{1,2,...,n−1}|{n}

{1,2,...,h}|{h+1,...,n}

and λU

, h = 1, . . . , n − 2 are

different in some cases. In the case α = 1, the copula function in (10) is an Archimedean copula with generator φ . Thus from Table 1 and Table 2 we can get the tail dependent coefficients of Clayton copula, Frank copula and Gumbel copula. In the case α ∈ (0, 1), the {1,2,...,n−1}|{n}

upper tail dependent coefficients λU , λU

and λUEDC are dependent on α. For example, the copula generated by

the operator of Frank copula is upper tail dependent, and it has arbitrary upper tail dependence coefficients by setting different values on α. {1,2,...,n−1}|{n}

{1,2,...,h}|{h+1,...,n}

φ (t)

λU

t −θ − 1  −θt  −1 − ln ee−θ −1

1−α

1−α

1

1−α 1+(n−1)α

1−α

1−α

1

1−α 1+(n−1)α

χn (α, θ1 ) χn−h (α, θ1 )

χn (α, θ1 )

(− lnt)θ

λU

1

λU

χn (α, θ1 )

2 − (1 + α) θ

λUEDC

1

(1+(n−1)α) θ

Table 1: The upper tail dependent coefficients for copula (10), here h < n − 1.

φ (t) t −θ − 1  −θt  −1 − ln ee−θ −1 (− lnt)θ

{1,2,...,h}|{h+1,...,n}

λL 1

(1 + α)− θ 0

λL 

1+(n−1)α 1+(n−h−1)α

0

0 0

− 1 θ

λLEDC 1

− (1+(n−1)α) θ 1 χn (α,− θ )

0 0

Table 2: The lower dependent coefficients for copula (10), here h ≤ n − 1.

4. Comparison results It is known that an Archimedean copula with generator φ is equal to an Archimedean copula with generator ξ , if and only if there exists α > 0 such that φ [−1] (t) = ξ [−1] (αt), t ≥ 0, see Alsina et al. (2006) for details. In this section, the identity between two copula functions in type (4) is discussed. Through comparing two copula functions in (4) with different pair of generators, one may catch the essential properties of the copula family. 10

φ ,ψ

Theorem 4.1. Let Cn

ξ ,ω

and Cn

be two copula functions with pairs of generators (φ , ψ), (ξ , ω) ∈ Φn+1 × Ψ, and ψ − φ ξ ,ω

φ ,ψ

and ω − ξ are increasing in [0,1]. Denote M = ψ ◦ φ [−1] and N = ω ◦ ξ [−1] . Then Cn (u) = Cn (u), u ∈ [0, 1]n , if and only if there exists λ > 0 such that ξ [−1] (t) = φ [−1] (λt),t ∈ [0, ∞), and N(t) =

1 M(λt),t ∈ [0, φ (0)/λ ]. λ

ξ ,ω

φ ,ψ

Proof. We start by proving necessity. For Cn (u) = Cn (u), u ∈ [0, 1]n , necessarily ! φ

n

φ (u(1) ) + ∑ ψ(u( j) )

[−1]

(22)

=ξ

!

n

ξ (u(1) ) + ∑ ω(u( j) ) ,

[−1]

j=2

j=2

(23)

for u1 , . . . , un ∈ [0, 1] satisfying φ (u(1) ) + ∑nj=2 ψ(u( j) ) ≤ φ (0). Applying φ to the both sides of (23), we get n

n

j=2

j=2

!

φ (u(1) ) + ∑ ψ(u( j) ) = φ ◦ ξ [−1] ξ (u(1) ) + ∑ ω(u( j) ) .

(24)

Letting A = φ ◦ ξ [−1] , a1 = ξ (u(1) ) ∈ [0, ξ (0)] and ai = ω(u(i) ) ∈ [0, ω(0)], i = 2, . . . , n, (24) can be rewritten as n

A(a1 ) + ∑ ψ ◦ ω

[−1]

j=2

n

!

(a j ) = A a1 + ∑ a j . j=2

(25)

Note that ω − ξ is increasing in [0,1]. From the proof of Lemma 2.1, one gets ω(u) ≤ ξ (u) for all u ∈ [0, 1]. Hence a1 = ξ (u(1) ) ≥ ω(u(1) ) ≥ ω(u(2) ) = a2 ≥ · · · ≥ ω(u(n) ) = an ≥ 0, i.e., ξ (0) ≥ a1 ≥ a2 ≥ · · · ≥ an ≥ 0. In order to prove that (22) holds, we first analyze the expression of A(t). Since φ , ξ ∈ Φn+1 , then φ (u), u ∈ [0, 1]

and ξ [−1] (v), v ∈ [0, ξ (0)] are continuous, strictly decreasing and differentiable, and the derivatives of φ (u), u ∈ [0, 1] and

ξ [−1] (v), v ∈ [0, ξ (0)] are strictly negative. Hence A(v) = φ ◦ ξ [−1] (v), v ∈ [0, ξ (0)] is also continuous and differentiable, and its derivative is strictly positive. Differentiating (25) with respect to a1 , one can get ! 0

0

A (a1 ) = A

n

a1 + ∑ a j .

(26)

j=2

Notice that this equality holds whenever ξ (0) ≥ a1 ≥ a2 ≥ · · · ≥ an ≥ 0. 0

0

0

Since φ , ξ ∈ Φn+1 , then φ (t), t ∈ (0, 1) and (ξ [−1] (t)) , t ∈ (0, ξ (0)) are continuous. Thus A is continuous on (0, ξ (0)). 0

Now we will prove that A (t),t ∈ (0, ξ (0)) is constant. Firstly, for n ≥ 2, letting ε > 0 be sufficiently small and setting a1 = ε 0

0

in (26), we get A (ε) = A (ε + ∑nj=2 a j ), for all ξ (0) ≥ ε ≥ a2 ≥ · · · ≥ an ≥ 0. Notice that ∑nj=2 a j ∈ [0, (n − 1)ε]. Setting 0

0

0

t = ∑nj=2 a j , we get A (ε) = A (ε +t) for all t ∈ [0, (n − 1)ε], i.e, A is constant on [ε, nε]. Secondly, set a1 = nε in (26). Then, 0

0

0

0

we get A (nε) = A (nε + t) for all t ∈ [0, (n − 1)nε], i.e, A is also constant on [nε, n2 ε]. Thus it yields that A is constant on 0

[ε, n2 ε]. By similar deduction, one can derive that A is constant on [ε, nk ε] for all k ∈ N+ . Since limk→∞ nk ε = ∞, n ≥ 2, 0

0

from the analysis above, we can verify that A is constant on [ε, ξ (0)) for any ε > 0. Letting ε → 0, one gets A is constant on 0

0

(0, ξ (0)). Then we conclude that A (t) = λ for all t ∈ (0, ξ (0)), where λ is a constant. Note that A (t) is strictly positive on (0, ξ (0)), then λ > 0. Since A(t) is continuous on [0, ξ (0)], we get A(t) = λt + γ for all t ∈ [0, ξ (0)]. Also, from A(0) = 0 we

have γ = 0. Thus A(t) = φ ◦ ξ [−1] (t) = λt,t ∈ [0, ξ (0)], which can be rewritten as ξ [−1] (t) = φ [−1] (λt),t ∈ [0, ξ (0)]. From 11

this equality, we easily get ξ (0) = φ (0)/λ . Also, if t ∈ (ξ (0), ∞), then ξ [−1] (t) = 0 = φ [−1] (λt). Hence, ξ [−1] (t) = φ [−1] (λt) holds for all t ∈ [0, ∞).

Next we will verify that H(t) = λ1 M(λt),t ∈ [0, φ (0)/λ ]. Substituting ξ [−1] (t) = φ [−1] (λt), t ∈ [0, ∞) into equation (23),

one obtains n

n

j=2

j=2

∑ ψ(u( j) ) = λ ∑ ω(u( j) ),

for all ui ∈ [0, 1],

which is equivalent to ψ(u) = λ ω(u), u ∈ [0, 1]. Thus using M = ψ ◦ φ [−1] and N = ω ◦ ξ [−1] , it can be rewritten as M(φ (u)) = λ H(ξ (u)) = λ N



 φ (u) , u ∈ [0, 1]. λ

Setting t = φ (u)/λ ∈ [0, φ (0)/λ ] in the above equation, we verify that N(t) = λ1 M(λt),t ∈ [0, φ (0)/λ ]. The proof of sufficiency is trivial, so we omit it.

5. Concluding remarks In this paper, we introduce a new family of copula functions to generalize the bivariate copula presented in Durante et al. (2007a) to multi-dimensional case, which is also a generalization of Archimedean copula family with two generators. The probabilistic structure of the copula function is given, which is easily applied for generating random numbers from the copula function. The advantage of the copula family is that it may have tail dependence. By comparing two copula functions in the family, the uniqueness is obtained.

Acknowledgments The authors thank the editor and two referees for their valuable comments and suggestions which lead to the improvement of the paper. Xie’s research was supported by the National Natural Science Foundation of China (Grants No. 11561047). Yang’s research was supported by the National Natural Science Foundation of China (Grants No. 11671021).

References Albrecher, H., Constantinescu, C., Loisel, S., 2011. Explicit ruin formulas for models with dependence among risks. Insurance: Mathematics and Economics 48, 265-270. Alsina, C., Frank, M.J., Schweizer, B., 2006. Associative Functions: Triangular Norms and Copulas. World Scientific, Hackensack. Bernardino, E. D., Rullière, D., 2016. On tail dependence coefficients of transformed multivariate Archimedean copulas. Fuzzy Sets and Systems 284, 89-112. Chaieb, L. L., Rivest, L. P., Abdous, B., 2006. Estimating survival under a dependent truncation. Biometrika 93, 655-669.

12

Charpentier, A., Segers, J., 2009. Tails of multivariate Archimedean copulas. Journal of Multivariate Analysis 100, 15211537. Cuadras, C.M., Augé, J., 1981. A continuous general multivariate distribution and its properties. Communications in Statistics-Theory and Methods 10, 339-353. Denuit, M., Dhaene, J., Goovaerts, M., Kaas, R., 2005. Actuarial Theory for Dependent Risks: Measures, Orders and Models. John Wiley and Sons Ltd, Chichester. Durante, F., 2014. Simulating from a family of generalized Archimedean copulas. In Melas, V. B., Mignani, S., Monari, P., Salmaso, L., editors, Topics in Statistical Simulation, Springer Proceedings in Mathematics and Statistics, pages 149-156, Springer, New York. Durante, F., Quesada-Molina, J.J., Sempi, C., 2007a. A generalization of the Archimedean class of bivariate copulas. Annals of the Institute of Statistical Mathematics 59, 487-498. Durante, F., Quesada-Molina, J.J., Úbeda-Flores, M., 2007b. On a family of multivariate copulas for aggregation process. Information Sciences 177, 5715-5724. Fernández-Sánchez, J., Nelsen, R.B., Quesada-Molina, J.J., Úbeda-Flores, M., 2016. Independence results for multivariate tail dependence coefficients. Fuzzy Sets and Systems 284, 129-137. Frahm, G., 2006. On the extremal dependence coefficient of multivariate distributions. Statistics and Probability Letters 76, 1470-1481. Frees, E.W., Valdez, E.A., 1998. Understanding relationships using copulas. North American Actuarial Journal 2, 1-25. Lakhal, L., Rivest, L. P., Abdous, B., 2008. Estimating survival and association in a semicompeting risks model. Biometrics 64, 180-188. Li, H., 2009. Orthant tail dependence of multivariate extreme value distributions. Journal of Multivariate Analysis 100, 243-256. Liebscher, E., 2008. Construction of asymmetric multivariate copulas. Journal of Multivariate Analysis 99, 2234-2250. Mai, J. F., Schenk, S., Scherer, M., 2016. Exchangeable exogenous shock models. Bernoulli 22, 1278-1299. Marshall, A.W., Olkin, I., 1967. A multivariate exponential distribution. Journal of the American Statistical Association 62, 30-44. McNeil, A.J., Frey, R., Embrechts, P., 2005. Quantitative Risk Management. Princeton University Press, Princeton. McNeil, A.J., Neˇslehov´a, J., 2009. Multivariate Archimedean copulas, d-monotone functions and l1 -norm symmetric distributions. The Annals of Statistics 37, 3059-3097. Nelsen, R.B., 2006. An Introduction to Copulas, Second Edition. Springer, New York. Sch¨onbucher, P., 2003. Credit Derivatives Pricing Models. John Wiley, Chichester. Wang, W., Wells, M.T., 2000. Model selection and semiparametric inference for bivariate failure-time data. Journal of the American Statistical Association 95, 62-76. 13

We introduce a new family of multivariate copula functions defined by two generators. The copula family is a generalization of Archimedean copula family. We extend the bivariate copula in Durante et al. (2007a) to multivariate case. The probabilistic structure of the copula function is given. Multivariate tail dependence and uniqueness are discussed.