Density of a Bivariate Copula

Source: R/BiCopPDF.R

This function evaluates the probability density function (PDF) of a given parametric bivariate copula.

BiCopPDF(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).

Value

A numeric vector of the bivariate copula density

  • of the copula family

  • with parameter(s) par, par2

  • evaluated at u1 and u2.

Details

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

BiCopPDF(u1, u2, obj)

can be used.

Author

Eike Brechmann

Examples

set.seed(123)
## simulate from a bivariate Student-t copula
cop <- BiCop(family = 2, par = -0.7, par2 = 4)
simdata <- BiCopSim(100, cop)

## evaluate the density of the bivariate t-copula
u1 <- simdata[,1]
u2 <- simdata[,2]
BiCopPDF(u1, u2, cop)
#>   [1] 1.7476289 1.1253686 4.3948771 0.8726650 1.5970650 4.0665279 1.5506525
#>   [8] 3.2518440 0.3420938 0.9234262 1.3371767 0.1861758 1.2740615 1.5294965
#>  [15] 0.7933051 0.3425504 1.0041005 4.5411878 1.9548186 1.0939714 1.7351468
#>  [22] 1.4652813 0.6291736 1.7040858 1.6721897 2.8457657 1.9823741 1.3391780
#>  [29] 2.9373786 2.9111003 1.2189760 1.2631258 2.0330900 0.8448330 1.9772725
#>  [36] 1.4537177 0.2037127 1.2349811 1.6541503 0.7310370 1.9794678 1.3927695
#>  [43] 2.0050030 0.3565457 3.2173480 2.6788939 1.7049721 0.9617950 1.7285330
#>  [50] 1.5889394 1.6010289 0.6209664 0.9527241 1.8776343 0.9331976 4.3537199
#>  [57] 4.4719155 1.6133001 0.5482457 1.6166834 1.6775950 1.0495470 0.5241108
#>  [64] 0.4516549 2.6485938 0.7553097 1.1889348 1.4670809 0.9331701 6.9320818
#>  [71] 1.5214964 1.0149675 2.5753882 0.4790395 1.9758710 2.0281979 1.1984456
#>  [78] 1.7362999 1.3836301 1.1894062 1.5046916 1.6979178 1.6588089 1.8752542
#>  [85] 1.2693172 1.4344133 1.0769339 1.4666279 1.6416774 1.8108982 2.4709521
#>  [92] 1.7982885 1.6135021 1.8222352 0.4068980 1.3195279 4.0104640 3.1903099
#>  [99] 1.5967170 2.2580077

## select a bivariate copula for the simulated data
fit <- BiCopSelect(u1, u2)
summary(fit)
#> Family
#> ------ 
#> No:    5
#> Name:  Frank
#> 
#> Parameter(s)
#> ------------
#> par:  -6.64
#> 
#> Dependence measures
#> -------------------
#> Kendall's tau:    -0.55 (empirical = -0.52, p value < 0.01)
#> Upper TD:         0 
#> Lower TD:         0 
#> 
#> Fit statistics
#> --------------
#> logLik:  35.54 
#> AIC:    -69.08 
#> BIC:    -66.47 
#> 
## and evaluate its PDF
round(BiCopPDF(u1, u2, fit), 3)
#>   [1] 1.832 1.090 4.311 0.719 1.792 3.616 1.658 3.277 0.163 1.108 1.486 0.125
#>  [13] 1.236 1.679 0.588 0.477 0.849 4.070 2.039 0.971 1.839 1.559 0.493 1.800
#>  [25] 1.788 3.064 2.168 1.330 2.795 2.779 1.294 1.220 2.113 0.712 2.070 1.487
#>  [37] 0.991 1.177 1.822 0.518 2.080 1.405 2.157 0.946 3.040 2.608 1.847 0.791
#>  [49] 1.986 1.788 1.734 0.536 0.837 2.115 0.759 3.578 4.262 1.718 0.410 1.793
#>  [61] 1.807 0.911 0.738 0.285 2.585 0.551 1.108 1.562 0.841 4.668 1.608 2.687
#>  [73] 2.549 0.272 2.062 2.108 1.120 1.898 1.393 1.108 1.625 1.856 1.833 1.963
#>  [85] 1.230 1.463 1.101 1.535 1.821 1.937 2.457 1.898 1.802 1.955 0.337 1.302
#>  [97] 4.257 3.082 1.734 2.298