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)

Arguments

u1,u2

Numeric vectors of equal length with values in [0,1].

family

An integer defining the bivariate copula family: 0 = independence copula 1 = Gaussian copula 2 = Student t copula (t-copula) 3 = Clayton copula 4 = Gumbel copula 5 = Frank

par

Copula parameter.

par2

Second parameter for bivariate t-copula; default: par2 = 0.

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

obj

BiCop object containing the family and parameter specification.

Value

A numeric vector of the second bivariate copula derivative with respect to deriv evaluated at u1 and u2 with parameter(s) par and par2.

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.

References

Schepsmeier, U. and J. Stoeber (2012). Derivatives and Fisher information of bivariate copulas. Statistical Papers. http://link.springer.com/article/10.1007/s00362-013-0498-x.

Examples

Run this code

## simulate from a bivariate t-copula
simdata <- BiCopSim(300, 2, -0.7, par2 = 4)

## second derivative of the bivariate t-copula with respect to the first parameter
u1 <- simdata[,1]
u2 <- simdata[,2]
BiCopDeriv2(u1, u2, 2, -0.7, par2 = 4, deriv = "par")

## estimate a bivariate copula from the data and evaluate its derivative
cop <- BiCopEst(u1, u2, family = 2)
BiCopDeriv2(u1, u2, cop, deriv = "par")

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