Derivatives of the h-Function of a Bivariate Copula

This function evaluates the derivative of a given conditional parametric bivariate copula (h-function) with respect to its parameter(s) or one of its arguments.

BiCopHfuncDeriv(
  u1,
  u2,
  family,
  par,
  par2 = 0,
  deriv = "par",
  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
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'')
23 = rotated Clayton copula (90 degrees)
`24` = rotated Gumbel copula (90 degrees)
`26` = rotated Joe copula (90 degrees)
`33` = rotated Clayton copula (270 degrees)
`34` = rotated Gumbel copula (270 degrees)
`36` = rotated Joe copula (270 degrees)
par: numeric; single number or vector of size length(u1); copula parameter.
par2: integer; single number or vector of size length(u1); second parameter for the t-Copula; default is par2 = 0, should be an positive integer for the Students's t copula family = 2.
deriv: Derivative argument
"par" = derivative with respect to the first parameter (default)
"par2" = derivative with respect to the second parameter (only available for the t-copula)
"u2" = derivative with respect to the second argument u2
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 conditional bivariate copula derivative

of the copula family,
with parameter(s) par, par2,
with respect to deriv,
evaluated at u1 and u2.

Details

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

BiCopHfuncDeriv(u1, u2, obj, deriv = "par")

can be used.

References

Schepsmeier, U. and J. Stoeber (2014). Derivatives and Fisher information of bivariate copulas. Statistical Papers, 55 (2), 525-542.
https://link.springer.com/article/10.1007/s00362-013-0498-x.

Author

Ulf Schepsmeier

Examples


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

## derivative of the conditional Student-t copula
## with respect to the first parameter
u1 <- simdata[,1]
u2 <- simdata[,2]
BiCopHfuncDeriv(u1, u2, cop, deriv = "par")
#>   [1] -0.598879576 -0.694092965  1.101459131 -0.579894656  0.085143765
#>   [6]  0.045263264 -0.124492828 -0.914662735  0.264757455 -0.809827056
#>  [11] -0.144991103 -0.316577613 -0.320545869 -0.179146455  0.366497913
#>  [16] -0.040173353 -0.369191559  0.014361201  0.592846659  0.359471850
#>  [21]  0.097449063  0.237241992  0.198754672  0.053242904 -0.754926562
#>  [26]  0.041462568  0.827360291  0.553869372 -0.490998434  0.190831164
#>  [31]  0.783682679  0.362917733  0.082694956 -0.249834704  0.101055634
#>  [36] -0.182089378  0.526243348  0.485013788 -0.222883303  0.440803641
#>  [41] -0.326582560 -0.555969332  0.065956789 -0.017284873  0.684402406
#>  [46] -0.257875035 -0.312641751  0.388477301  0.879453107 -0.022534107
#>  [51]  0.332678724 -0.608878940 -0.627556644 -0.089561930  0.498852955
#>  [56]  0.312101882 -1.081332576  0.748299526 -0.531491828  0.188438376
#>  [61]  0.361629504  0.358745853 -0.684002542  0.183017895 -0.343785307
#>  [66] -0.475266389 -0.339675659 -0.300649893 -0.234121611  0.049913733
#>  [71]  0.150418584  0.024234911  0.543959731  0.284837078 -0.472009315
#>  [76]  0.003428686  0.397520877  0.103865304  0.542623800  0.467659350
#>  [81]  0.280493267  0.247487710 -0.064071355  0.581023237  0.389754707
#>  [86]  0.626662199 -0.183429912 -0.196256827 -0.057137349 -0.087889506
#>  [91]  0.346188362  0.480466113  0.036183653 -0.117217329 -0.100535310
#>  [96]  0.249183916 -0.042171143  0.035366213 -0.316293586 -0.113213738

## estimate a Student-t copula for the simulated data
cop <- BiCopEst(u1, u2, family = 2)
## and evaluate the derivative of the conditional copula
## w.r.t. the second argument u2
BiCopHfuncDeriv(u1, u2, cop, deriv = "u2")
#>   [1]  1.79447650  1.31470857 13.03395287  0.86271682  1.23809501  2.02019394
#>   [7]  1.09848608  5.48491776  0.18740720  1.79948157  0.56911364  0.26109840
#>  [13]  0.89484041  1.17333442  0.50274380  0.03684953  0.66568767  1.72064677
#>  [19]  2.01643095  0.73752649  0.94338835  1.10066056  0.24283230  1.13387800
#>  [25]  2.09500861  1.15486819  2.81668724  1.25450701  2.76713621  2.02034154
#>  [31]  1.70821289  0.93373006  1.40199879  0.40717098  1.40692015  0.91574777
#>  [37]  3.50334420  1.04446990  1.32684648  0.55343129  1.65635905  1.31878627
#>  [43]  1.01066448  0.02232976  3.76128149  2.03093842  1.42168426  0.65291374
#>  [49]  2.79424972  1.22447834  1.33567334  0.85060968  1.01564132  0.83251806
#>  [55]  0.77236168  3.28769808 12.02997009  1.99958543  0.63784108  1.28120398
#>  [61]  1.43194157  0.69388625  1.45565408  0.15921374  2.18176190  0.61530785
#>  [67]  0.81399751  1.15659029  0.44459111  3.07430311  1.05949158  0.10664951
#>  [73]  2.56053697  0.25406270  1.83385232  1.25601348  0.89696876  0.83910056
#>  [79]  1.28947804  0.97209805  1.19471820  1.37493034  1.26104911  1.90545823
#>  [85]  0.97276630  1.48289536  0.46610225  1.03082755  1.25200565  0.94808684
#>  [91]  2.05443974  1.66980487  1.23515300  1.36177865  0.08808345  0.84517734
#>  [97]  1.05424206  1.67313352  1.32112674  1.53980743