VineCopula (version 2.2.0)

BiCopHfuncDeriv2: Second Derivatives of the h-Function of a Bivariate Copula

Description

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

Usage

BiCopHfuncDeriv2(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'') 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" = second derivative with respect to the first parameter (default) "par2" = second derivative with respect to the second parameter (only available for the t-copula) "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) "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).

Value

A numeric vector of the second-order 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

BiCopHfuncDeriv2(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. http://link.springer.com/article/10.1007/s00362-013-0498-x.

See Also

RVineGrad, RVineHessian, BiCopDeriv, BiCopDeriv2, BiCopHfuncDeriv, BiCop

Examples

# NOT RUN {
## 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 conditional bivariate t-copula
## with respect to the first parameter
u1 <- simdata[,1]
u2 <- simdata[,2]
BiCopHfuncDeriv2(u1, u2, cop, deriv = "par")

## 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
BiCopHfuncDeriv2(u1, u2, cop, deriv = "u2")

# }