Conditional distribution function (h-function) of a bivariate copula

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Description

This function evaluates the conditional distribution function (h-function) of a given parametric bivariate copula.

Usage

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BiCopHfunc(u1, u2, family, par, par2=0)

Arguments

u1,u2

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

family

An integer defining 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”)
14 = rotated Gumbel copula (180 degrees; “survival Gumbel”)
16 = rotated Joe copula (180 degrees; “survival Joe”)
17 = rotated BB1 copula (180 degrees; “survival BB1”)
18 = rotated BB6 copula (180 degrees; “survival BB6”)
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)

par

Copula parameter.

par2

Second parameter for bivariate copulas with two parameters (t, BB1, BB6, BB7, BB8; default: par2 = 0).

Details

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

h(u|v,θ) := F(u|v) = \partial C(u,v) / \partial v,

where C is a bivariate copula distribution function with parameter(s) θ. For more details see Aas et al. (2009).

Value

hfunc1

Numeric vector of the conditional distribution function (h-function) evaluated at u2 given u1, i.e., h(u2|u1,θ).

hfunc2

Numeric vector of the conditional distribution function (h-function) evaluated at u1 given u2, i.e., h(u1|u2,θ).

Author(s)

Ulf Schepsmeier

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

BiCopPDF, BiCopCDF, CDVineLogLik, CDVineSeqEst

Examples

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## Example 1: 4-dimensional C-vine model with mixed pair-copulas
data(worldindices)
Data = as.matrix(worldindices)[,1:4]
d = dim(Data)[2]
fam = c(5,1,3,14,3,2)

# sequential estimation
seqpar1 = CDVineSeqEst(Data,fam,type=1,method="itau")

# calculate the inputs of the second tree using h-functions
h1 = BiCopHfunc(Data[,1],Data[,2],fam[1],seqpar1$par[1])
h2 = BiCopHfunc(Data[,1],Data[,3],fam[2],seqpar1$par[2])
h3 = BiCopHfunc(Data[,1],Data[,4],fam[3],seqpar1$par[3])

# compare estimated parameters
BiCopEst(h1$hfunc1,h2$hfunc1,fam[4],method="itau")
seqpar1$par[4]

BiCopEst(h1$hfunc1,h3$hfunc1,fam[5],method="itau")
seqpar1$par[5]


## Example 2: 4-dimensional D-vine model with mixed pair-copulas

# sequential estimation
seqpar2 = CDVineSeqEst(Data,fam,type=2,method="itau")

# calculate the inputs of the second tree using h-functions
h1 = BiCopHfunc(Data[,1],Data[,2],fam[1],seqpar2$par[1])
h2 = BiCopHfunc(Data[,2],Data[,3],fam[2],seqpar2$par[2])
h3 = BiCopHfunc(Data[,3],Data[,4],fam[3],seqpar2$par[3])

# compare estimated parameters
BiCopEst(h1$hfunc2,h2$hfunc1,fam[4],method="itau")
seqpar2$par[4]

BiCopEst(h2$hfunc2,h3$hfunc1,fam[5],method="itau")
seqpar2$par[5]

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