derivative.psi: Computing psi, its inverse Psi and the k-th derivative of Psi

In ElliptCopulas: Inference of Elliptical Distributions and Copulas

Computing \psi, its inverse \Psi and the k-th derivative of \Psi

Description

The function \psi is used to estimate the generator of elliptical distribution. It depends on the parameter a, which reduces the bias of the estimator around zero. The functions f1 and f2 are already implemented in derivative.psi. They are required to compute higher derivatives of \Psi.

Usage

derivative.psi(x, a, d, k, inverse)

f1(x, d, k = 0)

f2(x, a, d, k = 0)

Arguments

x	a numeric value
a	a parameter a > 0 that reduces the bias of the estimator around zero
d	the dimension of the data
k	the order of derivative. If k = 0, then the original function value is returned. If k > 0, the value of its derivative is returned.
inverse	if inverse = TRUE, then the inverse of \Psi is of interest. Otherwise, the function \psi is used for the computation

Value

A numeric value \psi(x)^{(k)} if inverse = TRUE, otherwise \Psi(x)^{(k)}.

The functions f1 and f2 also return a numeric value

Functions

• f1(): f_1(x) = x^{2/d}

• f2(): f_2(x) = (x + a)^{d/2} - a^{d/2}

Note

The derivatives of \psi is not yet implemented. The function \psi is defined as \psi(x) = -a + (a^{d/2} + x^{d/2})^{2/d}. For any a > 0 and x > 0, it has an inverse. Let \Psi be the inverse function of \psi, then

\Psi(x) = ((x+a)^{d/2} - a^{d/2})^{2/d} = (f_1 \circ f_2)(x).

Author(s)

Victor Ryan, Alexis Derumigny

References

Ryan, V., & Derumigny, A. (2024). On the choice of the two tuning parameters for nonparametric estimation of an elliptical distribution generator arxiv:2408.17087.

See Also

derivative.tau and derivative.rho. vectorized_Faa_di_Bruno which is used for the computation of the derivatives.

Examples

# Return the 5-th derivative of the inverse of psi
derivative.psi(x = 1, a = 1, d = 3, k = 5, inverse = TRUE)

# Return psi
derivative.psi(x = 1, a = 1, d = 3, k = 0, inverse = FALSE)

ElliptCopulas documentation built on Sept. 11, 2024, 6:50 p.m.