Sklar’s theorem, copula products, and ordering results in factor models


 We consider a completely specified factor model for a risk vector X = (X
 1, . . ., Xd
 ), where the joint distributions of the components of X with a risk factor Z and the conditional distributions of X given Z are specified. We extend the notion of *-product of d-copulas as introduced for d = 2 and continuous factor distribution in Darsow et al. [6] and Durante et al. [8] to the multivariate and discontinuous case. We give a Sklar-type representation theorem for factor models showing that these *-products determine the copula of a completely specified factor model. We investigate in detail approximation, transformation, and ordering properties of *-products and, based on them, derive general orthant ordering results for completely specified factor models in dependence on their specifications. The paper generalizes previously known ordering results for the worst case partially specified risk factor models to some general classes of positive or negative dependent risk factor models. In particular, it develops some tools to derive sharp worst case dependence bounds in subclasses of completely specified factor models.


Introduction
A relevant class of distributions for modeling dependencies are factor models where each component of the underlying random vector X = (X , . . . , X d ) is supposed to depend on some common random factor Z through for some functions f i and a random vector (ε , . . . , ε d ) that is independent of Z . In this paper, we consider the case where Z is a real-valued random variable. If the bivariate distribution of (X i , Z) is speci ed and the distribution of X|Z = z is known for all i and z , then the distribution of X is fully speci ed. We denote this setting a completely speci ed factor model (CSFM). For applications to risk modeling, partially speci ed factor models (PSFMs) are introduced in Bernard et al. [5]. In these models, the distributions of (X i , Z) are speci ed. The joint distribution of (ε , . . . , ε d ) is, however, not prescribed. This means, that only the distributions of X i and Z as well as the copula D i = C X i ,Z of (X i , Z) are given. Then, the worst case distribution in the PSFM is determined by the conditionally comonotonic random vector X c Z = (F − X |Z (U), . . . , F − X d |Z (U)) , where U ∼ U( , ) is independent of Z , assuming generally a non-atomic underlying probability space (Ω, A, P) . If Z has a continuous distribution, the copula of X c Z is given by the upper product of the bivariate copulas D i , see [2]. Figure 1 On the left: a partially speci ed factor model with dependence speci cations D , . . . , D d and risk factor distribution function G . On the right: a completely speci ed factor model with dependence speci cations E , . . . , E d , conditional copula family C and factor distribution function G .
In standard factor models, the individual factors ε , . . . , ε d are assumed to be independent. Then, the distribution of X is completely speci ed and the components of X are conditionally independent given Z = z for all z . Further, the copula of X is then given by the conditional independence product of the bivariate speci cations D i , which is an extension of the bivariate copula product introduced in Darsow et al. [6] to arbitrary dimension, see [15].
In this paper, we introduce and study the * -product of copulas as an extension of the bivariate copula product considered in [8] to the multivariate case and to general factor distribution functions in order to model the copula of X = (f i (Z, ε i )) i for general dependence structures among (ε , . . . , ε d ) and also discontinuous Z . We provide a simple representation of a conditional distribution function by the corresponding univariate distribution functions and a generalized derivative of the associated copula. Then, we derive a Sklar-type theorem implying that the dependence structure of X is determined by the * -product of the dependence speci cations in the CSFM. Further, we establish a general continuity result for the * -product in dependence on all its arguments which is useful for corresponding approximation results. We study transformation properties of the * -product and introduce, as a counterpart of the upper product, the lower product of bivariate copulas in the two-and three-dimensional case.
In Section 3, we derive general lower and upper orthant ordering results for the * -product in dependence on the copula speci cations. This requires the consideration of integral inequalities like the rearrangement results of Lorentz [16] and Fan and Lorentz [11]. We extend and strengthen several recent results on the lower and upper orthant ordering of upper products to general * -products. In particular, we show that componentwise convexity of the conditional copulas plays an important role for the ordering of the * -products. We introduce the ≤ ∂ S -ordering on the set of bivariate copulas based on the Schur-ordering of copula derivatives allowing to derive a meaningful comparison criterion. We show that many well-known copula families satisfy this ordering.
Finally, in Section 4, we combine the * -product ordering results with the ordering of marginal distributions and obtain several general ordering results in CSFMs. As a consequence, this yields maximum elements and, thus, sharp bounds w.r.t. the lower and upper orthant ordering for classes of PSFMs as well as for classes of CSFMs with the conditional independence assumption.
The * -product of copulas in completely speci ed factor models for all x = (x , . . . , x d ) ∈ R d . The copula C is uniquely determined on the Cartesian product × d  (2) is positive for all ≤ t < t ≤ because ι G (t ) ≥ t > t ≥ ι − G (t) by Lemma A. 1

(iv). (b) If f is left-continuous and if the (ordinary) left-hand derivative f − (t ) := lim t t f (t )−f (t)
t −t exists, then ∂ G f (t ) exists for all G ∈ F . To see this, we know by (a) that ι G

) , then ∂ G f (t ) exists since f and ι G are left-continuous, see Lemma A.1(vi). (c) A useful transformation property of ∂ G is that
∂ G f (t) = ∂ G f (ι G (t)) = ∂ G f (G(x)) for all G ∈ F and for Lebesgue-almost all t , where x = G − (t) . This is a consequence of Lemma A. 1(v) considering the cases where G is continuous at x or has a jump discontinuity at x , compare equations (38) and (39) in the proof of Theorem 2.2.
The following result gives the representation of a conditional distribution function by the univariate marginals and the generalized partial derivative of the corresponding copula.

Theorem 2.2 (Representation of conditional distribution functions).
For F, G ∈ F , let X ∼ F and Z ∼ G be real random variables with copula C ∈ C , i.e., C = C X,Z . Then, the following statements hold true: (i) For all x ∈ R , there exists a G-null set Nx such that the conditional distribution function of X given Z = z evaluated at x is represented by for all z ∈ N c x . (ii) There exists a G-null set N such that F X|Z=z (x) = lim w x ∂ G C(F(w), G(z)) (5) for all x ∈ R and for all z ∈ N c .
The proof is given in the Appendix. (4) and (5), we make use of the left-hand limit in the de nition of the generalized di erential operator given by (2). If G has a discontinuity at z , then the operator ∂ G is the di erence quotient operator w.r.t. the second component of C between G(z) and G − (z) . If G is continuous at z , the operator ∂ G reduces to the ∂ − -operator denoting the left-hand partial derivative with respect to the second variable. Hence, if G is continuous for all z , then it holds that ∂ G = ∂ − . Denote by ∂ the operator which takes the partial derivative w.r.t. the second component of a multivariate function. Since copulas are almost surely partially di erentiable, see Nelsen [22,Theorem 2.2.7], it holds for all u , that ∂ − C(u, v) = ∂ C (u, v) for almost all v . (b) We point out that the right-hand expression in (4) is not necessarily right-continuous in x , and, thus, it does not generally de ne a distribution function in x . However, in the following de nition of the * -product as well as in most results of the paper, we integrate over the conditioning variable and, then, this representation of the conditional distribution function is appropriate.

Remark 2.3. (a) For the representation of the conditional distribution function in
In the following de nition, we extend the * -product introduced by Darsow et al. [6] for Markov structures, and, for arbitrary conditional dependencies, by Durante et al. [8] (for d = ) and [2] (for d ≥ ) to G ∈ F allowing also discontinuous factor distribution functions.
We need a measurability assumption which is implicitly assumed in the above mentioned literature by the de nition of the corresponding integrals. We call a family B = (B t ) t∈ [ , ] of d-copulas measurable if the The * -product of bivariate copulas is de ned in dependence on a measurable family B = (B t ) t∈ [ , ] of d-dimensional copulas and on a distribution function G ∈ F .
De nition 2.4 ( * -product of copulas). Note that the number d of bivariate copulas is typically clear from the context and therefore the simpli ed notation is used. We also sometimes use the notation D * B,G · · · * B,G D d := * B,G D i for the * -product of d bivariate copulas D . . . , D d w.r.t. to B and G . Note that for xed u , . . . , u d ∈ [ , ] the integrand in (6) is well-de ned as a consequence of Remark 2.1(b) because copulas are Lipschitz-continuous. The justi cation for the simpli ed notation in (iii) of the above de nition is due to Proposition 2.14.
As usual, we denote by Π d , M d , and W d , de ned by where we apply (3) for the second equality and use that B G t = B G ι G (t) which follows from Lemma A.1(v). The third equality follows from the transformation formula, see, e.g., [32,Theorem 2]. For the fourth equality, we use for xed (u , . . . , Since the last integral is a mixture of distribution functions, the product * B,G D i is a distribution function. The measurability of Fz(u , . . . , u d ) in z is a consequence of the measurability of B and, by (4) It remains to show that * B,G D i has uniform marginals.
where the rst equality holds due to the uniform marginals of the copula B G t , the second one is a consequence of the transformation formula and (3), and the last equality is given by Theorem 2.2 and the disintegration theorem.

. Sklar-type theorem for factor models
The following theorem describes the meaning of the notion of * -products. It is a version of Sklar's Theorem for completely speci ed factor models and states that the dependence structure of a random vector (X i ) ≤i≤d that follows a completely speci ed factor model, X i = f i (Z, ε i ) , is given by a * -product of the speci cations Conversely, for distribution functions F , . . . , F d+ ∈ F , bivariate copulas C , . . . , C d ∈ C and a measur- ] in (10) de nes a (d + )-dimensional distribution function F ,...,d+ with bivariate marginal distribution functions F i,d+ given by (9) and d-variate distribution function F ,...,d given by (11).
Proof. Due to Sklar's Theorem in the bivariate case, there exist C , . . . , C d ∈ C such that (9) holds for all (x , . . . , x d+ ) ∈ R d+ . The univariate marginal distribution functions of F ,...,d|F − d+ (t) are given by for all x ∈ R and for almost all t ∈ [ , ] , To show (11), we apply the disintegration theorem and obtain for all where for the second equality we use the representation in (10) and that lim , is a (d + )-dimensional distribution function with marginal distribution function of the rst d components given by and bivariate marginal distribution functions w.r.t. to the i-th and (d + )-st component given by The uniqueness properties follow directly from the uniqueness properties in Sklar's Theorem.
As a consequence of Sklar's theorem 2.7 for factor models, the conditional independence product, the upper product, and the lower product is characterized by conditional independence, conditional comonotonicity, and conditional countermonotonicity, respectively. Corollary 2.9. For ≤ i ≤ d and F i ∈ F , let X i ∼ F i be random variables on a non-atomic probability space. Then, for G ∈ F and D , . . . , D d ∈ C , the following statements hold true.
Throughout the following sections, the copula families B and C are assumed to be measurable.

. Basic properties of * -products
For a d-copula C , denote byC the corresponding survival function and by C its survival copula. Then, the survival function and the survival copula of the * -product are determined as follows.

Proposition 2.10 (Survival function and survival copula).
The survival function * B,G D i and the survival copula * B,G D i of the * -product * B,G D i are given by * B, Proof. Let (U , . . . , U d , Z) be a random vector such that U i is uniformly distributed on ( , ) , Z ∼ G , and for almost all t ∈ ( , ) and C U i ,Z = D i for all ≤ i ≤ d , compare Remark 2.8(a). Then, it holds by (11) where the third equality follows by the application of Sklar's Theorem for survival functions to the conditional survival function in the integrand, see, e.g., Georges et al. [12,Theorems 1 and 2] using that the i-th conditional marginal survival function is given byF The fourth equality is a consequence of Theorem 2.2. The second statement follows from the relationship C(u , . . . , u d ) =C( − u , . . . , − u d ) , (u , . . . , u d ) ∈ [ , ] d , between the survival copula C and the survival functionC of a copula C ∈ C d .
For some particular speci cations, the * -products simplify as follows.
Note that statements (i), (ii), and (v) in the above result are formulated w.r.t. continuous risk factor distribution functions and cannot be generalized to arbitrary G ∈ F . A counterexample can be constructed from the following example. Then, it holds that Π G D i = Π d ≠ M d using that ι G (t) = and ι − G (t) = for all t ∈ ( , ) . In fact, for Z ∼ G , it holds that P(Z = ) = , and, thus, the dependence speci cations C X i ,Z = D i = M do not yield any information on the X i and cannot force comonotonicity of (X , . . . , X d ) .
Next, we study the product * B,G D i in the case where D i = M for all i . We make use of ordinal sums de ned as follows.
Let J ⊂ N be a nite or countable subset of the natural numbers. Let (a k , b k ) k∈J be a family of pairwise disjoint, open subinterval of [ , ] and let (C k ) k∈J be a family of d-copulas. Then, the ordinal sum ( a k , b k , C k ) k∈J of (C k ) k∈J w.r.t. (a k , b k ) k∈J is de ned by where u = (u , . . . , u d ) ∈ [ , ] d , see, e.g., Mesiar and Sempi [17].
The following proposition characterizes ordinal sums by * -products.

Proposition 2.13 (Ordinal sums).
For G ∈ F , for a measurable family B = (B t ) t∈ [ , ] and a sequence (C k ) k∈J of d-copulas, and for pairwise disjoint open subintervals (a k , b k ) k∈J of ( , ) , the following statements are equivalent: which implies the assertion. Note that B G t is constant for t ∈ (ι − G (t), ι G (t)) .
Denote by A the closure of a set A ⊂ R . The following result justi es the simpli ed notation for the * -products where the argument G is omitted in the case that G is continuous, see De nition 2.4(iii). The proof is given in the Appendix.
Let d ≥ . Then, the following statements are equivalent: As a consequence of the above result, the * -product depends only on the closure of the range of the risk factor distribution G . Thus, the copula of a completely speci ed factor model is invariant under strictly increasing transformations of the factor variable.
The following result shows in which relevant cases the * -product attains the upper Fréchet copula.

Proposition 2.15 (Maximality).
For the * -product, the following statements hold true.
Proof. Statements (i) and (ii) follow from Proposition 2.11(i) and (ii). Statement (iii) is an extension of [2, Proposition 2.4(v)] to arbitrary G ∈ F . We give the proof in the Appendix. ) denotes the distribution of T w.r.t. λ . Let T P be the set of all T ∈ T such that T is bijective. Then, elements of T P are called shu es, see [9].
The following statement shows that simpli ed * -products are invariant under joint shu es of the factor variable Z assuming a continuous distribution function.

Proposition 2.16 (Invariance under shu es). For all T ∈ T P and C
given through is a bivariate copula. Furthermore, for simpli ed * -products with continuous factor distribution function and The proof is given in the Appendix.

. Continuity results for * -products
In this section, we derive continuity properties of the * -product w.r.t. to all its speci cations.
For the approximation of * -products w.r.t. the factor distribution, we need the following lemma. The proof is given in the Appendix.
Lemma 2.17. For Gn , G ∈ F , n ∈ N , the following statements hold true.
where each convergence is almost surely pointwise.
In the following example, we consider some typical approximations of distribution functions for which the corresponding transformations ι converge almost surely pointwise.

Then, Gn is a distribution function for all n with
Similarly to the above example, it holds that Ran(Gn) ⊆ { , n , n , . . . , } and ι Gn → ι G almost surely pointwise.
The following two counterexamples show that, in general, neither convergence in distribution (denoted by D − →) implies almost surely pointwise convergence of the corresponding transformations ι nor that the converse holds true.

Example 2.19 (Gn
. Let Gn = F N( , /n) be the distribution function of the normal distribution with mean and variance n . Then, For a continuity result of the * -product * B,G D i w.r.t. the bivariate dependence speci cations D i , we consider as slightly generalized version of the ∂-convergence for bivariate copulas in Mikusiński and Taylor [18].

Remark 2.22. a) Some typical approximations of copulas are the checkerboard, the checkmin and the Bernstein approximation, respectively. All these approximations are w.r.t. the ∂-convergence, see Mikusiński and
Taylor [18], and, thus, also w.r.

t. the ∂ -convergence. In contrast, the ∂ -convergence does not generally hold for the shu e-of-min approximation, see Mikusiński and Taylor [18, Example 4]. b) For a bivariate copula D , denote by D T with D T
for A, B ∈ C , see [30]. Note that for As a main result, we give su cient conditions for the continuity of the * -product w.r.t. all its arguments.

Theorem 2.23 (Continuity of * -products)
. ] be measurable families of d-copulas, and Gn , G ∈ F be distribution functions for all n ∈ N . If then it holds true that * B n ,Gn D i n → * B,G D i uniformly .
Due to the equicontinuity of copulas, the above * -products converge uniformly using Arzelà-Ascoli's theorem. Thus, the statement follows from the exchangeability of applying the limits and, again, from Arzelà-Ascoli's theorem. First, we show (17). Assume w.l.g. that and, thus, and for a subsequence (k l ) l∈N , then it follows from Lebesgue's di erential theorem, see, e.g., [4,Theorem 8.4.6], that and, since the partial derivative of a copula exists almost surely, that Altogether, this yields as k → ∞ , where we apply the dominated convergence theorem.
To show (15), let j ∈ { , . . . , d} and choose w.l.g. G k = G , B m = B , and D i n = Dn for all k, m, n ∈ N and i ≠ j . Let (G l ) l∈N be the discrete approximation of G given in Example 2.18(b). Then, the family (B G l t ) t is constant in t on the intervals ( κ− l , κ l ) , ≤ κ ≤ l , and each B G l t is Lipschitz continuous with Lipschitz constant .
Thus, for the Lebesgue measure λ on [ , ] , it holds that for all ε > and ≤ κ ≤ l , where the convergence follows from the assumption that D j n ∂ −→ D j . Then which implies that * B,G l D i n (u) → * B,G l D i (u) as n → ∞ for all l . Thus, the statement follows from * B,G l D i n l→∞ − −− → * B,G D i n uniformly, see (17). Statement (16) follows with the dominated convergence theorem.
In the following remark, we note that a weak approximation of the bivariate dependence speci cations or a weak approximation of the factor distribution does not guarantee the convergence of the corresponding * -products.

Remark 2.24. (a)
In general, the * -product * B,G D i is not continuous in D i w.r.t. weak convergence. A counterexample is given for the upper product and G ∈ F c in [2,Example 2.7].

For a counterexample, let (Gn)n be the approximation of G given by Example 2.19. Then, Gn
If the D i do not coincide for all i , then the * -products do not necessarily converge because, e.g., for the upper products, it holds that where the rst equality holds due to the continuity of Gn for all n , and the inequality is true because of the maximality property of the upper product, see Proposition 2.15(iii). The last equality follows from

. The lower product of bivariate copulas
In the following proposition, we provide basic properties for the lower product of bivariate copulas which are parallel to some results in [2] for the upper product. For a bivariate copula D ∈ C , de ne the re ected copulas D * and D * by Proposition 2.25. For D, E ∈ C and for a random vector (U , U , U ) the following statements hold true: (vi) In general, the lower product is neither commutative nor associative.
The proof is given in the Appendix.

Ordering results for * -products
In this section, we establish lower and upper orthant ordering results for the * -product * B,G D i w.r.t. the conditional copulas B and the bivariate speci cations D i . By the Sklar-representation theorem (Theorem 2.7) these results imply corresponding dependence ordering results for CSFM w.r.t. their speci cations.
In comparison to the ordering of * B,G D i w.r.t. the speci cations D i , an ordering w.r.t. the copula family B is a simple task and given by the following proposition which extends Durante et al. [8,Proposition 3].
where ≺ is one of the orders ≤ lo , ≤uo , and ≤c , respectively, then it holds true that * B,G D i ≺ * C,G D i for all G ∈ F and for all copulas Proof. The statement follows from the closure of these orders under mixtures (see Shaked  Another di culty is that, for xed i ∈ { , . . . , d} , ordering results for * d depends on D i through the (generalized) partial derivative ∂ G D i of D i . More precisely, a pointwise ordering of the integrands w.r.t. D i and E i , i.e., t)) ≤j≤d for all (u , . . . , u d ) ∈ [ , ] d and t ∈ ( , ) , is not possible: If we set u j = for all j ≠ i , then for all t implies D i = E i on [ , ] × Ran(G) and, thus, * d j= ,B,G D j = * d j= ,B,G E j .
In the remaining part of this section, we derive several lower and upper orthant ordering results for * d j= ,B,G D j w.r.t. the D i verifying integral inequalities based on the Schur-ordering, the sign-change ordering, and the lower orthant ordering, respectively.

. Ordering results for componentwise convex conditional copulas
Denote by ≺ S the Schur-ordering for functions, i.e., for integrable functions f , g : Here h * denotes the decreasing rearrangement of an integrable function h , i.e., the (essentially w.r.t. the Lebesgue measure λ) We say that a family As a basic integral inequality result, we make use of the following theorem on rearrangements from Fan and Lorentz [11,Theorem 1].

Theorem 3.3 (Ky Fan-Lorentz Theorem).
, be a family of continuous functions. Then, the following statements are equivalent: (ii) Φ with Φ(t, ·) := Φ t (·) satis es the following conditions for all ≤ t ≤ , ≤ a ≤ − δ , δ > , u k ≥ , k = , . . . , d , h ≥ and i ≠ j where those arguments are omitted which are the same in each expression: For a function f : be the di erence operator where ε > and where e i denotes the i-th unit vector w.r.t. the canonical base in R d . Then, f is said to be supermodular, respectively, directionally convex if ε i i ε j j f (x) ≥ for all x ∈ R d , for all ε i , ε j > , and for all ≤ i < j ≤ d , respectively, ≤ i ≤ j ≤ d . Note that in the literature, directionally convex functions are also called ultramodular or Wright convex.
Here, Condition (20) is supermodularity of Φ t for all t , condition (21) is convexity of Φ t in each component for all t . Functions that ful ll both conditions (20) and (21) are directionally convex. Motivated by Theorem 3.3, we consider the class C ccx d of componentwise convex d-copulas which is identical to the class of directionally convex copulas since copulas are supermodular. (22) is reversed, i.e., Φ has continuous second partial derivatives w.r.t. all variables, then conditions (20), (21), (22), and (23), respectively, are equivalent to

Remark 3.4. (a) As a consequence of the transformation formula, Theorem 3.3 also holds true if "decreasing" in (i) is substituted by "increasing" and the inequality in
respectively, see Lorentz [16].
In order to apply the Ky Fan-Lorentz Theorem to * -products, we consider an important class of bivariate copulas which are convex or concave in the second variable. For the next theorem, we need the following lemma. The proof is given in the Appendix.  Assume that B = (B t ) t∈ [ , ] is a continuous family of d-copulas. Then, the following statements are equivalent: (i) For all G ∈ F and for all CIS copulas D i , (ii) B ful lls conditions (21) and (22).
Since D i and E i are CIS, the functions f i and g i are decreasing; this yields f i ≺ S g i . Together with the boundedness of f i and g i it follows from the Ky Fan-Lorentz Theorem 3.3 that * B, because (B G t ) t ful lls conditions (21) and (22), see Lemma 3.6. This proves (i). The reverse direction follows in the same way as in the proof of the Ky Fan-Lorentz Theorem 3.3 (see Fan and Lorentz [11,Theorem 1]) because for all decreasing functions f i , g i : A similar result holds true w.r.t. the upper orthant ordering as follows. Assume that B = (B t ) t∈ [ , ] is a continuous family of d-copulas. Then, the following statements are equivalent: using the uniform marginal condition ∂ i B t (u) du i = u j and that ∂ i B t (u) is increasing in u i . For a discussion of componentwise convex copulas, see, e.g., Klement et al. [14] and Klement et al. [13].

15(iii). (c) The ordering results for comonotonic random vectors in Rüschendorf [24, Corollary 3(b)] and for random vectors with common CI copula in Müller and Scarsini [19, Theorem 4.5], respectively, are based on the application of the Ky Fan-Lorentz Theorem 3.3 to (conditional) quantile functions. In contrast, Theorem 3.7 follows from the Ky Fan-Lorentz Theorem 3.3 comparing conditional distribution functions w.r.t. the conditioning variable.
We make use of another integral inequality due to Lorentz [16] as follows.
Note that the above result also holds true if we replace the decreasing rearrangements f * i by the increasing rearrangements f i* of f i and condition (22) by (23).
As a consequence of the Lorentz Theorem 3.10, we obtain for continuous factor distribution functions G ∈ F c the following result concerning shu es. Proposition 3.11. Let D , . . . , D d ∈ C be CIS copulas.

] is a continuous family of d-copulas that ful lls condition (22), then it holds true that
Since D i is conditionally increasing, the decreasing rearrangement is given by g * i,u i (t) = ∂ D i (u i , t) for almost all t . Hence, Theorem 3.10 implies * B S T i (D i )(u) = B t (g i,u i (t)) ≤i≤d dt ≤ B t (g * i,u i (t)) ≤i≤d dt = * B D i (u) .
The second statement follows from the rst one with Proposition 2.16.

Remark 3.12. (a) Note that the speci cations on the right side of (24) are jointly shu ed. (b) A similar result to Proposition 3.11 holds true w.r.t. the upper orthant ordering. A generalization to arbitrary factor distribution functions G
To apply Lorentz's Theorem 3.10 to the ordering of * B,G D k w.r.t. D i , we introduce and study the orderings ≤ ∂ S,G and ≤ ∂ S on the set C of bivariate copulas.

De nition 3.13 (≤ ∂ S , Schur order for copula derivatives).
For G ∈ F and D, E ∈ C , de ne the Schur order for the partial copula derivative (w.r.t. the second variable) by For G ∈ F c , we abbreviate ≤ ∂ S,G by ≤ ∂ S .
The least element in C w.r.t to the ≤ ∂ S -order is given by the independence copula Π , i.e., it holds that Π ≤ ∂ S C for all C ∈ C . In contrast, a greatest element does not exist. However, M and W as well as every shu e of these copulas are maximal elements.

Proposition 3.14. Let D and E be bivariate copulas. Then D ≤ ∂ S E implies ζ (D T ) ≤ ζ (E T ) .
Proof. By de nition of the D -metric in (14) and by the transpose of a copula, we have that where the inequality follows from the Hardy-Littlewood-Polya theorem which states that f ≤ S g is equivalent to φ(f (t)) dt ≤ φ(g(t)) dt for all convex functions φ : R → R such that the expectations exist, see, e.g., [26,Theorem 3.21].
In general, D ≤ ∂ S E does not imply D ≤ ∂ S,G E even if E is a CIS copula, which is shown by the following counterexample.  Hence, we obtain for u ∈ ( , ] that ∂ G D * (u, ·) S ∂ G M (u, ·) and ∂ G D * (u, ·) ≠ ∂ G M (u, ·) . But this means that D * ≰ ∂ S,G M .
However, if both D and E are CIS (or CDS), then it can easily be veri ed that D ≤ ∂ S E yields D ≤ ∂ S,G E .
A relation of the ≤ ∂ S -ordering to the lower orthant ordering is given as follows. Note that we obtain from the de nition of the re ected copula E * of E in (18) that E * = ∂ S E , where, as usual, = ∂ S holds if ≤ ∂ S and ≥ ∂ S is ful lled.
For the increasing rearrangement g u * of ∂ D(u, t) , it similarly holds that for all u, v ∈ [ , ] , using that D and E are CIS. The reverse direction is given by (i).
Consider the class of bivariate copulas that are closer than E to the independence copula or equal to E w.r.t. the ≤ ∂ S -ordering. Due to the following result, the class C E has a least and a greatest element w.r.t. the lower orthant ordering given by a CDS and a CIS copula.

Proposition 3.17. There exist a unique CDS copula E ↓ ∈ C E and a unique CIS copula E
It holds that E ↓ = E * ↑ , where E * ↑ is de ned by (18), and Then, E ↑ is a bivariate copula, where the property of -increasingness follows for (u , v ) ≤ (u , v ) from Since ∂ E ↑ (u, ·) is a rearrangement of ∂ E(u, ·) , it holds that E = ∂ S E ↑ . Since ∂ E ↑ (u, t) = fu(t) for almost all t and fu is the essentially uniquely determined decreasing rearrangement of ∂ E(u, ·) , it follows that E ↑ is the uniquely determined CIS copula with E = ∂ S E ↑ . For the lower bound E ↓ , given by , the statement follows similarly, so (25) is proved. Since fu(t) dt = u for all u ∈ [ , ] , it follows that for all (u, v) ∈ [ , ] . Statement (26) follows with Lemma 3.16 (i).
In the following, we give some examples of ≤ ∂ S -ordered copula families.
Combining the Ky Fan-Lorentz Theorem 3.3 and Lorentz's Theorem 3.10, we get the following main result. Since E i is CIS, it holds that ∂ G E i (u i , ·) is decreasing. From the assumption that D i ≤ ∂ S,G E i , we obtain for the decreasing rearrangement g * i,u i of g i,u i that g * i,u i ≺ S ∂ G E i (u i , ·) . This yields

Theorem 3.20 (≤ ∂ S -ordering criterion). Let G ∈ F and let D i , E i ∈ C be bivariate copulas with E i CIS and D i ≤ ∂ S,G E i for all
where we apply Theorems 3.10 and 3.3 using that also the copulas (B G t ) t are componentwise convex and ful ll condition (22), see Lemma 3.6. Statement (ii) follows similarly to (i) applying formula (13) for the survival function of the * -product.
Since the independence copula coincides with its survival copula and is componentwise convex, we obtain the following result as a consequence of Theorem 3.20.  is no convexity condition w.r.t. B and B , 3 and 3.10, that  * B,G D i (u, . . . , u) ≥  * B,G E i (u, . . . , u) . So, for general G ∈ F \ F c and for a general continuous family B of componentwise convex d-copulas which ful lls condition (22), we have the following diagram:

. Upper product ordering results
To derive ordering results for upper and lower products of bivariate copulas, consider on C the sign change ordering and the symmetric sign change ordering de ned as follows.
For bivariate copulas D, E ∈ C , de ne the function fu,v : for almost all t ∈ ( , ) as the di erence of the partial derivatives of E and D w.r.t. the second variable for xed rst components u, v ∈ [ , ] .

De nition 3.24 (Sign change orderings).
The sign change ordering D ≤ ∂∆ E , respectively, the symmetric sign change ordering D ≤ s∂∆ E is de ned via the property that for all u, v , respectively, for all u = v , the function fu,v has no (−, +)-sign change.
The sign change orderings strengthen the standard bivariate dependence orderings. It holds true that see [2,Proposition 3.4]. Note that the lower and upper Fréchet copula are the least and greatest element, respectively, w.r.t. the ≤ ∂∆ -ordering, i.e., it holds that W ≤ ∂∆ D ≤ ∂∆ M for all D ∈ C . Examples of ≤ ∂∆ordered copula families are elliptical copulas and some families of Archimedean copulas, see [2]. Each of both conditions and implies D i ≥c E i , see [2,Proposition 3.6]. We generalize this result to arbitrary factor distributions as follows.

Theorem 3.25 (Sign-change ordering criterion for upper products).
Let G ∈ F be a distribution function and let D i , E i ∈ C , ≤ i ≤ d , be bivariate copulas. If either (28) or (29) holds, then it follows that Proof. Assume (28). For ≤ i ≤ d − and u i , v ∈ [ , ] , the functions f i , g i , h : ( , ) → [− , ] given by have a.s. no (−, +)-sign change. Then, also the piecewise averaged functions have a.s. no (−, +)-sign change. Thus, the assertion follows in the same way as the proof of [2, Proposition 3.6].
Under the assumption of (29), the statement follows similarly with [2, Lemma 3.2], using that the functions f G i , g G i , and h G i have a.s. no (+, −)-sign change.
Since we make use of it later on, we cite another concordance ordering criterion for upper products, based on the lower orthant ordering of the arguments.

Proposition 3.26 (≤ lo -ordering criterion for upper products).
For D , . . . , D d , E ∈ C , the following statements are equivalent: The result of Proposition 3.26 is given by [3, Theorem 1] even for the tighter supermodular ordering.

. Lower product ordering results
An ordering criterion similar to the sign change criterion for upper products in Theorem 3.25 holds true for lower products. Remember that, in general, the lower products M ∧ G D ∧ G E and W ∧ G D ∧ G E are -copulas only for continuous G . The symmetric copula D * associated with D ∈ C is de ned in (18).

Theorem 3.27 (Sign-change ordering criterion for lower products).
For bivariate copulas D , D , D ∈ C and G ∈ F , the following statements hold true: Proof. To show the lower orthant ordering in (i), let u = (u , u , u ) ∈ [ , ] . In the case that G ∈ F \ F c is discontinuous, set u = .
Consider the functions f , g, h : [ , ] → [− , ] de ned by Then f , g, h have no (−, +)-sign change and it holds that f (t) dt = g(t) dt . This yields the integral inequality where the rst equality follows from ∂ G M (u , t) = 1 {u >t} for almost all t and for arbitrary u ∈ [ , ] in the case that G is continuous, respectively, for u = if G is discontinuous. This yields M ∧ D ∧ D ≤ lo M ∧ D ∧ D in the continuous case and D ∧ G D ≤ lo D ∧ G D for arbitrary G .
For the upper orthant ordering in (i), we obtain analogously that

Statement (ii) follows analogously.
Similarly to the ≤ lo -ordering criterion for the concordance ordering of upper products in Proposition 3.26, we obtain a concordance-ordering result for lower products based on a ≤ lo -ordering criterion for the bivariate dependence speci cations.

Theorem 3.28 (≤ lo -ordering criterion for lower products).
Let D, E , E ∈ C be bivariate copulas. Then, the following statements are equivalent: Proof. Assume (i). To show the lower orthant ordering, let u = (u , u , u ) ∈ [ , ] . Then, it holds that where the rst inequality follows from the assumptions using that D * ≤ lo E if and only if D ≥ lo E * . The second inequality holds due to Jensen's inequality.
For the upper orthant ordering, we similarly obtain Assume (ii). Then, (i) follows from the closure of the lower orthant ordering under marginalization and from the marginalization property of * -products, see Proposition 2.11(iv).

. Ordering results for convex combinations
In Section 3.1, we have established that general lower orthant ordering results for * B,G D i in D i for xed D j , i ≠ j , are only possible if the conditional copulas B = (B t ) t ful ll the convexity condition (21). Remember that this convexity condition implies negative dependence of the bivariate marginals of B t . Motivated by Theorem 3.20 for componentwise convex conditional copulas and by Proposition 3.26 concerning a ≤ lo -ordering criterion for the upper product, the question arises for which * -products ordering results of the form hold true. Note that E is assumed to be a joint upper bound for the D i .
To partly answer this question, we generalize the necessary integral ordering condition in the Ky Fan-Lorentz Theorem 3.3 under an additional ordering assumption on the upper bound.  ≤ g i d , i , . . . , i d ∈ { , . . . , d} , such that f i ≺ S g i the integral inequality (19) holds true, then Φ ful lls the milder convexity condition where those components are omitted which are the same in each expression.
As a consequence, we obtain that lower orthant ordering results for * -products with a joint upper bound for all copulas also restrict the choice of conditional copulas.

Corollary 3.30. If for all CIS copulas
holds true, then B ful lls the milder convexity condition (31).
Proof. Let f i , g i be decreasing and bounded such that f i ≺ S g i and g i ≤ . . . ≤ g i d , i , . . . , i d ∈ { , . . . , d} . Assume w.l.g. that ≤ f i , g i ≤ . Then, there exist u , . . . , u d ∈ [ , ] and CIS copulas D i , E ∈ C with D i ≺ ∂ S E such that f i (t) = ∂ D i (u i , t) and g i (t) = ∂ E(u i , t) . Thus, the statement follows from Proposition 3.29. (31). In this case, inequality (32) is trivially ful lled because E i = M d whenever E i = E for all i . Note that for the upper product the non-trivial generalized inequality

Remark 3.31. (a) Due to Corollary 3.30, ordering results of the form (30) can not be obtained for all continuous families B = (B t ) t∈[ , ] of d-copulas. (b) The upper Fréchet copula M d ful lls the milder convexity condition
holds true whenever D i ≤ lo E (see Proposition 3.26).
Denote by co(M d , C ccx d ) the set of convex combinations of M d with elements of C ccx d . Then, we obtain the following result.
Then, for the simpli ed * -products, it holds true that * B D i ≤ lo * B E i .
Proof. The copula B is of the form B = aM d + ( − a)C , for some a ∈ [ , ] , where C ∈ C ccx d ful lls the convexity condition (21). Thus, the statement follows from Theorem 3.20 and from (33) using that D i ≺ ∂ S E implies D i ≤ lo E , see Lemma 3.16.
Note that in the above result, E i = E for i ∈ { , . . . , d} is a joint upper bound for the copulas D , . . . , D d .

Ordering results for completely speci ed factor models
In this section, we combine the ordering results on * -products in Section 3 with the ordering of the univariate marginal distributions. This leads to lower and upper orthant as well as concordance ordering results for CSFMs and, thus, to bounds w.r.t. these orderings in classes of CSFMs and PSFMs, respectively.
Suppose that X = (X , . . . , X d ) with X i = f i (Z, ε i ) and Y = (Y , . . . , Y d ) with Y i = g i (Z , ε i ) are ddimensional random vectors that follow a completely speci ed factor model with factor distribution function G = F Z and G = F Z , respectively, such that Ran(G) = Ran(G ) . Then the corresponding copulas are given by the * -products , and C G t = C Y|Z =G − (t) , see Theorem 2.7. Further, by Sklar's Theorem, the corresponding distribution functions are given by using that Ran(G) = Ran(G ) , see Proposition 2.14.
We establish conditions on the conditional copula families B and C assumed generally to be measurable, on the dependence speci cations D i and E i , and on the distributions of the components X i and Y i to infer lower orthant, upper orthant and concordance comparison results for X and Y .
The following proposition compares CSFMs where the bivariate dependence speci cations D i and E i coincide.

Proposition 4.1 (Ordering conditional copulas).
Assume that D i = E i for all i . Then, the following statements hold true. Figure 3 The setting in Section 4: completely speci ed factor models with dependence speci cations D i and B = (B t ) t as well as E i and C = (C t ) t , ≤ i ≤ d , and with factor distribution function G and G , respectively, such that Ran(G) = Ran(G ) .
Proof. The statements follow from Proposition 3.2 for xed marginal distributions together with Sklar's Theorem (respectively, Sklar's Theorem for survival functions) for xed conditional copulas using that In the remaining part of this section, we also establish ordering conditions w.r.t. the dependence speci cations D i and E i .
For the following theorem, we need a family of componentwise convex conditional (survival) copulas that lies between B and C . Then, we obtain a general ordering condition in dependence on the bivariate speci cations, the conditional copulas and the marginal distributions. (i) If B satis es condition (22) and if B t ∈ C ccx d for all t , then (ii) If B satis es condition (23) and if B t ∈ C ccx d for all t , then (iii) If B and B satisfy condition (22) and (23), respectively, and if B t , B t ∈ C ccx d for all t , then Proof. To show (i), we obtain from Proposition 3.2 and Theorem 3.20 that * B, Then, the statement follows with Sklar's Theorem. Statements (ii) and (iii) follow analogously.
Since the independence copula and its associated survival copula are componentwise convex, we obtain as a consequence of the above theorem ordering results for the standard factor model.  Figure 4 Classes C E i = {C ∈ C | C ≤ ∂ S E i } of bivariate copulas generated by the copulas E i ∈ C , i = , . . . , d , via the ≤ ∂ S -ordering. Note that M , Π , and W denote the upper Fréchet copula, the independence copula, and the lower Fréchet copula, respectively. The copulas E i ↑ and E i ↓ are the uniquely determined copulas that are CIS and CDS, respectively, such that As a consequence of Proposition 3.14, it holds for all In the following remark, we determine sharp bounds for some relevant classes of CSFMs including classes of standard factor models with bounded bivariate speci cation sets.

Remark 4.4.
Let F i ∈ F for all i . Denote by ≺ one of the orderings ≤ lo and ≤uo . For E i ∈ C , denote by For a risk factor Z ∼ G , G ∈ F c , consider the class  Figure 4. Then, for all ξ ∈ X f , it holds that In PSFMs, the conditional copulas are not speci ed. For the comparison of upper bounds in classes of PSFMs, we note that the worst case distribution in a PSFM w.r.t. the orthant orders is obtained when the conditional copula speci cations attain the upper Fréchet copula.
Proof. From Proposition 3.2 and Theorem 3.25 we obtain that * B, Then (i) follows with Sklar's Theorem. Statements (ii) and (iii) follow analogously.
Similarly, we obtain for lower bounds in the two-and three-dimensional case the following result.
Proof. For G ∈ F c , we obtain from Theorem 3.27 and Proposition 3.
. Similarly, B = W = (W ) . Then, we obtain that and, thus, (X , X ) ≤ lo (Y , Y ) . For the upper orthant and concordance ordering, the statements follow analogously.
Note that the same results hold true if the inequality signs ≤ ∂∆ and ≤ s∂∆ in Theorem 4.5 and Theorem 4.6 (with D = E = W ) are reversed.
For classes of partially speci ed internal factor models (PSIFMs) where the rst component of the risk vector in the PSFM coincides with (an increasing function of) the factor variable, see [3], we obtain the following results. Note that in this class, the rst bivariate dependence speci cation is given by the upper Fréchet copula M .
Proof. From Proposition 3.2 and Theorem 3.26, we obtain that * B Thus, the statement follows with Sklar's Theorem. Statements (ii) and (iii) follow analogously.
For lower bounds in the three-dimensional case, we obtain the following result.

Conclusion
In this paper, we obtain some general ordering results for factor models w.r.t. the speci cations of the joint distributions of the components with the risk factor variable. The results generalize the upper product ordering results in [2,3] to general conditional dependence structures and are based essentially on a version of Sklar's theorem as well as on classical ordering results based on rearrangements. The results in this paper allow to determine worst case distributions w.r.t. the orthant orderings for classes of CSFMs as well as in subclasses of PSFMs for any d ≥ and, similarly, of best case distributions for d = , . Related ordering results w.r.t. the stronger supermodular and the directionally convex ordering need di erent techniques and are the subject of a subsequent study.

Con ict of interest statement:
Authors state no con ict of interest.
(ii), (iii) and (iv) follow from the de nition of G − and G − , respectively, considering the cases where G is discontinuous and constant around x , respectively.
(v) is a consequence of (ii). (vi): This follows from the left-continuity of G − and G − . (vii): To show the left-continuity of ι G at G(y) , let (tn) n∈N be strictly increasing in [ , ] with limit G(y) > . Then, we have as n → ∞ applying (ii), (i), and (iv). To show the right-continuity of ι − G at G − (y) , let (tn) n∈N be strictly decreasing in [ , ] with limit G − (y) < . Then, we obtain similarly that (viii): Consider the distribution functions G and H de ned by Then ι G and ι H are given by So, ι G is not left-continuous at t = and ι H is not right-continuous at t = .

Proof of Theorem 2.2.
Consider the set Ic := (z , z ) | z < z , G is continuous on (z , z ) of open intervals on which G is continuous, and denote by Is := {{z} | z ∈ R} the set of one-point sets. Note that each element of Ic is the intersection of an open interval in R and the preimage of ( , ) under G . We show that for all (z , z ) ∈ Ic and {z} ∈ Is . Since G has at most countably many jump discontinuities, every open interval (y , y ) ⊂ R can be written as a disjoint union of at most countably many elements of Ic and Is . Then, (36) and (37) for all open intervals (y , y ) ⊂ R . Hence, the integrands coincide for G-almost all z , which yields (i).
To show (36), let (z , z ) ∈ Ic . Assume w.l.g. that t := G(z ) < G − (z ) =: t . Then we obtain from the disintegration theorem and Sklar's Theorem that where the third equality follows from the disintegration theorem applied on copulas. For the fourth equality, we use that the left-hand derivative and the derivative of the copula w.r.t. the second component coincide for Lebesgue-almost all s . The fth equality follows from ι G (s) = s = ι − G (s) and ι − G (s − ε) = s − ε for all s ∈ (t , t ) and ε ∈ ( , s − t ) because G is continuous at G − (s) and G − (s − ε) , respectively, see Lemma A.1(ix). The sixth equality holds by de nition of the di erential operator in (2), and the last equality is a consequence of the transformation formula.
To show (37), assume for z ∈ R w.l.g. that G(z) > G − (z) . Then we obtain where we use G(z) > G − (z) and apply Lemma A.1(v) for the third equality. For the fourth equality, we use the left-continuity of ι − G , see Lemma A.1(vi). The last equality follows with the de nition of the operator ∂ G in (2). To show statement (ii) of Theorem 2.2, denote by Q the rational numbers. Due to part (i) it holds that for all x ∈ Q and for all z outside the G-null set N := x∈Q Nx . Then we obtain for x ∈ R that for all z ∈ N c . For z ∈ N c , the function Hz is by de nition right-continuous. Since C is a -copula and thus -increasing, Hz is non-decreasing. Further, Hz(−∞) = and Hz(∞) = . Hence, Hz(x) = lim w x ∂ G C(F(w), G(z)) coincides with F X|Z=z (x) for all x ∈ R and for all z ∈ N c . This proves the assertion.
In the general case that Ran(G ) = Ran(G ) , it holds that Ran(G ) and Ran(G ) only di er in a Lebesgue-null set because distribution functions have at most countably many jump discontinuities. Hence, the rst part implies that ι G (t) = ι G (t) for Lebesgue-almost all t . where the second equality holds because ∂ G D (·, t) is increasing for all t , the third equality follows from (3) and the transformation formula, see, e.g., [32,Theorem 2], and the last equality is a consequence of Theorem 2.2 and the disintegration theorem.