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

Victor Ryan, Alexis Derumigny

TL;DR

The paper tackles nonparametric estimation of the elliptical distribution generator $g$ using Liebscher’s kernel-based estimator with two tuning parameters, a bandwidth $h$ and a center parameter $a$. It derives explicit asymptotic MSE expressions, provides closed-form optimal choices for $h$ and $a$, and develops data-driven procedures to select these tuning parameters, including direct optimization of the AMSE tilde and a two-step adaptive algorithm. The methodology extends to estimating derivatives of $g$ via a triangular-inversion scheme, with bias/variance results that justify the estimators and guide bandwidth selection. A comprehensive simulation study demonstrates the practical performance and robustness of the adaptive tuning approach, and the authors discuss extensions to unknown nuisance parameters and potential alternatives like MISE-based criteria. Overall, the work delivers practical, theory-backed tools for accurate, dimension-agnostic nonparametric estimation of elliptical-generation mechanisms with applications in finance and copula modeling.

Abstract

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.

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

TL;DR

The paper tackles nonparametric estimation of the elliptical distribution generator using Liebscher’s kernel-based estimator with two tuning parameters, a bandwidth and a center parameter . It derives explicit asymptotic MSE expressions, provides closed-form optimal choices for and , and develops data-driven procedures to select these tuning parameters, including direct optimization of the AMSE tilde and a two-step adaptive algorithm. The methodology extends to estimating derivatives of via a triangular-inversion scheme, with bias/variance results that justify the estimators and guide bandwidth selection. A comprehensive simulation study demonstrates the practical performance and robustness of the adaptive tuning approach, and the authors discuss extensions to unknown nuisance parameters and potential alternatives like MISE-based criteria. Overall, the work delivers practical, theory-backed tools for accurate, dimension-agnostic nonparametric estimation of elliptical-generation mechanisms with applications in finance and copula modeling.

Abstract

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 -- as usual in kernel smoothing -- and an additional parameter that control the behavior near the center of the distribution. We give an explicit expression for the asymptotic MSE at a point , and derive explicit expressions for the optimal tuning parameters and . Estimation of the derivatives of the generator is also discussed. A simulation study shows the performance of the new methods.
Paper Structure (24 sections, 8 theorems, 146 equations, 7 figures, 1 algorithm)

This paper contains 24 sections, 8 theorems, 146 equations, 7 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. MSE and optimal choice of h
  3. Estimation of the derivatives of the generator
  4. Data-driven choice of the tuning parameters
  5. Estimation of AMSEtilde
  6. Direct optimization of AMSEtilde
  7. Simulation study
  8. Influence of the tuning parameters on the MISE of Liebscher's estimator hat g_ n,h,a in dimension 3
  9. Comparison of the first-step estimator with the new estimator (adaptive choice of the tuning parameters)
  10. Extensions and open problems
  11. Unknown mu and Sigma
  12. Estimation of g by minimization of the integrated MSE (MISE)
  13. Proof of Theorem \ref{['thm:MSE']}
  14. Expression for the bias
  15. Expression for the variance
  16. ...and 9 more sections

Key Result

lemma 1

For every $k \geq 0$, there exists $k+1$ functions $\alpha_{0,k}(\xi), \dots, \alpha_{k,k}(\xi)$ such that and the functions $\alpha_{i,k}$ do not depend on $g$, but only on $a$ and $d$. Furthermore, $\alpha_{k,k}(\xi) \neq 0$.

Figures (7)

  • Figure 1: MISE of Liebscher's estimator $\widehat{g}_{n,h,a}$
  • Figure 2: MISE of Liebscher's estimator $\widehat{g}_{n,h,a}$ as a function of the sample size $n$ and the tuning parameter $h$. $a$ is chosen as the best tuning parameter for each pair $(n, h)$. All axes are in log scale; labels are given in the form 1.0e+01 with the meaning $1.0 \times 10^{01}$.
  • Figure 3: Heatmap of the optimal tuning parameter ${ a_{\textnormal{opt}} }$ as a function of the sample size $n$ and the tuning parameter $h$. The red curve represents the best $h$ as a function of $n$. All axes are in log scale; labels are given in the form 1.0e+01 with the meaning $1.0 \times 10^{01}$.
  • Figure 4: Decomposition of the MISE of Liebscher's estimator $\widehat{g}_{n,h,a}$ in terms of bias and variance, as a function of the tuning parameter $h$ for $n = 5000$ and $a = 1$.
  • Figure 5: MISE of Liebscher's estimator $\widehat{g}_{n,h,a}$ as a function of the tuning parameters $a$ and $h$, for a sample size of $5000$ using the Gaussian generator and for different dimensions.
  • ...and 2 more figures

Theorems & Definitions (16)

  • lemma 1
  • lemma 2
  • lemma 3
  • proof
  • lemma 4
  • proof
  • lemma 5
  • proof
  • lemma 6
  • proof
  • ...and 6 more