derivative.psi {ElliptCopulas} | R Documentation |
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 |
d |
the dimension of the data |
k |
the order of derivative.
If |
inverse |
if |
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)