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Statistical Inference for Quasi-Infinitely Divisible Distributions via Fourier Methods

Vladimir Panov, Anton Ryabchenko

TL;DR

The paper develops a Fourier-based statistical framework for quasi-infinitely divisible distributions, extending the Lévy–Khintchine representation to signed measures and enabling semiparametric inference for mixtures of a normal component with an extra QID tail. It proves that, for subfamilies with supersmooth nonparametric parts, key parameters ($\sigma$, $\lambda^*$, $\gamma^*$) converge at polynomial rates, and the nonparametric density component can be estimated with a related polynomial rate. A four-step estimation procedure is proposed for the parametric part, together with kernel-based recovery of the continuous component, and the authors validate the method through numerical studies on two-component normal mixtures, a Bart Simpson-like distribution, and a Student–normal convolution model. The results show favorable performance in large samples and provide a principled alternative to EM in certain QID settings, supporting practical deployment in statistical inference for non-ID models with quasi-Lévy structure.

Abstract

This study focuses on statistical inference for the class of quasi-infinitely divisible (QID) distributions, which was recently introduced by Lindner, Pan and Sato (2018). The paper presents a Fourier approach, based on the analogue of the L{é}vy-Khintchine theorem with a signed spectral measure. We prove that for some subclasses of QID distributions, the considered estimates have polynomial rates of convergence. This is a remarkable fact when compared to the logarithmic convergence rates of similar methods for infinitely divisible distributions, which cannot be improved in general. We demonstrate the numerical performance of the algorithm using simulated examples.

Statistical Inference for Quasi-Infinitely Divisible Distributions via Fourier Methods

TL;DR

The paper develops a Fourier-based statistical framework for quasi-infinitely divisible distributions, extending the Lévy–Khintchine representation to signed measures and enabling semiparametric inference for mixtures of a normal component with an extra QID tail. It proves that, for subfamilies with supersmooth nonparametric parts, key parameters (, , ) converge at polynomial rates, and the nonparametric density component can be estimated with a related polynomial rate. A four-step estimation procedure is proposed for the parametric part, together with kernel-based recovery of the continuous component, and the authors validate the method through numerical studies on two-component normal mixtures, a Bart Simpson-like distribution, and a Student–normal convolution model. The results show favorable performance in large samples and provide a principled alternative to EM in certain QID settings, supporting practical deployment in statistical inference for non-ID models with quasi-Lévy structure.

Abstract

This study focuses on statistical inference for the class of quasi-infinitely divisible (QID) distributions, which was recently introduced by Lindner, Pan and Sato (2018). The paper presents a Fourier approach, based on the analogue of the L{é}vy-Khintchine theorem with a signed spectral measure. We prove that for some subclasses of QID distributions, the considered estimates have polynomial rates of convergence. This is a remarkable fact when compared to the logarithmic convergence rates of similar methods for infinitely divisible distributions, which cannot be improved in general. We demonstrate the numerical performance of the algorithm using simulated examples.

Paper Structure

This paper contains 16 sections, 7 theorems, 72 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Quasi-infinitely divisible distributions
  3. Definition and spectral representation
  4. Examples
  5. Fourier-based estimation procedure
  6. Semiparametric inference for the model \ref{['model1']}
  7. Convergence rates
  8. Numerical study
  9. Two-component normal mixture
  10. Bart Simpson distribution
  11. Another example of model \ref{['model1']}
  12. Proofs
  13. Proof of Theorem \ref{['thm1']}
  14. Proof of Theorem \ref{['thm2']}
  15. Proof of Remark \ref{['remark2']}
  16. ...and 1 more sections

Key Result

Lemma 1

Consider the mixture $\mu = p\mu_1 + (1-p)\mu_2$, where $p \in (1/2, 1)$, $\mu_1$ is a QID distribution with triplet $(\gamma, \sigma^2, \nu)$, and $\mu_2$ is some distribution on $\mathbb{R}.$ Suppose that there exists a finite signed measure $\Lambda$ on $\mathbb{R}$ such that its Fourier transfor where $\phi_1$ and $\phi_2$ are the characteristic functions of $\mu_1$ and $\mu_2$, If this measur

Figures (9)

  • Figure 1: The plot of $\operatorname{Re}(\log(\phi(u)))$ (orange line) and plots of $N=20$ realisations of its estimate $\operatorname{Re}(\log(\phi_n(u)))$ (grey lines).
  • Figure 2: Boxplots for the estimates of $\sigma_1^2$ and $p$ obtained by the EM algorithm (blue boxes) and our approach (green boxes) for the two-component normal mixture model.
  • Figure 3: Plots of the true function $g^{\circ}(x)$ (orange line) and its estimates, obtained by the EM algorithm (blue line) and our inference method (green line) for the two-component normal mixture model.
  • Figure 4: Histogram from the modified Bart Simpson model and the graph of the true density $g(x)$.
  • Figure 5: Boxplots for the estimates of $\sigma_1^2$ and $p$ obtained by the EM algorithm (blue boxes) and our approach (green boxes) for the modified Bart Simpson model.
  • ...and 4 more figures

Theorems & Definitions (18)

  • Definition 1
  • Lemma 1
  • Corollary 1
  • proof
  • Example 1
  • Corollary 2
  • proof
  • Example 2
  • Example 3
  • Lemma 2
  • ...and 8 more