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Copula families and proved coverage

Every constructor in this catalogue returns a ProbabilityTheory.Copula d: a probability measure with proved uniform coordinate marginals. There are no admissibility axioms, sorry proofs, or unverified CDF formulas used as measures. Import Copula for everything, or use the modules below.

Archimedean families

The convention is the decreasing inverse generator C(u) = ψ(∑ᵢ φ(uᵢ)), with ψ(φ(u)) = u on (0,1]. BivariateGenerator checks nonnegativity, monotonicity, convexity and inverse identities. Its copula constructor proves all classical copula conditions before invoking the measure representation theorem. A zero coordinate is handled separately and gives CDF zero.

HasArchimedeanGenerator C g identifies a generator in a specified dimension; IsArchimedean C asserts existence of such a generator. These predicates do not extend a bivariate admissibility proof to higher dimensions.

Family Constructor Parameters Dimension Proved formula or identity
Independence independence d None Any finite d Product CDF; exponential generator in dimension two
Clayton clayton d θ hθ θ > 0 Any finite d (1 + ∑ᵢ(uᵢ^(-θ)-1))^(-1/θ); Archimedean identification; parameter limits; bivariate PQD, CI, CDF TP2 and actual MTP2 density; tail pair (2^(−1/θ),0)
Clayton, negative branch claytonNegative θ hθ hn −1 ≤ θ < 0 2 max(0,u^(−θ)+v^(−θ)−1)^(−1/θ); lower bound at −1; NQD, CD and failure of CDF TP2 and of MTP2 density for all negative parameters; tail pair (0,0)
Gumbel–Hougaard gumbel θ hθ θ ≥ 1 2 exp(-((−log u)^θ+(−log v)^θ)^(1/θ)); independence at 1; pointwise comonotonicity as θ→∞; increasing lower-orthant order; max-stability; CDF TP2 for all θ≥1
Joe joe θ hθ θ ≥ 1 2 joe_cdf_full gives the Table 1 CDF on the closed square; independence at 1; pointwise comonotonicity as θ→∞; exact tail pair (0,2−2^(1/θ))
Frank frank θ hθ θ > 0 2 frank_cdf_full gives the logarithmic CDF on the closed square, including zero axes
Ali–Mikhail–Haq amh θ hmin hmax −1 ≤ θ ≤ 1 2 cdf_amh proves uv/(1−θ(1−u)(1−v)) on the closed square; at θ=1 this is Clayton(1); CI/PQD and CDF TP2 iff θ≥0, CD/NQD iff θ≤0; MTP2 density at θ=0,1 and excluded for θ<0 (interior positive range pending); lower-orthant order iff θ≤η; tail pair (0,0) below θ=1 and (1/2,0) at θ=1
Frank, negative branch frankNegative θ hθ θ < 0 2 frankNegative_cdf_full gives the reflected CDF and frankNegative_cdf_source proves the printed logarithmic CDF on the closed square
BB1 / Clayton–Gumbel bb1 θ hθ δ hδ θ > 0, δ ≥ 1 2 bb1_cdf_full on the closed square; δ = 1 recovers Clayton
BB6 / Joe–Gumbel bb6 θ hθ δ hδ θ ≥ 1, δ ≥ 1 2 bb6_cdf_full on the closed square; δ = 1 recovers Joe
Nelsen 2 nelsen2 θ hθ θ ≥ 1 2 nelsen2_cdf_full on the closed square; lower bound at 1 and pointwise upper bound at θ→∞; increasing lower-orthant order; exact tail pair (0,2−2^(1/θ)); non-PQD/non-CI and no CDF TP2 or MTP2 density; CD exactly at θ=1
Nelsen 8 nelsen8 θ hθ θ ≥ 1 2 nelsen8_cdf_full proves the printed rational CDF on the closed square; θ = 1 is W and θ→∞ tends pointwise to Clayton at one; exact tail pair (0,0); increasing lower-orthant order; non-PQD/non-CI and no CDF TP2 or MTP2 density; CD exactly at θ=1
Nelsen 7 nelsen7 θ θ : I 2 max(0,θuv+(1−θ)(u+v−1)); CD; increasing LO; exact xi, rho and tau; CDF TP2 and MTP2 density iff θ=1; W and Π endpoints
Nelsen 12 nelsen12 θ hθ θ ≥ 1 2 nelsen12_cdf_full on the closed square; Clayton at one and pointwise comonotonicity as θ→∞; increasing lower-orthant order; exact lower/upper tail coefficients; CI and MTP2 density at θ=1; CDF TP2, PQD and non-CD for all θ≥1
Nelsen 14 nelsen14 θ hθ θ ≥ 1 2 nelsen14_cdf_full on the closed square; Clayton at one; pointwise comonotonicity as θ→∞; exact lower/upper tail coefficients 1/2 and 2−2^(1/θ); CI and MTP2 density at θ=1; CDF TP2, PQD and non-CD for all θ≥1
Genest–Ghoudi / Nelsen 15 genestGhoudi θ hθ θ ≥ 1 2 genestGhoudi_cdf_full on the closed square; lower bound at 1; pointwise comonotonicity as θ→∞; exact lower/upper tail coefficients 0 and 2−2^(1/θ); never CI or TP2; CD iff θ=1

The shared transformation g.outerPower θ hθ sends ψ(t) to ψ(t^(1/θ)), for θ ≥ 1. Concavity of the power map and convexity of the decreasing generator prove validity. This supplies Gumbel, BB1, and BB6 without repeating their measure constructions. claytonGenerator_copula proves that the bivariate generator construction agrees with the existing gamma-frailty Clayton measure.

The additional transformation g.innerPower θ hθ raises ψ to θ and composes φ with u ↦ u^(1/θ). It preserves convexity for θ≥1 and accommodates finite zeros. All bivariate Archimedean copulas are proved exchangeable by IsArchimedean.isExchangeable.

Modules: Copula.Archimedean.Basic, Copula.Archimedean.Exponential, Copula.Archimedean.Power, Copula.Archimedean.Clayton, Copula.Families.Gumbel, Copula.Families.Joe, Copula.Families.Frank, Copula.Families.FrankNegative, and Copula.Families.AMH.

Extreme-value families

IsExtremeValue C is the max-stability identity C(u₁^t,…,u_d^t) = C(u)^t for every t > 0, on the entire closed cube. The identity is proved for every family in this table.

Family Constructor Parameters Dimension
Independence independence d None Any finite d
Comonotonic / upper Fréchet bound comonotonic d None Any finite d
Logistic / Gumbel–Hougaard gumbel θ hθ θ ≥ 1 2
Marshall–Olkin marshallOlkin α β α, β : I, including 0 and 1 2
Cuadras–Augé cuadrasAuge α α : I 2
Asymmetric logistic / Tawn tawn θ hθ α β θ ≥ 1, α, β : I 2
Common-shock extension commonShock d a a : Fin d → I Any finite d

cdf_marshallOlkin gives min(u^α,v^β) u^(1-α) v^(1-β), including all coordinate and parameter boundaries. Cuadras–Augé is the equal-weight subfamily. The all-zero and all-one Marshall–Olkin parameters give independence and comonotonicity respectively. Rank.MarshallOlkin proves the full-parameter Spearman rho formula 3αβ/(2α+2β−αβ). Rank.MarshallOlkinConditional identifies the conditional CDF almost everywhere, and Rank.MarshallOlkinXi proves directional Chatterjee xi 2α²β/(3α+β−2αβ). Both coefficient formulas include the independence axes and singular positive-weight laws.

CDF TP2 is preserved by maxProduct in dimension two. Consequently the Gumbel–Hougaard, Tawn, Marshall–Olkin and Cuadras–Augé CDFs are TP2 for all admissible parameters, including singular common-shock laws. This is distinct from MTP2 of a Lebesgue density.

The reusable maxProduct C D a construction has CDF C(uᵢ^aᵢ) D(uᵢ^(1-aᵢ)). It uses independent samples, transformed power marginals, and coordinatewise maxima. Zero weights use a constant zero sample. It preserves extreme-value stability when both inputs have it. Tawn uses Gumbel and independence as its two inputs. gumbel_cdf_full and tawn_cdf_full state the named CDF formulas on the whole closed square, with explicit grounded values on the zero axes; tawn_cdf_positive records the analytic expression where both logarithms are defined. The exact reductions tawn_zero_left, tawn_zero_right, and tawn_shape_one give independence on both weight axes and at shape one; tawn_one_one recovers Gumbel at unit weights. These include zero-coordinate endpoints. The construction is also useful with inputs that are not extreme-value copulas.

ExtremeValue.Diagonal derives the power diagonal t^κ, with 1≤κ≤2, directly from bivariate max-stability. TailDependence.ExtremeValue computes κ and both tail coefficients for Gumbel, Marshall–Olkin, Cuadras–Augé and Tawn, including all admitted endpoints. Rank.PowerDiagonal gives closed forms for their Spearman footrule and Blomqvist beta. These results also have generic versions for arbitrary power-diagonal copulas; see tail dependence and rank coefficients.

Modules: Copula.Transform.Power, Copula.Transform.MaxProduct, Copula.ExtremeValue.Basic, Copula.Families.MarshallOlkin, Copula.Families.Gumbel.

Elliptical Gaussian scale mixtures

The reusable construction samples Z ~ N(0,R) and an independent mixing variable T, and forms s(T) Z. The scale is measurable and positive almost surely. Atomless coordinate marginals and the Sklar factorization are proved. No density, finite moments, or nonsingularity of R is required.

R must be positive semidefinite with diagonal one. It is a dispersion parameter; a covariance matrix of the mixed law need not exist. These are Gaussian scale mixtures, a subclass of elliptical laws. The package does not claim a characterization of all elliptical distributions or a radial/characteristic function characterization of this class.

Family Constructor Mixing law / scale Extra parameters
Gaussian gaussian R hR hdiag Unmixed centered Gaussian None
Student-t studentT R hR hdiag ν hν G ~ Gamma(ν/2, rate ν/2), s(G)=1/√G Every real ν > 0
Cauchy cauchy R hR hdiag Student-t with ν = 1 None
Symmetric variance-gamma varianceGamma R hR hdiag κ hκ G ~ Gamma(κ, rate κ), s(G)=√G κ > 0
Symmetric Laplace laplace R hR hdiag Variance-gamma with κ = 1 None
Generalized slash slash R hR hdiag q hq E ~ Exp(1), s(E)=exp(E/q) q > 0; ordinary slash at 1
Normal–lognormal normalLognormal R hR hdiag τ T ~ N(0,τ²), s(T)=exp(T) τ : NNReal

Every row supports all finite dimensions, including zero. studentT_one and varianceGamma_one identify the Cauchy and Laplace special cases. These are stochastic constructions, with uniform marginal and Sklar theorems; elementary copula CDFs, densities, tail coefficients, and moment formulas are not asserted. In particular an identity dispersion matrix need not give an independent copula when coordinates share a random scale.

Modules: Copula.Elliptical.ScaleMixture, Copula.Families.StudentT, Copula.Families.ScaleMixtures, and the existing Gaussian modules.

Exception for tail coefficients: for the bivariate Student-t copula with correlation r ∈ (−1, 1] and any real ν > 0, Copula.Elliptical.StudentTTail proves λ_L = λ_U = 2 t_{ν+1}(−√((ν+1)(1−r)/(1+r))) (equivalently the angular form ∫_{arccos(r)/2}^{π/2} cos^ν / ∫_0^{π/2} cos^ν), with the Student-t distribution function of Copula.Families.StudentT.Distribution. Copula.Families.StudentT.TailMonotone shows that this coefficient is strictly increasing in r ∈ [−1, 1], strictly decreasing in ν > 0 for r ∈ (−1, 1), tends to 0 as ν → ∞ (the Gaussian limit) and to 1 − arccos(r)/π as ν → 0⁺; Copula.Families.StudentT.Marginal proves that every margin of studentTLaw R ν (unit diagonal) is the Student-t law with density studentTPDF ν, so the Student-t copula is the copula of the multivariate t distribution with t_ν margins: C(T_ν(x₁), …, T_ν(x_d)) = P(X ≤ x).

Polynomial and mixture families

Family Constructor Range / formula Dimension
Farlie–Gumbel–Morgenstern fgm θ hθ abs θ ≤ 1; uv(1+θ(1-u)(1-v)) 2
Fréchet mixture frechet a b ha hb hab a,b ≥ 0, a+b ≤ 1; aM+bW+(1-a-b)Π 2
Mardia mardia θ hθ abs θ ≤ 1; Fréchet weights θ²(1+θ)/2, θ²(1-θ)/2 2
Countermonotonic / lower Fréchet bound countermonotonic max(0,u+v-1) 2

FGM's rectangle increment is factored and proved nonnegative for the full parameter interval. Its actual copula measure has an MTP2 density exactly when the parameter is nonnegative; the negative exclusion rules out all density versions, not just the displayed polynomial. Mardia's endpoints -1,0,1 are respectively the lower Fréchet bound, independence, and the upper Fréchet bound.

finiteMixture C w hw hsum accepts any finite list of copulas and nonnegative weights summing to one; its CDF is the corresponding weighted sum. mix C D a provides the two-component interface with a : I. These mixture constructors work in every finite dimension. The existing reflect and reindex APIs also apply to every new family.

Modules: Copula.Mixture, Copula.Families.FGM, Copula.Families.Frechet.

The additional ordinal-sum construction places two bivariate copulas on successive intervals. It includes split endpoints, regional CDF formulas, recovery, exact componentwise ordering, exchangeability and PQD closure. Its lower and upper tail limits are inherited from the first and last nonempty blocks. These constructions are not additional named rows in the Ansari–Rockel family count. Their probability law and rho, tau, footrule and common-split concordance formulas are also proved, with sharp fixed-split bounds and a unique optimal split for independent components. The converse is constructive: every interior diagonal fixed point yields two rescaled component copulas, with proved reconstruction and uniqueness at that split. This includes singular laws and does not require density formulas.

Vine constructions

Copula.Vine combines bivariate families into C-, D-, and regular vines in every finite dimension. RVineStructure supplies variable orders and regular attachment paths. Pair copulas can be fixed or vary measurably with conditioning values, including singular laws. Proximity and marginal preservation are proved. The direct simplified C-vine API additionally has its explicit CDF recursion and all-independence identity. See the vine guide for the constructors, conventions, and exact proved coverage.

Additional checked family properties

Fréchet and Mardia have exact CI/CD and Lebesgue-density classifications, including singular endpoints. Both have a TP2 Lebesgue density exactly at independence. See positive dependence.

Nelsen 7 has xi=1-theta and exact logarithmic rho and tau for all theta in [0,1], a checked conditional CDF, and exact Schur comparison in both directions, with parameter order reversed. Its only CI member is independence; every member is CD.

Formal statementSource and proofProbabilityTheory.Copula.chatterjeeXi_nelsen7
Formal statementSource and proofProbabilityTheory.Copula.nelsen7_rho
Formal statementSource and proofProbabilityTheory.Copula.nelsen7_tau
Formal statementSource and proofProbabilityTheory.Copula.schurBothLE_nelsen7_iff

Scope and next extensions

This catalogue contains 29 named families/special cases; overlapping classes are not counted twice. It distinguishes proved analytic CDFs from stochastic constructions. General multivariate Archimedean admissibility, BB7/BB8, Galambos, Hüsler–Reiss, Pickands representations, and further family-specific dependence formulas remain future work. The rank API includes closed forms for all six coefficients of FGM, Fréchet and Mardia, on their full parameter domains. The Fréchet and Mardia tau formulas are (a−b)(a+b+2)/3 and θ³(θ²+2)/3; their xi formulas are (a−b)²+ab and θ⁴(1+3θ²)/4, with all singular boundaries included. The rank documentation lists the complete formulas. The ordering API proves exact FGM parameter ordering, FGM Schur order by absolute parameter, and comparison results for mixtures and extremal copulas. The tail-dependence API gives both tail limits for FGM, Fréchet, Mardia, Gumbel, Marshall–Olkin, Cuadras–Augé and Tawn, together with the independence and Fréchet-bound benchmarks. Tail formulas for the other analytic and elliptical families remain future work. The bivariate validity theorem does not establish the new Archimedean families in higher dimensions, even for ranges known to be valid mathematically.

The complete Ansari–Rockel index tracks all 38 paper families, including those not yet implemented. Its property and formula tables are reference targets, not additional proved constructors or theorems.

Plackett, Raftery and asymmetrized families

Family Constructor Range / formula Dimension
Plackett plackett θ hθ θ > 0; (S − √(S² − 4uvθ(θ−1)))/(2(θ−1)), S = 1 + (θ−1)(u+v), Π at θ = 1 2
Raftery raftery θ h0 h1 0 ≤ θ < 1; M + (1−θ)/(1+θ)(uv)^{1/(1−θ)}(1 − max(u,v)^{−(1+θ)/(1−θ)}) 2
Khoudraji asymmetrization khoudraji C a b a, b : I; u^{1−a}v^{1−b}C(u^a, v^b) 2

Plackett and Raftery are proved to be copulas through the derivative criterion rectangle_nonneg_of_hasDerivAt (vertical sections differentiable on (0,1) with a partial derivative nondecreasing in the other variable); neither has a singular component. Plackett: constant cross-product ratio, positive ordering in θ with limits W and M, exchangeability, radial symmetry, β = (√θ−1)/(√θ+1), ρ = (θ+1)/(θ−1) − 2θ log θ/(θ−1)² and tail independence. Kendall's tau has no elementary closed form; it is proved to equal (θ+1)/(θ−1) − 2θρ/(θ−1)² + 4(θ+1)√θ/(θ−1)² ∫₀¹ √(v(1−v)) arctan((1−(θ+1)v)/(2√θ√(v(1−v)))) dv (kendallTau_plackett_eq_spearmanRho_arctan, via the double-integral forms kendallTau_plackett_eq_integral and kendallTau_plackett_eq_rational); since C_θ has full support, τ and ρ are strictly increasing in θ, vanish exactly at θ = 1, and tend to ±1 as θ → ∞, 0⁺. Raftery: Nelsen's closed form, C_0 = Π, C_θ → M, PQD, λ_L = 2θ/(1+θ), λ_U = 0, Blomqvist's beta, ρ = θ(4−3θ)/(2−θ)², τ = 2θ/(3−θ). Khoudraji's construction is maxProduct C Π ![a, b]; Marshall–Olkin and Tawn are its instances for M and Gumbel, it preserves max-stability, order, PQD and NQD, and K_{a,b}(M) is exchangeable iff a = b or ab = 0.

Modules: Copula.Families.Plackett (Basic, Order, Spearman, Tail, Kendall, KendallOrder, KendallArctan), Copula.Order.StrictKendall, Copula.Families.Raftery, Copula.Families.RafterySpearman, Copula.Families.RafteryKendall, Copula.Families.Khoudraji.

Mathematical references

The formal statements and their checked proofs in this repository specify the exact implemented coverage; the references also discuss results beyond it.

Library revision: fe53ea2f · Lean 4.34.0