derivative.tau: Computing tau and its k-th derivative

In ElliptCopulas: Inference of Elliptical Distributions and Copulas

Computing \tau and its k-th derivative

Description

The function \tau is used to compute \alpha_{i,k}, which is required to compute the derivatives of the generator of elliptical distribution. The functions f3 and f4 are already implemented in derivative.tau. These functions are needed for computing higher derivatives of \tau.

Usage

derivative.tau(x, a, d, k)

f3(x, d, k = 0)

f4(x, a, d, k = 0)

Arguments

x	a numeric vector
a	a parameter a > 0 that reduces the bias of the estimator around zero
d	the dimension of the data
k	the order of derivatives for f3 and f4

Value

A numeric vector \tau^{(k)}(x_1), ..., \tau^{(k)}(x_N) where N = length(x).

The functions f3 and f4 also return a numeric value

Functions

• f3(): f_3(x) = x^{(d-2)/d}

• f4(): f_4(x) = a^{d/2} + x^{d/2}

Note

The function \tau is defined as follows: \tau(x) = x^{(d-2)/2}/\psi^{\prime}(x), where \psi^{\prime}(x) = x^{d/2 - 1}(a^{d/2} + x^{d/2})^{2/d - 1}. The definition of \psi is already described in derivative.tau. Therefore, by the definition of f_3 and f_4, the function \tau is actually \tau(x) = (f_3 \circ f_4)(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.psi and derivative.rho. vectorized_Faa_di_Bruno which is used for the computation of the derivatives.

Examples

# Return the 5-th derivative of tau at x = 1
derivative.tau(x = 1, a = 1, d = 3, k = 5)

# Return the value of tau at x = 1.
derivative.tau(x = 1, a = 1, d = 3, k = 0)

# Vectorized version
derivative.tau(x = c(1,3), a = 1, d = 3, k = 5)

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