BiCopHinv {VineCopula}

Inverse Conditional Distribution Function of a Bivariate Copula

Description

Evaluate the inverse conditional distribution function (inverse h-function) of a given parametric bivariate copula.

Usage

BiCopHinv(u1, u2, family, par, par2 = 0, obj = NULL, check.pars = TRUE)

BiCopHinv1(u1, u2, family, par, par2 = 0, obj = NULL, check.pars = TRUE)

BiCopHinv2(u1, u2, family, par, par2 = 0, obj = NULL, check.pars = TRUE)

Arguments

u1, u2

numeric vectors of equal length with values in [0,1].

family

integer; single number or vector of size length(u1); defines the bivariate copula family:
0 = independence copula
1 = Gaussian copula
2 = Student t copula (t-copula)
3 = Clayton copula
4 = Gumbel copula
5 = Frank copula
6 = Joe copula
7 = BB1 copula
8 = BB6 copula
9 = BB7 copula
10 = BB8 copula
13 = rotated Clayton copula (180 degrees; ⁠survival Clayton'') \cr `14` = rotated Gumbel copula (180 degrees; ⁠survival Gumbel”)
16 = rotated Joe copula (180 degrees; ⁠survival Joe'') \cr `17` = rotated BB1 copula (180 degrees; ⁠survival BB1”)
18 = rotated BB6 copula (180 degrees; ⁠survival BB6'')\cr `19` = rotated BB7 copula (180 degrees; ⁠survival BB7”)
20 = rotated BB8 copula (180 degrees; “survival BB8”)
23 = rotated Clayton copula (90 degrees)
'24' = rotated Gumbel copula (90 degrees)
'26' = rotated Joe copula (90 degrees)
'27' = rotated BB1 copula (90 degrees)
'28' = rotated BB6 copula (90 degrees)
'29' = rotated BB7 copula (90 degrees)
'30' = rotated BB8 copula (90 degrees)
'33' = rotated Clayton copula (270 degrees)
'34' = rotated Gumbel copula (270 degrees)
'36' = rotated Joe copula (270 degrees)
'37' = rotated BB1 copula (270 degrees)
'38' = rotated BB6 copula (270 degrees)
'39' = rotated BB7 copula (270 degrees)
'40' = rotated BB8 copula (270 degrees)
'104' = Tawn type 1 copula
'114' = rotated Tawn type 1 copula (180 degrees)
'124' = rotated Tawn type 1 copula (90 degrees)
'134' = rotated Tawn type 1 copula (270 degrees)
'204' = Tawn type 2 copula
'214' = rotated Tawn type 2 copula (180 degrees)
'224' = rotated Tawn type 2 copula (90 degrees)
'234' = rotated Tawn type 2 copula (270 degrees)

par

numeric; single number or vector of size length(u1); copula parameter.

par2

numeric; single number or vector of size length(u1); second parameter for bivariate copulas with two parameters (t, BB1, BB6, BB7, BB8, Tawn type 1 and type 2; default: par2 = 0). par2 should be an positive integer for the Students's t copula family = 2.

obj

BiCop object containing the family and parameter specification.

check.pars

logical; default is TRUE; if FALSE, checks for family/parameter-consistency are omitted (should only be used with care).

Details

The h-function is defined as the conditional distribution function of a bivariate copula, i.e.,

h_1(u_2|u_1;\boldsymbol{\theta}) := P(U_2 \le u_2 | U_1 = u_1) = \frac{\partial C(u_1, u_2; \boldsymbol{\theta})}{\partial u_1},

h_2(u_1|u_2;\boldsymbol{\theta}) := P(U_1 \le u_1 | U_2 = u_2) = \frac{\partial C(u_1, u_2; \boldsymbol{\theta})}{\partial u_2},

where (U_1, U_2) \sim C, and C is a bivariate copula distribution function with parameter(s) \boldsymbol{\theta}. For more details see Aas et al. (2009).

If the family and parameter specification is stored in a BiCop() object obj, the alternative version

BiCopHinv(u1, u2, obj),
BiCopHinv1(u1, u2, obj),
BiCopHinv2(u1, u2, obj)

can be used.

Value

BiCopHinv returns a list with

hinv1

Numeric vector of the inverse conditional distribution function (inverse h-function) of the copula family with parameter(s) par, par2 evaluated at u2 given u1, i.e., h_1^{-1}(u_2|u_1;\boldsymbol{\theta}).

hinv2

Numeric vector of the inverse conditional distribution function (inverse h-function) of the copula family with parameter(s) par, par2 evaluated at u1 given u2, i.e., h_2^{-1}(u_1|u_2;\boldsymbol{\theta}).

BiCopHinv1 is a faster version that only calculates hinv1; BiCopHinv2 only calculates hinv2.

Author(s)

Ulf Schepsmeier, Thomas Nagler

References

Aas, K., C. Czado, A. Frigessi, and H. Bakken (2009). Pair-copula constructions of multiple dependence. Insurance: Mathematics and Economics 44 (2), 182-198.

See Also

BiCopHfunc(), BiCopPDF(), BiCopCDF(), RVineLogLik(), RVineSeqEst(), BiCop()

Examples

# inverse h-functions of the Gaussian copula
cop <- BiCop(1, 0.5)
hi <- BiCopHinv(0.1, 0.2, cop)

# or using the fast versions
hi1 <- BiCopHinv1(0.1, 0.2, cop)
hi2 <- BiCopHinv2(0.1, 0.2, cop)
all.equal(hi$hinv1, hi1)
all.equal(hi$hinv2, hi2)

# check if it is actually the inverse
cop <- BiCop(3, 3)
all.equal(0.2, BiCopHfunc1(0.1, BiCopHinv1(0.1, 0.2, cop), cop))
all.equal(0.1, BiCopHfunc2(BiCopHinv2(0.1, 0.2, cop), 0.2, cop))