elliptselect: elliptselect
| elliptselect | R Documentation |
elliptselect
Description
This functions selects optimal tuning parameters for improved kernel estimation of the generator of an elliptical distribution.
Usage
elliptselect(n, q, pseq, aseq)
Arguments
n |
The sample size. |
q |
The total dimension. |
pseq |
Candidate values for the |
aseq |
Candidate values for the |
Details
When using the function elldistrest for estimating an elliptical generator g_{\mathcal{R}}
based on a kernel k with bandwidth h_{n}, the function
\psi(t) = -a + \left (a^{q/2} + t^{q/2} \right )^{2/q},
and the shrinkage function (for q > 3)
\alpha(t) = 1 - \frac{1}{t^{\delta} + 1},
this function selects h_{n}, \delta and a in the following way.
Use the normal generator g_{\mathcal{R}}(t) = e^{-t/2}/(2 \pi)^{q/2} as reference generator, and define
\Psi(t) = \frac{\pi^{q/2}}{\Gamma(q/2)} \left (\psi^{-1}(t) \right )^{\prime} \left (\psi^{-1}(t) \right )^{q/2 - 1} g_{\mathcal{R}} \left (\psi^{-1}(t) \right ),
as well as
h_{n}^{\text{opt}} = \left \{\frac{\left (\int_{-1}^{1} k^{2}(t) dt \right ) \left (\int_{0}^{\infty} \alpha(t)^{-1} \Psi(t) dt \right )}{\left (\int_{-1}^{1} t^{2} k(t) dt \right )^{2} \left (\int_{0}^{\infty} \left (\alpha(t)^{2} \Psi^{\prime \prime}(t) \right )^{2} dt \right )} \right \}^{1/5} n^{-1/5}.
When q = 2, take \alpha(t) = 1 (there is no need for shrinkage), and take h_{n}^{\text{opt}}. The value of a does not matter.
When q > 2, specify a grid of candidate \delta-values in (3/4 - 1/q,1) and a grid of a-values in (0, \infty).
For each of these candidate values, compute the corresponding optimal (AMISE) bandwidth h_{n}^{\text{opt}}.
Take the combination of parameters that minimizes (a numerical approximation of) the (normal reference) AMISE given in equation (20) of De Keyser & Gijbels (2024).
Value
A list with elements "Opta" containing the optimal a, "Optp" containing the optimal \delta, and "Opth" containing the optimal h_{n}.
References
De Keyser, S. & Gijbels, I. (2024). Hierarchical variable clustering via copula-based divergence measures between random vectors. International Journal of Approximate Reasoning 165:109090. doi: https://doi.org/10.1016/j.ijar.2023.109090.
See Also
elldistrest for improved kernel estimation of the elliptical generator of an elliptical distribution,
ellcopest for improved kernel estimation of the elliptical generator of a meta-elliptical copula,
phiellip for estimating the \Phi-dependence between k random vectors having a meta-elliptical copula.
Examples
q = 4
n = 1000
opt_parameters = elliptselect(n,q,seq((3/4)-(1/q)+0.01,1-0.01,len = 200),
seq(0.01,2,len = 200))
