VineCopula: BiCopDeriv2 – R documentation

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BiCopDeriv2

Second Derivatives of a Bivariate Copula Density

Description

This function evaluates the second derivative of a given parametric bivariate copula density with respect to its parameter(s) and/or its arguments.

Usage

BiCopDeriv2(
  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`	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"` = 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)
`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

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.

Value

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.

Author(s)

Ulf Schepsmeier, Jakob Stoeber

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.

Examples

## 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")

## 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")

VineCopula

Statistical Inference of Vine Copulas

v2.4.1

GPL (>= 2)

Authors

Thomas Nagler [aut, cre], Ulf Schepsmeier [aut], Jakob Stoeber [aut], Eike Christian Brechmann [aut], Benedikt Graeler [aut], Tobias Erhardt [aut], Carlos Almeida [ctb], Aleksey Min [ctb, ths], Claudia Czado [ctb, ths], Mathias Hofmann [ctb], Matthias Killiches [ctb], Harry Joe [ctb], Thibault Vatter [ctb]

Initial release