Nelsen Table 4.1: one-parameter Archimedean families

Reference: Roger B. Nelsen, An Introduction to Copulas, second edition, Springer, 2006, Table 4.1 (Section 4.2). The numbering below is Nelsen's and agrees with the A01--A22 numbering of Ansari and Rockel.

Conventions. Nelsen writes C(u,v) = phi^[-1](phi(u) + phi(v)) with a decreasing generator phi. The library uses the inverse generator psi (field toFun of BivariateGenerator, convex on [0, infinity)) and the generator phi (field invFun). Ranges are the ranges of Nelsen's Table 4.1 (as recorded in docs/ansari-rockel.md).

Status legend. "in library" means a compiled, proved copula constructor exists for the stated range. All 22 families of the table are now in the library, each on Nelsen's full parameter range (limiting parameter values that are not members of the family, such as theta = 0 for #9, are noted where they are proved as limits or special cases).

#	Generator phi(t)	Range of theta	Status	Module and main declarations
1	(t^(-theta) - 1)/theta	theta >= -1, theta != 0	in library (theta > 0 and -1 <= theta < 0; theta = 0 is independence)	Copula.Families.Clayton, Copula.Archimedean.Clayton (claytonGenerator), Copula.Families.Clayton.Negative (claytonNegative)
2	(1 - t)^theta	theta >= 1	in library	Copula.Families.Nelsen (nelsen2, nelsen2_cdf_full)
3	ln((1 - theta(1 - t))/t)	-1 <= theta < 1 (the endpoint theta = 1 is Clayton(1) in this library)	in library	Copula.Families.AMH (amh, cdf_amh)
4	(-ln t)^theta	theta >= 1	in library	Copula.Families.Gumbel (gumbel)
5	-ln((e^(-theta t) - 1)/(e^(-theta) - 1))	theta != 0	in library (theta > 0 and theta < 0 separately; theta = 0 is independence)	Copula.Families.Frank, Copula.Families.FrankNegative
6	-ln(1 - (1 - t)^theta)	theta >= 1	in library	Copula.Families.Joe (joe, joe_cdf_full)
7	-ln(theta t + (1 - theta))	0 < theta <= 1	in library (also theta = 0 as W)	Copula.Families.Nelsen7 (nelsen7)
8	(1 - t)/(1 + (theta - 1) t)	theta >= 1	in library	Copula.Families.Nelsen8 (nelsen8)
9	ln(1 - theta ln t)	0 < theta <= 1	in library (whole range)	Copula.Families.NelsenTable.N9 (nelsen9, cdf_nelsen9)
10	ln(2 t^(-theta) - 1)	0 < theta <= 1	in library (full range; inner power of AMH(-1))	Copula.Families.NelsenTable.N10 (nelsen10, cdf_nelsen10)
11	ln(2 - t^theta)	0 < theta <= 1/2	in library (whole range; non-strict, pseudo-inverse clamped at ln 2; lambda_L = 0, lambda_U = 0; limit Pi at theta -> 0)	Copula.Families.NelsenTable.N11 (nelsen11, cdf_nelsen11), Copula.TailDependence.NelsenTable, NelsenTableUpper, Copula.Families.NelsenTable.LimitsZero (tendsto_nelsen11_zero)
12	(1/t - 1)^theta	theta >= 1	in library	Copula.Families.Nelsen (nelsen12, BB1 with theta = 1)
13	(1 - ln t)^theta - 1	theta > 0	in library (whole range; theta = 1 is independence)	Copula.Families.NelsenTable.N13 (nelsen13, cdf_nelsen13, nelsen13_one)
14	(t^(-1/theta) - 1)^theta	theta >= 1	in library	Copula.Families.Nelsen (nelsen14, BB1)
15	(1 - t^(1/theta))^theta	theta >= 1	in library (Genest--Ghoudi)	Copula.Families.Nelsen (genestGhoudi)
16	(theta/t + 1)(1 - t)	theta >= 0	in library (whole range; theta = 0 is W; theta -> infinity gives Pi/(Sigma - Pi) = Clayton(1); lambda_L = 1/2 for theta > 0)	Copula.Families.NelsenTable.N16 (nelsen16, cdf_nelsen16, nelsen16_zero), Copula.Families.NelsenTable.Limits (tendsto_nelsen16_atTop), Copula.TailDependence.NelsenTable
17	-ln(((1 + t)^(-theta) - 1)/(2^(-theta) - 1))	theta != 0	in library (both signs at once; theta = -1 is independence; limits C_infinity = M and C_{-infinity} = max(0, (uv + u + v - 1)/2) (family 7 at 1/2, see below); lambda_L = lambda_U = 0)	Copula.Families.NelsenTable.N17 (nelsen17, cdf_nelsen17, nelsen17_neg_one), Copula.Families.NelsenTable.LimitsInfinity (tendsto_nelsen17_atTop, tendsto_nelsen17_atBot)
18	e^(theta/(t - 1))	theta >= 2	in library (whole range; non-strict; theta -> infinity gives M; lambda_L = 0, lambda_U = 1)	Copula.Families.NelsenTable.N18 (nelsen18, cdf_nelsen18), Copula.Families.NelsenTable.Limits (tendsto_nelsen18_atTop), Copula.TailDependence.NelsenTable
19	e^(theta/t) - e^theta	theta > 0	in library (whole range)	Copula.Families.NelsenTable.N19 (nelsen19, cdf_nelsen19)
20	e^(t^(-theta)) - e	theta > 0	in library (whole range)	Copula.Families.NelsenTable.N20 (nelsen20, cdf_nelsen20)
21	1 - (1 - (1 - t)^theta)^(1/theta)	theta >= 1	in library (whole range; non-strict; theta = 1 is W; lambda_L = 0, lambda_U = 2 - 2^(1/theta); limit M at infinity)	Copula.Families.NelsenTable.N21 (nelsen21, cdf_nelsen21, nelsen21_one), Copula.TailDependence.NelsenTable, NelsenTableLower, Copula.Families.NelsenTable.LimitsInfinity (tendsto_nelsen21_atTop)
22	arcsin(1 - t^theta)	0 < theta <= 1	in library (whole range; non-strict; lambda_L = 0, lambda_U = 0; corrected CDF, see below; limit Pi at theta -> 0)	Copula.Families.NelsenTable.N22 (nelsen22, cdf_nelsen22), Copula.TailDependence.NelsenTable, NelsenTableUpper, Copula.Families.NelsenTable.LimitsZero (tendsto_nelsen22_zero)

Non-strict generators (#11, #18, #21, #22) are built with BivariateGenerator.ofClamp (Copula.Archimedean.Clamp): the pseudo-inverse is F(min t a) for the inverse F of the generator on [0, a], a = phi(0). The families whose convexity needs a second-derivative argument (#11, #13,

17, #18) prove it with convexOn_of_hasDerivWithinAt2_nonneg; #16 and #21 use

elementary convexity (Euclidean norm, convex antitone function of a concave function), and #22 is an inner power of its theta = 1 member.

Correction for #22. With a = 1 - u^theta and b = 1 - v^theta, Table 4.1 prints C(u,v) = max((1 - a sqrt(1 - b^2) - b sqrt(1 - a^2))^(1/theta), 0). This is correct only where phi(u) + phi(v) <= pi/2, equivalently a^2 + b^2 <= 1; outside that region the copula is zero while the printed expression is positive (it tends to one for small u = v). cdf_nelsen22 states the corrected formula with this case distinction.

Generic theory for BivariateGenerator (diagonal formula, strict inequality C(u,u) < u, generator recovery phi(psi(s)) = s) lives in Copula.Archimedean.Theory and Copula.Archimedean.Diagonal; shared convexity lemmas for the new families are in Copula.Archimedean.TheoryConvex.

Kendall's tau, tail coefficients and quadrant dependence

Kendall's tau is proved from Nelsen's Corollary 5.1.4 (tau = 1 + 4 int_0^1 phi/phi'), using BivariateGenerator.kendallTau_eq_of_hasDerivAt (Copula.Archimedean.KendallTauGenerator) for an explicit differentiable generator. Tail coefficients use Corollary 5.4.3; quadrant dependence uses Theorem 4.4.2 with Pi (Copula.Archimedean.Concordance). The first table lists the results from the earlier round; the complete matrix is at the end of this page.

#	Kendall's tau	Tails (lambda_L, lambda_U) and quadrant dependence	Lean
1	theta/(theta + 2) (both signs)	Clayton increasing in theta > 0	kendallTau_clayton, kendallTau_claytonNegative, lowerOrthantLE_clayton
5	1 - (4/theta)(1 - D_1(theta)), D_1(theta) = (1/theta) int_0^theta t/(e^t - 1) dt (both signs)		kendallTau_frank_debye, kendallTau_frankNegative_debye, debyeOne_neg
6	1 + (4/theta) int_0^1 (1 - (1-t)^theta) log(1 - (1-t)^theta) / (1-t)^(theta-1) dt		kendallTau_joe
7	2 - 2/theta - 2(1 - theta)^2 log(1 - theta)/theta^2 (-1 at theta = 0)	lambda_U = 0	kendallTau_nelsen7, kendallTau_nelsen7_zero, hasUpperTailDependence_nelsen7
8	(theta - 4)/(3 theta)		kendallTau_nelsen8
9	1 - (4/theta) int_0^1 t(1 - theta log t) log(1 - theta log t) dt	NQD	kendallTau_nelsen9, isNQD_nelsen9
10		(0, 0); NQD	hasLowerTailDependence_nelsen10, hasUpperTailDependence_nelsen10, isNQD_nelsen10
11		lambda_U = 0	hasUpperTailDependence_nelsen11
13	1 - (4/theta) int_0^1 t((1 - log t) - (1 - log t)^(1-theta)) dt	(0, 0); PQD for theta >= 1, NQD for theta <= 1	kendallTau_nelsen13, hasLowerTailDependence_nelsen13, hasUpperTailDependence_nelsen13, isPQD_nelsen13, isNQD_nelsen13
15	(2 theta - 3)/(2 theta - 1)		kendallTau_genestGhoudi
16	4 theta - 1 - 4 theta log((1 + theta)/theta) + 4(1 - theta) sqrt(theta) arctan(1/sqrt(theta)) (theta > 0)	lambda_U = 0	kendallTau_nelsen16, hasUpperTailDependence_nelsen16
17		(0, 0)	hasLowerTailDependence_nelsen17, hasUpperTailDependence_nelsen17
18	1 - 4/(3 theta)		kendallTau_nelsen18
19	1 - (4/theta) int_0^1 t^2 (1 - e^(theta - theta/t)) dt	PQD	kendallTau_nelsen19, isPQD_nelsen19
20	1 - (4/theta) int_0^1 t^(theta+1) (1 - e^(1 - t^(-theta))) dt	PQD	kendallTau_nelsen20, isPQD_nelsen20
21		lambda_U = 2 - 2^(1/theta)	hasUpperTailDependence_nelsen21
22		lambda_U = 0	hasUpperTailDependence_nelsen22

Limit of family 17 at theta -> -infinity. With r = (1 + u)(1 + v)/2 the CDF tends to max(1, r) - 1 = max(0, (uv + u + v - 1)/2), which is the member theta = 1/2 of family 7 and not W (for example, at u = v = 0.8 the limit is 0.62, while W(0.8, 0.8) = 0.6); this is proved by the two-sided bound nelsen17_bounds_neg.

Complete property matrix

Every family of Table 4.1 (plus the named Gumbel, Frank, Joe, AMH members) is classified for quadrant dependence over its full parameter range (Copula.Archimedean.QuadrantClassification, one *_quadrant theorem per family; negative statements are proved from tail coefficients, non-strictness of the generator, or explicit witnesses). Kendall's tau is proved for all families (closed form where one exists, otherwise the explicit integral of Corollary 5.1.4). Blomqvist's beta is proved in closed form for all 22 families (Copula.Archimedean.BlomqvistTable, BlomqvistTableN; generic BivariateGenerator.blomqvistBeta_copula). Spearman's rho is proved wherever a closed or one-dimensional form exists; for the other families only the generic formula BivariateGenerator.spearmanRho_copula (rho = 12 int int C - 3) is available and no elementary closed form is known.

#	PQD	NQD	Kendall's tau (new in this round)	Spearman's rho
1 Clayton	theta > 0	-1 <= theta < 0	kendallTau_clayton[Negative]	generic
2	never	iff theta = 1	kendallTau_nelsen2	4 Gamma(1/theta+1)^2/Gamma(2/theta+1) - 3 (spearmanRho_nelsen2)
3 AMH	iff theta >= 0	iff theta <= 0	kendallTau_amh	series, dilogarithm form for abs theta < 1, 4 pi^2 - 39 at 1, 33 - 48 log 2 at -1 (SpearmanRhoAMH*)
4 Gumbel	yes	iff theta = 1	kendallTau_gumbel	generic
5 Frank	theta > 0	theta < 0	kendallTau_frank_debye	1 - (12/theta)(D_1 - D_2) (spearmanRho_frank_debye, spearmanRho_frankNegative_debye)
6 Joe	yes	iff theta = 1	kendallTau_joe	generic
7	iff theta = 1	yes	kendallTau_nelsen7	nelsen7_rho
8	never	iff theta <= 2	kendallTau_nelsen8	generic
9	never	yes	kendallTau_nelsen9	one-dimensional integral (spearmanRho_nelsen9)
10	never	yes	kendallTau_nelsen10	generic
11	never	yes	kendallTau_nelsen11	generic
12	yes	never	kendallTau_nelsen12	generic
13	iff theta >= 1	iff theta <= 1	kendallTau_nelsen13	generic
14	yes	never	kendallTau_nelsen14	generic
15	never	iff theta = 1	kendallTau_genestGhoudi	generic
16	iff theta >= 1	iff theta = 0	kendallTau_nelsen16	generic
17	iff theta >= -1	iff theta <= -1	kendallTau_nelsen17	generic
18	never	never	kendallTau_nelsen18	generic
19	yes	never	kendallTau_nelsen19	generic
20	yes	never	kendallTau_nelsen20	generic
21	never	iff theta = 1	kendallTau_nelsen21	generic
22	never	yes	kendallTau_nelsen22	generic

Several families are neither PQD nor NQD on part of their range (#2 and #15 and #21 for theta > 1,

8 for theta > 2, #16 for 0 < theta < 1, and #18 throughout). Family #17 changes sign at

theta = -1, not at 0. "Generic" means Spearman's rho is available only as the double integral of the CDF; Spearman's rho of the other families has no known elementary closed form.

Library revision: fe53ea2f · Lean 4.34.0