Quantitative analysis of non-exchangeability in bivariate copulas: Sharp bounds, statistical tests and mixing constructions

Quan titativ e analysis of non-exc hangeabilit y in biv ariate copulas: Sharp b ounds, statistical tests and mixing constructions Man uel ´ Ub eda-Flores Departmen t of Mathematics, Univ ersity of Almer ´ ıa, 04120 Almer ´ ıa, Spain mubeda@ual.es Marc h 27, 2026 Abstract A biv ariate random v ector ( X , Y ) is exc hangeable if ( X , Y ) and ( Y , X ) share the same distribution, whic h in copula terms amoun ts to C ( u, v ) = C ( v , u ). Building on the axiomatic framework of [F. Duran te, E.P . Klemen t, C. Sempi, M. ´ Ub eda-Flores (2010). Measures of non-exc hangeabilit y for biv ariate random vectors. Statistical P ap ers 51(3), 687–699], we develop three original con tributions. W e deriv e sharp upper bounds on the non-exc hangeability measure µ p ( C ) in terms of the Sch w eizer and W olff dependence measure and Sp earman’s ρ . W e prov e the exact scaling identit y µ p ( αC + (1 − α ) C t ) = | 2 α − 1 | µ p ( C ) for all p ∈ [1 , + ∞ ], enabling explicit prescription of an y target degree of non- exc hangeability . Finally , w e prop ose and analyse a nonparametric p erm utation test for H 0 : C = C t , pro ve its consistency , and v alidate its finite-sample performance via Monte Carlo sim ulation. MSC2020: 60E05, 62H20. Keywor ds : copula; non-exchangeabilit y; measures of asso ciation; measures of asymmetry . 1 In tro duction Consider a pair of real-v alued random v ariables ( X , Y ) defined on a common probability space. Let F X and F Y denote their resp ective marginal distribution functions (d.f.’s), and let F X,Y and F Y ,X represen t the join t d.f.’s of the vectors ( X , Y ) and ( Y , X ). F ollo wing [2], the random vector ( X , Y ) is classified as exchange able pro vided that F X = F Y and F X,Y = F Y ,X . As p oin ted out by Nelsen [11], while the concept of exchangeabilit y has b een extensiv ely explored in the statistical literature, there has b een a notable lack of researc h regarding the sp ecific mechanisms through which identically distributed v ariables deviate from this prop ert y , as w ell as the underlying dep endence structures that arise in suc h cases. Ho wev er, sev eral studies [1, 4, 7, 9, 11] inv estigate the non-exchangeabilit y of biv ariate distributions, particularly fo cusing on their implementation within v arious statistical frameworks. Inspired b y these developmen ts, the present work aims to pro vide an axiomatic foundation for quan- tifying the degree of non-exchangeabilit y in biv ariate vectors comp osed of contin uous and identically distributed random v ariables. Sp ecial emphasis is placed on the significance and application of these measures within the context of copula theory . The paper is organised as follows. Section 2 reviews the necessary bac kground on copulas and the axiomatic construction of non-exchangeabilit y measures. Section 3 establishes sharp upp er bounds on 1 µ p ( C ) via the Sch weizer and W olff measure and Sp earman’s ρ . Section 4 studies the class M 1 of maxi- mally non-exchangeable copulas, provides the exact concordance v alues of the extremal family M θ , and c haracterises the feasible pairs ( µ ∞ , ρ ) achiev able by con vex com binations. Section 5 prov es the scaling la w µ p ( αC + (1 − α ) C t ) = | 2 α − 1 | µ p ( C ) and its extension to mixing with symmetric copulas. Section 6 dev elops the p erm utation test and its theoretical prop erties. Section 7 presen ts the Monte Carlo study . Finally , Section 8 collects the conclusions. 2 Preliminaries In order to establish a rigorous framew ork for our analysis, this section pro vides the necessary mathemat- ical background that underpins the rest of this study . W e b egin b y reviewing the fundamental prop erties of copulas, which serve as the essen tial to ols for mo deling the dep endence structure of multiv ariate dis- tributions indep enden tly of their marginals. Building up on these definitions, we recall the axiomatic construction of non-exc hangeability measures. 2.1 Basic definitions and prop erties of copulas W e first review several fundamen tal concepts regarding copulas. F or a complete surv ey , see [5, 10]. F ormally , a (biv ariate) c opula is a function C : [0 , 1] 2 → [0 , 1] that satisfies the following tw o condi- tions: (C1) Boundary c onstr aints : F or every t ∈ [0 , 1], C ( t, 0) = C (0 , t ) = 0 and C ( t, 1) = C (1 , t ) = t . (C2) The 2 -incr e asing pr op erty : F or any u, v , u ′ , v ′ ∈ [0 , 1] suc h that u ≤ u ′ and v ≤ v ′ , the C - volume of the rectangle [ u, u ′ ] × [ v , v ′ ] is non-negativ e, i.e., C ( u ′ , v ′ ) − C ( u, v ′ ) − C ( u ′ , v ) + C ( u, v ) ≥ 0 . According to Sklar’s the or em [13], for any pair of con tin uous random v ariables ( X, Y ), their joint d.f. F X,Y can b e uniquely represen ted for all x, y ∈ [ −∞ , + ∞ ] as F X,Y ( x, y ) = C ( F X ( x ) , F Y ( y )) , where C is a copula (for a complete pro of of this result, see [14]). Essen tially , C corresp onds to a biv ariate distribution function on [0 , 1] 2 with uniform marginals. F or an y copula C ∈ C —where C denotes the set of all copulas—, the following inequality holds: W ( u, v ) ≤ C ( u, v ) ≤ M ( u, v ) for all ( u, v ) ∈ [0 , 1] 2 , where W ( u, v ) = max( u + v − 1 , 0) and M ( u, v ) = min( u, v ) are the F r´ echet–Ho effding lower and upp er b ounds , representing p erfect negative and p ositiv e dep endence (countermonotonicit y and comonotonicity), respectively . F urthermore, the product copula Π( u, v ) = uv characterizes the case where X and Y are indep enden t. The exc hangeabilit y of a random pair can b e elegan tly c haracterized through its asso ciated copula as follows: Let ( X , Y ) b e a vector of contin uous random v ariables with copula C . Then X and Y are exc hangeable if, and only if, they are iden tically distributed ( F X = F Y ) and C is symmetric , i.e., C ( u, v ) = C ( v , u ) for all ( u, v ) ∈ [0 , 1] 2 . F or an y C ∈ C , we define its transp ose as C t ( u, v ) = C ( v , u ); and, additionally , the survival c opula ˆ C is giv en b y ˆ C ( u, v ) = u + v − 1 + C (1 − u, 1 − v ) for all ( u, v ) ∈ [0 , 1] 2 . The following prop erties describ e ho w transformations of the random v ariables affect the underlying copula: Let X and Y be contin uous random v ariables with marginals F X , F Y and copula C . Then: 2 (a) The copula representing the vector ( Y , X ) is the transp ose copula C t . (b) If g : R → R is a strictly increasing transformation, the copula of ( g ( X ) , g ( Y )) remains C . (c) If g : R → R is a strictly decreasing transformation, the copula of ( g ( X ) , g ( Y )) becomes the surviv al copula ˆ C . 2.2 Measures of non-exc hangeabilit y The de gr e e of non-exchange ability of a copula C is quantified b y µ p ( C ) := d p ( C, C t ) , p ∈ [1 , + ∞ ] , where d p is the L p distance on C . Explicitly , for p < ∞ , µ p ( C ) = Z [0 , 1] 2   C ( u, v ) − C ( v , u )   p du dv ! 1 /p , while µ ∞ ( C ) = sup ( u,v ) ∈ [0 , 1] 2 | C ( u, v ) − C ( v , u ) | . It is prov ed in [6] that µ p satisfies a natural set of axioms (b oundedness; n ullity if, and only if, C is symmetric; in v ariance under transp osition and under strictly monotone transformations; and con tinuit y with resp ect to w eak con vergence), and that the maximum v alue is K µ ∞ = 1 3 and K µ p =  2 · 3 − p ( p + 1)( p + 2)  1 /p for p < ∞ . In what follows, all measures µ p are used in their normalised form, so that µ p ( C ) ∈ [0 , 1] for every C ∈ C . 3 Upp er b ounds for µ p in terms of kno wn measures One of the most suggestiv e observ ations in [6, Remark 1] is that the non-exchangeabilit y measure µ p ( C ) is b ounded ab o ve b y the Sch w eizer and W olff dep endence measure σ p ( C ), given b y σ p ( C ) := c p · d p ( C, Π) =  (2 p + 2)! 2 ( p !) 2  1 /p Z [0 , 1] 2   C ( u, v ) − uv   p d u d v ! 1 /p for p ∈ [1 , ∞ ) and σ ∞ ( C ) := c ∞ · d ∞ ( C, Π) = 4 sup ( u,v ) ∈ [0 , 1] 2 | C ( u, v ) − uv | (see [12]), although the optimal constan t w as not determined, nor w as the result extended to other classical measures. The ob jectiv e of this section is precisely to address this gap and to establish explicit and optimal upp er b ounds for µ p ( C ) as a function of Sp earman’s ρ , giv en b y ρ ( C ) = 12 Z [0 , 1] 2 C ( u, v ) d u d v − 3 (see [10]). The key insigh t is that the (concordance) measures quantify the global proximit y of C to the extreme copulas M and W , whereas µ p ( C ) measures the asymmetry of C with resp ect to the diagonal. These tw o notions are not indep enden t: a copula close to M or W has little margin to b e asymmetric, as these copulas are themselves symmetric. Precisely formalizing this intuitiv e principle is the common thread of this section. 3 3.1 Bound via the Sc h weizer and W olff dep endence measure W e b egin b y pro viding a precise form ulation —with an optimal constan t— of the b ound mentioned without detail in the reference article. The primary to ol is the triangle inequalit y in the metric space ( C , d p ). Prop osition 1. L et C ∈ C . Then µ p ( C ) ≤ 2 c p σ p ( C ) , (1) for al l p ∈ [1 , + ∞ ] . Mor e over, the c onstant 2 /c p is sharp for every p ∈ [1 , + ∞ ] . Pr o of. The change of v ariables ( u, v ) 7→ ( v , u ) preserv es the Leb esgue measure on [0 , 1] 2 and the supremum o ver [0 , 1] 2 , so the map C 7→ C t is an isometry of ( C , d p ) for ev ery p ∈ [1 , + ∞ ]. In particular, Π t = Π giv es d p (Π , C t ) = d p (Π , C ), and the triangle inequality in ( C , d p ) yields µ p ( C ) = d p ( C, C t ) ≤ d p ( C, Π) + d p (Π , C t ) = 2 d p ( C, Π) = 2 c p σ p ( C ) . No w, let B ∈ C be any copula with B  = B t , and define, for ε > 0, C ε ( u, v ) := uv + ε 2  B ( u, v ) − B ( v , u )  . (2) Since B and B t b oth ha ve uniform marginals, Z 1 0 [ B ( u, v ) − B ( v , u )] d v = u − u = 0 for ev ery u , and similarly in u , so C ε has uniform marginals. F or ε small enough, C ε is 2-increasing, e.g., its density 1 + ( ε/ 2) ∂ 2 ∂ u∂ v ( B − B t )( u, v ) is non-negative, hence a copula. By construction, C ε − Π = ( ε/ 2)( B − B t ) is an tisymmetric, so C ε ( u, v ) + C ε ( v , u ) = 2 uv for all ( u, v ) ∈ [0 , 1] 2 . Therefore C ε ( u, v ) − C ε ( v , u ) = ε ( B ( u, v ) − B ( v , u )), and µ p ( C ε ) = d p ( C ε , C t ε ) = ε d p ( B , B t ) = ε µ p ( B ) , (3) d p ( C ε , Π) = ε 2 d p ( B , B t ) = ε 2 µ p ( B ) , (4) where (4) uses the fact that C ε − Π is an tisymmetric and hence d p ( C ε , Π) p = ( ε/ 2) p d p ( B , B t ) p . F rom (3) and (4), µ p ( C ε ) σ p ( C ε ) = ε µ p ( B ) c p · ( ε/ 2) µ p ( B ) = 2 c p for every ε > 0. The b ound (1) is therefore sharp. Remark 1. The c opula C ε define d in (2) satisfies µ p ( C ε ) /σ p ( C ε ) = 2 /c p exactly , for every ε > 0 and every B  = B t . The key pr op erty that makes the triangle ine quality an e quality in d p is that C ε − Π is antisymmetric, which for c es Π to lie exactly on the d p -ge o desic b etwe en C ε and C t ε . 4 3.2 Bound via Sp earman’s ρ Sp earman’s ρ is related to σ 1 . This allows the previous b ound to b e conv erted into a direct expression in terms of ρ , with the additional adv an tage that ρ is easily estimable from data. Prop osition 2. F or every C ∈ C , µ 1 ( C ) ≤ σ 1 ( C ) 6 ≤ 1 − | ρ ( C ) | 3 . (5) In p articular, if C is r adial ly symmetric, i.e., C = ˆ C , then µ 1 ( C ) ≤ (1 − ρ ( C )) / 6 . Pr o of. The left inequality follows from Prop osition 1 with p = 1. F or the second inequality , write f ( u, v ) = C ( u, v ) − uv and split the integral b y sign: σ 1 ( C ) 12 = Z { f ( u,v ) ≥ 0 } f ( u, v ) d u d v + Z { f ( u,v ) < 0 } | f ( u, v ) | d u d v , ρ ( C ) 12 = Z { f ( u,v ) ≥ 0 } f ( u, v ) d u d v − Z { f ( u,v ) < 0 } | f ( u, v ) | d u d v . Adding and subtracting, the p ositiv e part of f satisfies Z { f ( u,v ) ≥ 0 } f ( u, v ) d u d v = σ 1 ( C ) + ρ ( C ) 24 and is b ounded ab o v e b y Z [0 , 1] 2 [ M ( u, v ) − Π( u, v )] d u d v = 1 12 . Similarly , the negativ e part satisfies Z { f ( u,v ) < 0 } | f ( u, v ) | d u d v = σ 1 ( C ) − ρ ( C ) 24 and is b ounded ab o v e b y Z [0 , 1] 2 [Π( u, v ) − W ( u, v )] d u d v = 1 12 . Hence: σ 1 ( C ) + ρ ( C ) 24 ≤ 1 12 , σ 1 ( C ) − ρ ( C ) 24 ≤ 1 12 , whic h giv e σ 1 ( C ) ≤ 2 − ρ ( C ) and σ 1 ( C ) ≤ 2 + ρ ( C ) resp ectiv ely . Com bining: σ 1 ( C ) ≤ min { 2 − ρ ( C ) , 2 + ρ ( C ) } = 2(1 − | ρ ( C ) | ) , whence the result follows. Remark 2. Pr op osition 2 has a twofold interpr etation. On one hand, str ong p ositive or ne gative c on- c or danc e limits non-exchange ability: the closer | ρ ( C ) | is to 1 , the smal ler µ 1 ( C ) must b e. In the extr eme, | ρ ( C ) | = 1 implies C ∈ { M , W } , b oth symmetric, so µ 1 ( C ) = 0 . On the other hand, the ine quality serves as a statistic al c omp atibility c ondition: estimate d values ˆ µ n and ˆ ρ n fr om data that violate (5) signal sampling err or or mo del missp e cific ation. 5 4 Maximally non-exc hangeable copulas Let M 1 b e the set M 1 :=  C ∈ C : µ ∞ ( C ) = 1 3  , the class of maximal ly non-exchange able copulas with resp ect to µ ∞ , since K µ ∞ = 1 / 3 is the supremum of µ ∞ o ver C [6]. A key elemen t of this class is the copula M θ , given b y M θ ( u, v ) = min  u, v , ( u − 1 + θ ) + + ( v − θ ) +  , θ ∈ [0 , 1 / 3] , for all ( u, v ) ∈ [0 , 1] 2 , where ( x ) + = max { 0 , x } (see [6]). F or this copula we hav e ρ ( M θ ) = 1 − 6 θ + 6 θ 2 and τ ( M θ ) = (1 − 2 θ ) 2 , where τ represents the Kendall’s τ measure of concordance, giv en b y τ ( C ) = 4 Z [0 , 1] 2 C ( u, v ) d C ( u, v ) − 1 (see [10]). In particular, ρ ( M 1 / 3 ) = − 1 / 3, τ ( M 1 / 3 ) = 1 / 9, and µ ∞ ( M 1 / 3 ) = 1 / 3. Note that M 1 / 3 ∈ M 1 and b oth concordance measures are non-zero. This shows that the class M 1 con tains copulas with non- trivial concordance, and that the relationship b etw een maximal non-exc hangeability and concordance is more subtle than a simple zero-concordance statemen t. Theorem 2 of [6] guarantees that for each θ ∈ [0 , 1] there exists a copula with µ ∞ -v alue exactly θ / 3: it suffices to take C θ = θ C 1 + (1 − θ ) C s , where C 1 ∈ M 1 and C s is any symmetric copula ( C t s = C s , so µ ∞ ( C s ) = 0). W e also hav e ρ ( C θ ) = θ ρ ( C 1 ) + (1 − θ ) ρ ( C s ) . In particular, as θ gro ws from 0 to 1, the non-exchangeabilit y µ ∞ ( C θ ) increases monotonically from 0 to 1 / 3, while ρ ( C θ ) interpolates linearly b et ween its v alues at C s and C 1 . Given target v alues µ 0 ∈ [0 , 1 / 3] and r 0 ∈ [ − 1 , 1], this allo ws us to construct a copula with µ ∞ = µ 0 and ρ = r 0 pro vided there exist C 1 ∈ M 1 and C s symmetric with θ ρ ( C 1 ) + (1 − θ ) ρ ( C s ) = r 0 , θ = 3 µ 0 . This is a linear equation in ρ ( C s ) (once C 1 is fixed), solv able whenev er r 0 lies in the range of the right- hand side ov er all symmetric C s . Since ρ ( C s ) can b e an y v alue in [ − 1 , 1], the solv able range of r 0 is the in terv al [ θ ρ ( C 1 ) − (1 − θ ) , θ ρ ( C 1 ) + (1 − θ )]. F or θ = 3 µ 0 and C 1 = M 1 / 3 (with ρ ( C 1 ) = − 1 / 3): r 0 ∈  − 3 µ 0 · 1 3 − (1 − 3 µ 0 ) , − 3 µ 0 · 1 3 + (1 − 3 µ 0 )  =  2 µ 0 − 1 , 1 − 4 µ 0  . This characterises the feasible range of ρ for giv en µ 0 when C 1 = M 1 / 3 , and can b e used as a diagnostic in mo del fitting. 5 Scaling of µ p under mixing In Copula Theory , mixing a copula with its transp ose has a natural statistical interpretation: if ( X , Y ) has copula C and ( Y , X ) has copula C t , then C α = αC + (1 − α ) C t is the copula of a vector that behav es lik e ( X , Y ) with probability α and like ( Y , X ) with probability 1 − α . This construction yields a one- parameter family that interpolates contin uously b et w een C (at α = 1), its transp ose C t (at α = 0), and the symmetric copula ( C + C t ) / 2 (at α = 1 / 2), at which p oin t all asymmetry is eliminated. A natural question arises: how do es the non-exchange ability me asur e µ p change as α varies? The follo wing result is the core result of this section. It establishes that µ p b eha v es p erfectly linearly along the family { C α } . 6 Prop osition 3. F or every C ∈ C , α ∈ [0 , 1] , and p ∈ [1 , + ∞ ] , we have µ p ( C α ) = | 2 α − 1 | µ p ( C ) . (6) Pr o of. The starting p oint is to compute the antisymmetric part of C α . Since C α ( u, v ) − C α ( v , u ) =  αC ( u, v ) + (1 − α ) C ( v , u )  −  αC ( v , u ) + (1 − α ) C ( u, v )  = (2 α − 1)  C ( u, v ) − C ( v , u )  . (7) The antisymmetric part of C α is exactly that of C , multiplied p oin t wise by the scalar (2 α − 1). W e study t wo cases: 1. Case p ∈ [1 , + ∞ ). T aking the absolute v alue, raising to the p o w er p , and integrating: µ p ( C α ) p = Z [0 , 1] 2 | C α ( u, v ) − C α ( v , u ) | p d u d v = | 2 α − 1 | p Z [0 , 1] 2 | C ( u, v ) − C ( v , u ) | p d u d v = | 2 α − 1 | p µ p ( C ) p . T aking the p -th ro ot gives µ p ( C α ) = | 2 α − 1 | µ p ( C ). 2. Case p = + ∞ . W e hav e µ ∞ ( C α ) = sup ( u,v ) ∈ [0 , 1] 2 | C α ( u, v ) − C α ( v , u ) | = | 2 α − 1 | sup ( u,v ) ∈ [0 , 1] 2 | C ( u, v ) − C ( v , u ) | = | 2 α − 1 | µ ∞ ( C ) , since the constan t factor | 2 α − 1 | ≥ 0 comm utes with the suprem um. Therefore, the result follows. The elegance of Prop osition 3 lies in the fact that iden tity (7) is p oin twise: it holds at every ( u, v ) ∈ [0 , 1] 2 individually . This mak es the scaling la w (6) v alid simultaneously for all L p norms with p ∈ [1 , + ∞ ], with no additional integration or conv ergence argumen t required. This is in contrast to the b ounds of Section 3, whic h are sp ecific to eac h v alue of p . Prop osition 3 has an immediate and practically useful consequence: given any target v alue µ 0 ∈ [0 , µ p ( C )], one can construct a copula with exactly that degree of non-exchangeabilit y , as the following result shows. Corollary 4. L et C ∈ C with µ p ( C ) > 0 and let µ 0 ∈ [0 , µ p ( C )] . Then the mixe d c opula C α with α = 1 2  1 + µ 0 µ p ( C )  ∈  1 2 , 1  satisfies µ p ( C α ) = µ 0 . Pr o of. Setting | 2 α − 1 | µ p ( C ) = µ 0 with α ∈ [1 / 2 , 1] (so that 2 α − 1 ≥ 0) gives 2 α − 1 = µ 0 /µ p ( C ), i.e., α = (1 + µ 0 /µ p ( C )) / 2. Since 0 ≤ µ 0 ≤ µ p ( C ), this v alue satisfies α ∈ [1 / 2 , 1]. Remark 3. Cor ol lary 4 pr ovides a key building blo ck for statistic al mo del fitting: to c onstruct a c opula with pr escrib e d µ p -value e qual to an empiric al estimate ˆ µ n , take any c opula C with µ p ( C ) ≥ ˆ µ n and mix it with p ar ameter α = (1 + ˆ µ n /µ p ( C )) / 2 . In p articular, taking C ∈ M 1 (maximal ly non-exchange able, with µ ∞ ( C ) = 1 / 3 ) and any tar get ˆ µ n ∈ [0 , 1 / 3] , the c opula C α with α = (1 + 3 ˆ µ n ) / 2 satisfies µ ∞ ( C α ) = ˆ µ n exactly. 7 Remark 4. The c onvexity of | 2 α − 1 | gives, for α 1 , α 2 ∈ [0 , 1] and λ ∈ [0 , 1] : µ p ( C λα 1 +(1 − λ ) α 2 ) ≤ λ µ p ( C α 1 ) + (1 − λ ) µ p ( C α 2 ) . The non-exchange ability of a mixtur e do es not exc e e d the weighte d aver age of the non-exchange abilities of the extr eme mixtur es: mixing is “asymmetry-r e ducing” in the c onvex sense. No w w e verify that the family { C α } α ∈ [0 , 1] is coherent with the five axioms (B1)–(B5) defining a measure of non-exc hangeability in [6]. Let µ b e any measure of non-exchangeabilit y . Then the function φ : [0 , 1] → [0 , K µ ] defined b y φ ( α ) := µ ( C α ) = | 2 α − 1 | µ ( C ) satisfies: 1. Boundedness (B1): φ ( α ) ≤ K µ for all α , since | 2 α − 1 | ≤ 1 and µ ( C ) ≤ K µ . 2. Zero iff symmetric (B2): φ ( α ) = 0 ⇔ α = 1 / 2 (when µ ( C ) > 0), and C 1 / 2 = ( C + C t ) / 2 is symmetric. 3. Symmetry under transp osition (B3): µ (( C α ) t ) = µ ( C 1 − α ) = | 2(1 − α ) − 1 | µ ( C ) = | 2 α − 1 | µ ( C ) = φ ( α ). 4. Inv ariance under surviv al copula (B4): Using ˆ C t = ( ˆ C ) t (since ˆ C t ( u, v ) = ( ˆ C ) t ( u, v )), one computes ˆ C α ( u, v ) = u + v − 1 + C α (1 − u, 1 − v ) = α ˆ C ( u, v ) + (1 − α ) ( ˆ C ) t ( u, v ) , so µ ( ˆ C α ) = | 2 α − 1 | µ ( ˆ C ) = | 2 α − 1 | µ ( C ) = φ ( α ) b y (B4) applied to C . 5. Contin uit y (B5): F or α, β ∈ [0 , 1], d ∞ ( C α , C β ) = | α − β | d ∞ ( C, C t ), so α 7→ C α is Lipsc hitz in d ∞ . Com bined with the con tinuit y of µ , φ is contin uous. Prop osition 3 mixes a copula C with its o wn transpose C t . A natural generalisation is to mix C with an arbitrary symmetric copula S ∈ C (satisfying S t = S ). Prop osition 5. L et C ∈ C , S ∈ C with S t = S , and α ∈ [0 , 1] . Define D α := αC + (1 − α ) S . Then, for every p ∈ [1 , + ∞ ] : µ p ( D α ) = α µ p ( C ) . Pr o of. The antisymmetric part of D α is: D α ( u, v ) − D α ( v , u ) = α  C ( u, v ) − C ( v , u )  + (1 − α ) [ S ( u, v ) − S ( v , u )] | {z } = 0 , since S is symmetric. The identit y holds p oin twise for every ( u, v ) ∈ [0 , 1] 2 , so for p ∈ [1 , + ∞ ): µ p ( D α ) p = Z [0 , 1] 2   α  C ( u, v ) − C ( v , u )    p d u dv = α p µ p ( C ) p , and taking the p -th ro ot gives µ p ( D α ) = α µ p ( C ). F or p = ∞ , the constant α ≥ 0 comm utes with the suprem um, giving µ ∞ ( D α ) = α µ ∞ ( C ). Remark 5. Comp aring Pr op osition 5 with Pr op osition 3 r eve als a structur al distinction b etwe en the two mixing op er ations: • Mixing C with a symmetric c opula S : µ p ( D α ) = α µ p ( C ) , which is linear in α and r e aches zer o only at α = 0 . 8 • Mixing C with its tr ansp ose C t : µ p ( C α ) = | 2 α − 1 | µ p ( C ) , which is V-shap ed in α and r e aches zer o at the midp oint α = 1 / 2 . The differ enc e r efle cts a ge ometric distinction: mixing with S simply dilutes the asymmetric p art of C , while mixing with C t simultane ously intr o duc es a c anc el lation b etwe en C and its “mirr or image”. Both ar e sp e cial c ases of the gener al c onvex c ombination β C + γ C t + (1 − β − γ ) S ( β + γ ≤ 1 , S symmetric), for which µ p = | β − γ | µ p ( C ) , a formula that c ontains b oth r esults as p articular c ases ( γ = 0 gives Pr op osition 5; β + γ = 1 , β = α gives Pr op osition 3). As a direct consequence of Prop osition 5, the map α 7→ µ p ( D α ) = α µ p ( C ) is strictly increasing on [0 , 1] (when µ p ( C ) > 0), with µ p ( D 0 ) = 0 (the symmetric copula S has zero non-exchangeabilit y) and µ p ( D 1 ) = µ p ( C ). This is in contrast to the V-shap e of α 7→ µ p ( C α ), and the tw o families together span all intermediate v alues µ 0 ∈ [0 , µ p ( C )]. Corollary 6. F or any µ 0 ∈ [0 , µ p ( C )] , µ p ( D α ) = µ 0 with α = µ 0 /µ p ( C ) . 6 A nonparametric test of non-exc hangeabilit y based on copulas The preceding sections ha v e dev elop ed an axiomatic theory of non-exchangeabilit y and established its connections with classical concordance measures, the Sch weizer and W olff dep endence measure, and the geometry of the copula space. A natural question is ho w to detect non-exchangeabilit y from data: giv en an observed sample, can we decide whether the copula of the underlying biv ariate distribution is symmetric? This section addresses that question b y constructing a nonparametric hypothesis test whose statistic is the empirical analogue of the measure µ p itself. The approac h is conceptually clean: instead of reducing the problem to a comparison of marginal or joint empirical distributions, we measure directly ho w far the empirical copula is from its o wn transpose—the natural definition of asymmetry in the copula framew ork. This gives the test a direct in terpretive connection to the theoretical dev elopment of the pap er. The hypothesis is: H 0 : C = C t (exc hangeability) v ersus H 1 : C  = C t (non-exc hangeability) . Under H 0 , the copula is symmetric and ( X , Y ) is exchangeable; under H 1 , µ p ( C ) > 0 and the ordering of the comp onents matters. Detecting this difference has practical consequences: if H 0 is plausible, the practitioner ma y restrict attention to symmetric copula families (Gaussian, Clayton, F rank, Gum b el), substan tially simplifying mo del selection; if H 1 is supp orted by the data, asymmetric families m ust b e considered instead. 6.1 The test statistic Let { ( X i , Y i ) } n i =1 b e an i.i.d. sample from a contin uous biv ariate vector ( X , Y ) with marginals F X , F Y and copula C . Since the copula is inv ariant to strictly increasing transformations of the marginals, the test need not estimate F X and F Y separately . Instead, we work with the pseudo-observ ations ˆ U i := rank( X i ) n + 1 , ˆ V i := rank( Y i ) n + 1 , i = 1 , . . . , n, whic h approximate samples from Uniform(0 , 1) under an y contin uous marginals. The division by n + 1 rather than n av oids b oundary effects and is standard in the empirical copula literature. 9 Let C n denote the empiric al c opula [3] based on the pseudo-observ ations ( ˆ U i , ˆ V i ), whic h is defined by C n ( u, v ) = 1 n n X i =1 1 ( ˆ U i ≤ u, ˆ V i ≤ v ) where 1 S denotes the c haracteristic function of a set S . The test statistic is the empirical counterpart of µ p ( C ) p , scaled by n to pro duce a non-trivial asymp- totic b eha viour. Definition 1. F or p ∈ [1 , + ∞ ) , the test statistic is T n := n Z [0 , 1] 2 | C n ( u, v ) − C n ( v , u ) | p d u d v . (8) Remark 6. The c ase p = + ∞ , i.e., T ( ∞ ) n = √ n · sup ( u,v ) | C n ( u, v ) − C n ( v , u ) | , admits an analo gous tr e atment and is exclude d her e for br evity. In practice, the in tegral in (8) is approximated by a Riemann sum ov er a uniform grid of step 1 /G on each co ordinate: T n ≈ n G 2 G X j =1 G X k =1 | C n ( j /G, k /G ) − C n ( k /G, j /G ) | p . The in tuition behind the scaling b y n is the following. Under H 0 , C n ( u, v ) − C n ( v , u ) fluctuates around zero at rate n − 1 / 2 , so n R | C n − C t n | p div erges at rate n 1 / 2 . Under H 1 , C n ( u, v ) − C n ( v , u ) conv erges to the non-zero function C ( u, v ) − C ( v , u ), so T n /n → µ p ( C ) p > 0 and T n div erges at the faster rate n . This rate separation— √ n under the null, n under alternativ es—is what driv es the consistency of the test and is made precise in Subsection 6.2. 6.2 Theoretical prop erties 6.2.1 Distributional b ehaviour under H 0 Under the null hypothesis, the quantit y C n − C t n v anishes in the limit, but its fluctuations at scale √ n are describ ed b y the empiric al c opula pr o c ess . This pro cess, first studied systematically in [3] and [8], is defined as G n ( u, v ) := √ n ( C n ( u, v ) − C ( u, v )) and conv erges weakly in ℓ ∞ ([0 , 1] 2 ) to a cen tred Gaussian pro cess G C whose cov ariance structure dep ends on the copula C . The following result makes precise how T n inherits this w eak con vergence. Theorem 7. Assume H 0 : C = C t and that C satisfies the r e gularity c onditions for we ak c onver genc e of the empiric al c opula pr o c ess. Then T n √ n = √ n Z [0 , 1] 2 | C n ( u, v ) − C n ( v , u ) | d u d v D − − → L C := Z [0 , 1] 2 | G C ( u, v ) − G C ( v , u ) | d u d v . (9) Pr o of. By [8], √ n ( C n − C ) ⇝ G C w eakly in ℓ ∞ ([0 , 1] 2 ). Under H 0 , C = C t , so for every ( u, v ) ∈ [0 , 1] 2 : √ n  C n ( u, v ) − C n ( v , u )  = √ n  C n ( u, v ) − C ( u, v )  − √ n  C n ( v , u ) − C ( v , u )  ⇝ G C ( u, v ) − G C ( v , u ) . The map f 7→ R [0 , 1] 2 | f ( u, v ) | du dv is a contin uous functional on ℓ ∞ ([0 , 1] 2 ) (it is Lipsc hitz with constan t 1), so the contin uous mapping theorem applies and gives (9). 10 Remark 7. We note thr e e observations on The or em 7. (i) The the or em establishes that T n / √ n has a finite limiting distribution, which implies that T n itself diver ges to + ∞ under H 0 at r ate √ n . This is not a deficiency of the test: the critic al values gr ow at r ate √ n (The or em 8), while under H 1 the statistic gr ows at the faster r ate n , and it is this asymptotic sep ar ation that makes the test c onsistent. (ii) The limiting distribution L C dep ends on the unknown c opula C thr ough the pr o c ess G C . This dep endenc e is unavoidable for a test that is sensitive to al l forms of asymmetry, and it pr events the use of universal critic al tables. It is pr e cisely for this r e ason that the p ermutation c alibr ation of Subse ction 6.2.3 is ne e de d in pr actic e. (iii) An analo gous r esult holds for any p ∈ [1 , + ∞ ) : the functional f 7→ ( R [0 , 1] 2 | f | p ) 1 /p is also c ontinuous on ℓ ∞ ([0 , 1] 2 ) , and the c ontinuous mapping the or em gives c on ver genc e of ( T n / √ n ) 1 /p to ( R | G C − G t C | p ) 1 /p . 6.2.2 Consistency against fixed alternativ es The following result shows that the test detects every fixed asymmetric copula with probability tending to one, regardless of the significance lev el c hosen. Theorem 8. Under H 1 : C  = C t , for any se quenc e of critic al values c n,α = O P ( √ n ) : P ( T n > c n,α ) n →∞ − − − → 1 . Pr o of. Under H 1 , the Glivenk o–Can telli theorem for the empirical copula [3] gives sup ( u,v ) | C n ( u, v ) − C ( u, v ) | → 0 almost surely , and therefore T n n = Z [0 , 1] 2 | C n ( u, v ) − C n ( v , u ) | p d u d v a.s. − − − → Z [0 , 1] 2 | C ( u, v ) − C ( v , u ) | p d u d v = µ p ( C ) p > 0 . Hence T n div erges almost surely at rate n . Since c n,α = O P ( √ n ) by Theorem 7, w e ha ve c n,α /n → 0 in probabilit y , and P ( T n > c n,α ) = P ( T n /n > c n,α /n ) → 1. Remark 8. The c onsistency ar gument r eve als a ple asing c onne ction with the the ory of the pr e c e ding se ctions: the limit µ p ( C ) p > 0 is pr e cisely the p -th p ower of the non-exchange ability me asur e of C , and the c ondition C  = C t is e quivalent to µ p ( C ) > 0 . The test is ther efor e c onsistent against exactly the alternatives wher e the non-exchange ability me asur e is p ositive—a p erfe ct alignment b etwe en the the or etic al me asur e and the empiric al test. 6.2.3 P erm utation calibration Since the limiting distribution L C of T n / √ n dep ends on the unkno wn copula C , critical v alues cannot b e tabulated in adv ance. A standard solution is to approximate the n ull distribution by co ordinate p erm utation: under H 0 , sw apping the tw o co ordinates of any observ ation do es not c hange the joint distribution, so a sample obtained by randomly exchanging ( ˆ U i , ˆ V i ) and ( ˆ V i , ˆ U i ) for eac h i indep enden tly has, under H 0 , the same distribution as the original. This leads to the following resampling scheme. 11 Definition 2. F or e ach r eplic ate b = 1 , . . . , B , dr aw indep endent Bernoul li (1 / 2) variables ε ( b ) 1 , . . . , ε ( b ) n (indep endently of the data) and form the r esample d observations  e U ( b ) i , e V ( b ) i  := ( ( ˆ U i , ˆ V i ) if ε ( b ) i = 1 , ( ˆ V i , ˆ U i ) if ε ( b ) i = 0 . Compute the p ermute d statistic T ( b ) n fr om the r esample d observations { ( e U ( b ) i , e V ( b ) i ) } n i =1 using (8) . The test r eje cts H 0 at level α if T n > ˆ c ( B ) n,α , wher e ˆ c ( B ) n,α is the empiric al (1 − α ) -quantile of { T (1) n , . . . , T ( B ) n } . The theoretical justification of this sc heme rests on the exchangeabilit y of the sample under H 0 : since ( ˆ U i , ˆ V i ) d = ( ˆ V i , ˆ U i ) under H 0 , the distribution of the original observ ation is identical to that of the p ermuted version, making each p erm uted statistic T ( b ) n an indep endent draw from the same null distribution as T n . Prop osition 9. Under H 0 , P  T n > ˆ c ( B ) n,α  n,B →∞ − − − − − → α. Pr o of. Under H 0 : C = C t , the joint distribution of ( ˆ U i , ˆ V i ) is in v arian t under the swap ( ˆ U i , ˆ V i ) 7→ ( ˆ V i , ˆ U i ). Applying this swap indep enden tly to each observ ation with probability 1 / 2 preserves the join t distribution of the entire sample. Hence, conditionally on the data, eac h T ( b ) n has the same distribution as T n under H 0 . The T ( b ) n are also indep enden t of eac h other (giv en the data), so by the strong la w of large num b ers, the empirical distribution of { T (1) n , . . . , T ( B ) n } conv erges almost surely to the n ull distribution of T n as B → ∞ . In particular, ˆ c ( B ) n,α con verges to the true (1 − α )-quantile c n,α of T n under H 0 , and P ( T n > c n,α ) → α as n → ∞ by Theorem 7. Remark 9. The p ermutation scheme of Definition 2 has sever al pr actic al advantages. It is exact for any finite n and B (in the sense that it c ontr ols the typ e-I err or at level α without any asymptotic ap- pr oximation, pr ovide d the exchange ability under H 0 holds exactly). It is distribution-fr e e: the r esampling distribution adapts automatic al ly to the unknown C , without any p ar ametric assumption. And it is c om- putational ly che ap: e ach r esample d statistic T ( b ) n is c ompute d fr om the same empiric al c opula matrix, mer ely with r ows and c olumns r andomly tr ansp ose d. In pr actic e, B = 999 or B = 9999 r eplic ates suffic e for a r eliable appr oximation of the (1 − α ) -quantile at standar d signific anc e levels. 7 Mon te Carlo sim ulation study W e complement the theoretical results of the preceding sections with a Monte Carlo study that examines t wo asp ects of the p erm utation test describ ed in Section 6: its ability to maintain the nominal lev el under symmetric copulas, and its p o wer to detect asymmetry as the degree of non-exchangeabilit y and the sample size v ary . All experiments use the test statistic with p = 1, a Riemann grid of G = 35 p oin ts per co ordinate, and B = 299 p erm utation replicates p er test. This choice of B is standard for a nominal level of α = 0 . 05: it guaran tees that the empirical p -v alue is a m ultiple of 1 / 300, with the exact v alue 1 / 300 ≈ 0 . 003 b eing the smallest achiev able. The Mon te Carlo exp erimen ts use 100 replications to estimate the empirical level and 80 replications to estimate p o wer. 12 7.1 Empirical lev el under the n ull hypothesis W e ev aluate the empirical level of the test under three symmetric copulas with n = 100: the Gaussian copula w ith linear correlation ρ = 0 . 5, the Clayton copula w ith parameter θ = 2, and the F arlie-Gumbel- Morgenstern (FGM) copula with parameter θ = 0 . 5. In all cases H 0 is true and the nominal level is α = 0 . 05 (T able 1). Copula P arameter Empirical level Gaussian ρ = 0 . 5 0.050 Cla yton θ = 2 0.040 F GM θ = 0 . 5 0.060 Av erage 0.050 T able 1: Empirical level under H 0 ( n = 100, 100 Mon te Carlo replications, B = 299 p ermutations, α = 0 . 05). All three empirical levels are compatible with the nominal α = 0 . 05. With 100 Monte Carlo replications, the standard deviation of the estimated lev el is p 0 . 05 · 0 . 95 / 100 ≈ 0 . 022, so deviations up to ± 0 . 044 are within one standard deviation and do not indicate miscalibration. The test main tains its size across three symmetric copula families with qualitatively differen t tail b eha viour: the Gaussian copula has symmetric ligh t tails, Clayton has strong lo wer-tail dep endence, and FGM has weak dep endence throughout. 7.2 P o wer against the M θ family T o ev aluate pow er, w e generate samples from the copula M θ with three v alues of θ ∈ { 1 / 6 , 1 / 4 , 1 / 3 } , corresp onding to degrees of non-exc hangeability µ ∞ ( M θ ) ∈ { 1 / 6 , 1 / 4 , 1 / 3 } . Recall τ ( M 1 / 3 ) = 1 / 9 ≈ 0 . 11 and ρ ( M 1 / 3 ) = − 1 / 3 ≈ − 0 . 33, so even the maximally non-exc hangeable member of the family has non- trivial concordance (see T able 2). Asymmetry n = 50 n = 100 n = 200 n = 400 θ = 1 / 6 ( µ ∞ = 1 / 6) 1.000 1.000 1.000 1.000 θ = 1 / 4 ( µ ∞ = 1 / 4) 0.988 1.000 1.000 1.000 θ = 1 / 3 ( µ ∞ = 1 / 3) 0.650 0.988 1.000 1.000 T able 2: Empirical p o wer of the test by sample size and degree of asymmetry ( α = 0 . 05, 80 Mon te Carlo replications, B = 299 p erm utations). The pow er is essen tially one for mo derate asymmetry ( θ = 1 / 6 and 1 / 4) ev en at n = 50. F or the maximally non-exc hangeable case ( θ = 1 / 3, µ ∞ = 1 / 3), the p o wer gro ws rapidly with n , reac hing 0 . 988 at n = 100 and 1 . 000 at n = 200. This slightly low er p ow er at small n relative to the other tw o cases is consisten t with the concordance structure of M 1 / 3 : its Kendall’s τ is 1 / 9 ≈ 0 . 11 (small but non-zero) and its Sp earman’s ρ is − 1 / 3 (negativ e), indicating that the asymmetry manifests in an unusual directional pattern that requires somewhat larger samples to detect reliably . 13 7.3 Illustrativ e example ( n = 300 ) T able 3 summarises the test output for three sp ecific samples of size n = 300, illustrating the tw o qualitativ ely different outcomes: non-rejection under a symmetric copula, and rejection with strong evidence under t wo asymmetric copulas. Scenario T n Critical v alue p -v alue Reject H 0 ˆ τ ˆ ρ Gaussian ρ = 0 . 6 (symmetric) 3.34 7.67 0.930 No 0 . 41 0 . 58 M θ , θ = 1 / 6 27.56 6.77 < 0 . 001 Y es 0 . 58 0 . 76 M θ , θ = 1 / 3 26.61 10.01 < 0 . 001 Y es 0 . 10 − 0 . 33 T able 3: T est results for n = 300 ( B = 399 p erm utations, α = 0 . 05). The empirical concordance v alues ˆ τ and ˆ ρ are rep orted for context. In the symmetric case (Gaussian), T n = 3 . 34 is well b elo w the critical v alue 7 . 67, and the p -v alue of 0 . 930 provides no evidence against H 0 . In the tw o asymmetric cases, T n far exceeds the critical v alue and p < 0 . 001. The con trast b et w een the t wo M θ scenarios is instructive: for θ = 1 / 6, the empirical ˆ τ = 0 . 58 is large and p ositiv e, making the asymmetry “hidden” behind a strong concordance signal; the test nev ertheless detects it with ov erwhelm- ing significance. F or θ = 1 / 3, the empirical ˆ τ ≈ 0 . 10 and ˆ ρ ≈ − 0 . 33 are close to the exact theoretical v alues τ ( M 1 / 3 ) = 1 / 9 ≈ 0 . 11 and ρ ( M 1 / 3 ) = − 1 / 3 ≈ − 0 . 33. The small τ and negative ρ reflect the distinctiv e concordance structure of M 1 / 3 , and the test detects its asymmetry with the same confidence as for θ = 1 / 6. 8 Conclusions W e ha ve sharp ened the b ound betw een the non-exc hangeabilit y measure and the Sc hw eizer and W olff dep endence measure with an explicit optimal constan t, v alid for all L p norms sim ultaneously , and es- tablished a chain of inequalities connecting non-exchangeabilit y with Sp earman’s ρ that serv es b oth as a theoretical b ound and as a mo del-diagnostic to ol. W e hav e also computed the exact Kendall and Sp ear- man concordance v alues of the extremal copula family of the reference article, showing that maximal non-exc hangeability and non-zero concordance are compatible. 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