derivative.tau {ElliptCopulas} | R Documentation |
\tau
and its k
-th derivativeThe 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
.
derivative.tau(x, a, d, k)
f3(x, d, k = 0)
f4(x, a, d, k = 0)
x |
a numeric vector |
a |
a parameter |
d |
the dimension of the data |
k |
the order of derivatives for |
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
f3()
: f_3(x) = x^{(d-2)/d}
f4()
: f_4(x) = a^{d/2} + x^{d/2}
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)
.
Victor Ryan, Alexis Derumigny
Ryan, V., & Derumigny, A. (2024). On the choice of the two tuning parameters for nonparametric estimation of an elliptical distribution generator arxiv:2408.17087.
derivative.psi
and derivative.rho
.
vectorized_Faa_di_Bruno
which is used for the computation
of the derivatives.
# 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)