Bounds for the Clayton copula

Abstract. We provide two upper bounds on the Clayton copula Cθ(u1, . . . , un) if θ > 0 and n > 2 and a lower bound in the case θ ∈ [−1, 0) and n > 2. The obtained bounds provide a nice probabilistic interpretation related to some negative dependence structures and also allow defining three new two-dimensional copulas, which tighten the classical Fréchet–Hoeffding bounds for the Clayton copula when n = 2.

The Clayton copula is interesting as it can model various kinds of dependence, ranging from comonotonicity in the limit as θ → ∞, independence if θ ↓ 0 (also if θ ↑ 0) and countermonotonicity if θ = −1 [7, p. 168]. This copula is often used in modelling when data shows asymmetry and lower tail dependence; see, e.g. [9,16] and [17] in finance, [1] and [18] in insurance, [2] in multiple test theory, among many other applications. For some other facts on the role of Clayton copulas in the Archimedean copula families, see, e.g. [12] and [13].
The interest in obtaining sharper than the classical Fréchet-Hoeffding bounds (see, e.g. [15, p. 30]) for the Clayton copula is partially motivated by the investigations of Dindienė and Leipus [5] who wondered whether, given a sequence of random variables X 1 , X 2 , . . . , such that for any integer n 1 and (x 1 , . . . , x n ) ∈ R n , if where F i (x i ) = P(X i x i ), i = 1, . . . , n, then there exists a κ > 0 such that that is, random variables X 1 , X 2 , . . . are upper extended negatively dependent (or have the UEND property) (see Section 3.1). It is well known (see [15,Cor. 4.6.3]) that C θ (u) is bounded from below (resp. above) by the independence copula Π if θ > 0 (resp. θ ∈ [−1, 0)): In this paper, we provide two upper bounds on C θ (u) if θ > 0 and a lower bound in the case θ ∈ [−1, 0). The first bound (see Theorem 1 below) yields that random variables X 1 , X 2 , . . . satisfying (1) for any n 1 are pairwise UEND, i.e. P(X i > x i , X j > x j ) (1 + θ)P(X i > x i )P(X j > x j ) for any i = j. However, the full UEND property requires further investigations, in particular, a sharper lower bound in Lemma 2 is needed.
Our first result is the following theorem for the case θ > 0: Theorem 1. Let θ > 0 and n 2. If u = (u 1 , . . . , u n ) ∈ [0, 1] n , then Remark 1. When n = 2 and θ > 0, the above inequality is simply with no indication why the two smallest arguments u (1) and u (2) appear in the general case. This is essentially due to the upper bound in Lemma 2 (see also Remark 3 below).
is a copula (in fact, Archimedean).
The rest of the paper is organized as follows: Section 2 contains the proofs of the stated results. Section 3 provides the details of the connection of the obtained bounds and pairwise UEND property of a sequence of random variables joined by Clayton copula (see Section 3.1) and describes when the new bounds are superior to the classical Fréchet-Hoeffding bounds (see Section 3.2).

Proofs
In this section, the proofs of Theorems 1, 2 and 3 are split into several lemmas for easier readability. In particular, Lemmas 1, 2, and 3 provide ingredients for the proof of Theorem 1. The proof of Proposition 1 is given at the end.
Proof. The non-negativity of the numerator above and its strict positivity on (1, +∞) n is a simple consequence of the majorization theory (see [10,Ch. 3]). Indeed, the function andx is not a permutation ofx. Clearly, for any To prove the stated upper bound in (7), for any Notice that, due to the symmetry of f n , we can assume that x = (x [1] , . . . , x [n] ). Then and since all x i 1. Therefore, by (8), ∂ 2 f n (x)/∂x 1 ∂x 2 0, implying that ∂f n /∂x 1 is nonincreasing in x 2 , and so This means that f n is nonincreasing in Remark 3. Some comments about the choice of x [1] and x [2] are in order. One can try, more generally, taking ( . . , n} in the denominator of the fraction in (7). Then since, clearly, ln Lemma 3. Let θ > 0. Then for any u = (u 1 , . . . , u n ) ∈ (0, 1) n , n 2, Proof. Write C θ (u) = exp{H θ (u)}, so that where ln u (1) ln u (2) .
Proof of Proposition 1. First observe that both T 1,θ and T 2,θ satisfy the required boundary conditions of a copula: and It remains to show that both T 1,θ and T 2,θ are 2-increasing, i.e. for any rectangle 0 a 1 a 2 1 and 0 b 1 b 2 1, Split the square [0, 1] 2 into four non-overlapping (except for touching boundaries) squares: (for the choice of ν j , see Section 3.2 (i) and (ii)). Then the intersections A 0 ∩ A i,j , i = 1, 2, 3; j = 1, 2, are again rectangles (possibly line segments or even empty) and where ν j c 1,j c 2,j 1 and ν j d 1,j d 2,j 1, then also and since the function z θ (x; a, b) := xb 1+θ ln x − xa 1+θ ln x for e −1/θ a b 1 and x ∈ [e −1/θ , 1] is nondecreasing in x, which follows from Hence both T 1,θ and T 2,θ are bivariate copulas.

Discussion
In this section we discuss an application of the obtained bounds to certain negative dependence structures mentioned in Section 1 as well as give a comparison with the classical Fréchet-Hoeffding bounds.

Connection to UEND structures
We now discuss some of the applications of the obtained bounds. For n = 2, Theorems 1 and 3 yield an interesting probabilistic interpretation, related to certain dependence structures. More explicitly, suppose random variables X 1 and X 2 are distributed according to laws F 1 and F 2 , respectively, and satisfy (1) with n = 2. Then, by (3) and (4), for θ > 0, where F i := 1 − F i , i = 1, 2, i.e. the variables X 1 , X 2 are both positively dependent and upper extended negatively dependent (UEND) (see [8]). Similarly, if −1 θ < 0, then and, similarly, variables X 1 , X 2 are both upper extended positively dependent (UEPD) and negatively dependent. Note that the mentioned UEND and UEPD properties are much easier to verify for classical Farley-Gumbel-Morgenstern, Frank or Ali-Mikhail-Haq copulas (see [5]). Unfortunately, extending the UEND property for θ > 0 (resp. UEPD for θ ∈ [−1, 0)) to n 3 random variables X 1 , . . . , X n with mutual distribution function generated by the Clayton copula (see (1)) requires a sharper lower bound (resp. upper bound) on C θ than provided by the independence copula Π. Indeed, e.g. for n = 3 and any (x 1 , . . . , x n ) ∈ R n , we have, by Sklar's theorem and inclusion-exclusion principle, Now since Clayton copula is Archmedean and (3) holds, we can write Using this and Theorem 1 for two of the three terms in the second sum of (15), we get https://www.mii.vu.lt/NA since, by Theorem 1, Of course, by symmetry, we can replace the term F 1 (x 1 )(F 2 (x 2 ) + F 3 (x 3 )) of the last inequality by but it still dominates the product , are close to zero. A sharper upper bound on the joint survival function could be possible provided a better lower bound in Lemma 2 is obtained. This is left for future research.
Note that the extended negative dependence concept has been demonstrated to be important in proving limit theorems of probability theory such as the strong law of large numbers (see, e.g. [3,11]), showing some max-sum equivalence properties for heavytailed distributions (see, e.g. [4,5]) or obtaining precise large deviations (see [8]).