This function evaluates the second derivative of a given parametric bivariate copula density with respect to its parameter(s) and/or its arguments.
BiCopDeriv2(
u1,
u2,
family,
par,
par2 = 0,
deriv = "par",
obj = NULL,
check.pars = TRUE
)numeric vectors of equal length with values in \([0,1]\).
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)
Copula parameter.
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.
Derivative argument "par" = second derivative with respect to
the first parameter (default)"par2" = second derivative with respect to
the second parameter (only available for the t-copula) "u1" = second derivative with respect to
the first argument u1 "u2" = second derivative with respect to
the second argument u2 "par1par2" = second derivative with respect to
the first and second parameter (only available for the t-copula)
"par1u1" = second derivative with respect to
the first parameter and the first argument "par2u1" = second derivative with respect to the
second parameter and the first argument (only available for the t-copula) "par1u2" = second derivative with respect to
the first parameter and the second argument "par2u2" = second derivative with respect to
the second parameter and the second argument
(only available for the t-copula)
BiCop object containing the family and parameter
specification.
logical; default is TRUE; if FALSE, checks
for family/parameter-consistency are omitted (should only be used with
care).
A numeric vector of the second-order bivariate copula derivative
of the copula family
with parameter(s) par, par2
with respect to deriv
evaluated at u1 and u2.
If the family and parameter specification is stored in a BiCop()
object obj, the alternative version
BiCopDeriv2(u1, u2, obj, deriv = "par")can be used.
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.
## simulate from a bivariate Student-t copula
set.seed(123)
cop <- BiCop(family = 2, par = -0.7, par2 = 4)
simdata <- BiCopSim(100, cop)
## second derivative of the Student-t copula w.r.t. the first parameter
u1 <- simdata[,1]
u2 <- simdata[,2]
BiCopDeriv2(u1, u2, cop, deriv = "par")
#> [1] 5.52809763 -6.76374650 -31.83591292 -7.01735211 12.12433505
#> [6] 15.34033040 7.73510022 -0.94670323 -1.41601296 -10.02843789
#> [11] -8.68029395 0.95588885 -0.43138922 9.42578602 -6.56519892
#> [16] -1.92568441 -5.59326067 -7.43485156 8.75046881 -4.39491258
#> [21] 0.54810296 6.50816869 -6.08440441 8.24368107 -1.88422962
#> [26] -6.83775896 -2.49916761 -0.50757932 24.59738707 24.02868314
#> [31] -8.20068903 -0.40169484 14.00199683 -7.33235759 14.39322430
#> [36] 1.62021137 0.20592689 -1.63022993 12.18709188 -6.38424843
#> [41] 15.64016667 0.52662869 -0.08898609 -2.14329519 18.97063160
#> [46] 23.47566683 11.93263827 -5.92514078 -8.04998550 12.10185339
#> [51] 9.66701779 -6.18525476 -7.12666182 -6.03618458 -6.23950226
#> [56] 43.13666702 -26.40456885 -2.31934111 -5.05991291 11.85207116
#> [61] 10.63559810 -5.09236098 -5.21402757 -3.46821079 23.50426791
#> [66] -6.53208680 -2.60973050 6.55672714 -7.58773112 24.26485948
#> [71] 6.35521025 -13.51591361 18.47383323 -3.78259121 12.86727622
#> [76] 9.52164910 -2.02966281 -3.73662609 0.64114966 -2.45665241
#> [81] 7.97389045 12.63800575 12.57201860 8.05718034 -0.18516031
#> [86] -0.54918861 -8.48560057 5.11986176 12.45511274 -0.16930071
#> [91] 21.40568958 9.99918150 12.22886338 13.95705626 -2.90314950
#> [96] -0.70096338 -36.15097718 12.04052543 9.84430985 16.15624508
## estimate a Student-t copula for the simulated data
cop <- BiCopEst(u1, u2, family = 2)
## and evaluate its second derivative w.r.t. the second argument u2
BiCopDeriv2(u1, u2, cop, deriv = "u2")
#> [1] -50.9297502 -20.3997269 -9647.0491441 -3.0619298 -19.2131620
#> [6] 439.0305273 -11.9305604 -981.0602729 6.0432357 -49.2532624
#> [11] 31.4704288 3.2419182 -4.6891806 -16.3300434 5.5496307
#> [16] 11.5078191 2.3379621 1415.4822777 -69.4183935 0.4417732
#> [21] 14.9629162 -13.0874751 11.3172255 -9.4718976 -76.0198858
#> [26] 288.6360881 -166.1668958 -18.9079400 -151.8387268 -24.3486367
#> [31] -43.1482782 -6.4339824 -17.7634868 10.6928234 -21.0956436
#> [36] -1.8977339 -1493.6473702 -10.4407591 -23.0412676 4.0813051
#> [41] -40.4913757 -21.9430547 40.5151457 22.4774720 -359.9646048
#> [46] -51.4848808 -27.6384419 2.4087037 -161.6757427 -18.6545135
#> [51] -23.4287291 -0.9156089 -7.9151776 62.7708305 -1.0671406
#> [56] -76.9350918 -7856.8717329 -67.1375847 3.5570211 -21.0306386
#> [61] -28.1863945 1.7354084 -28.9232194 8.1934421 -74.6844705
#> [66] 2.8909110 -1.8476828 -15.4994178 12.0408783 3385.9824223
#> [71] -10.0727446 269.7842111 -128.6626053 7.0941952 -53.9328817
#> [76] -0.8665695 -5.1601313 34.0996573 -20.6058487 -7.7898285
#> [81] -17.1704758 -25.2604303 -19.7325242 -59.7799324 -7.9521905
#> [86] -30.2167805 19.4505062 -9.3929417 -19.4740699 23.2683910
#> [91] -66.4589642 -42.2406065 -18.9331405 -22.8359080 9.5854430
#> [96] -1.2060011 2401.2319963 152.7789732 -22.7630506 -18.1517880