On the choice of the two tuning parameters for nonparametric estimation of an elliptical distribution generator

Elliptical distributions are a simple and flexible class of distributions that depend on a one-dimensional function, called the density generator. In this article, we study the non-parametric estimator of this generator that was introduced by Liebscher (2005). This estimator depends on two tuning parameters: a bandwidth $h$ -- as usual in kernel smoothing -- and an additional parameter $a$ that control the behavior near the center of the distribution. We give an explicit expression for the asymptotic MSE at a point $x$, and derive explicit expressions for the optimal tuning parameters $h$ and $a$. Estimation of the derivatives of the generator is also discussed. A simulation study shows the performance of the new methods.