Statistical inference based on band-limited kernels: Rational-infinitely divisible distributions and beyond

Technical Summary: Statistical Inference Based on Band-Limited Kernels for Rational-Infinitely Divisible Distributions

Problem Statement
This paper addresses the problem of statistical inference for mixture distributions of the form $\mu = \omega\mu_d + (1-\omega)\mu_c$, where $\mu_d$ is a discrete component, $\mu_c$ is an absolutely continuous component, and $\omega \in (0,1]$. The study focuses specifically on the class $\mathcal{Q}$ of rational-infinitely divisible distributions (also known as quasi-infinitely divisible distributions). A distribution $\mu$ belongs to $\mathcal{Q}$ if its characteristic function is the ratio of the characteristic functions of two infinitely divisible distributions. While statistical estimation for standard infinitely divisible distributions is well-established, inference for the broader class $\mathcal{Q}$, particularly for mixtures of type (1.1), has received limited attention. The authors aim to develop non-parametric estimators for the mixing weight $\omega$, the discrete component $\mu_d$ (specifically its atoms and probabilities), the continuous density $\mu_c$, and the associated quasi-Lévy measure $\nu$.

Methodology
The proposed framework relies on band-limited kernels, functions whose Fourier transforms have compact support. The methodology leverages two distinct theoretical pillars:

1. Almost Periodic Functions: The characteristic function of the discrete component $\mu_d$ belongs to the class of almost periodic functions ($AP(\mathbb{R}, \mathbb{C})$). By convolving the empirical characteristic function with a band-limited kernel, the authors exploit the Riemann-Lebesgue lemma, which ensures that the convolution of the continuous component vanishes at high frequencies, while the discrete component retains its structure.
2. Inverse Fourier Methods: For estimating the quasi-Lévy measure, the authors utilize the Lévy-Khintchine representation adapted for class $\mathcal{Q}$, which involves a signed (quasi) Lévy measure. They employ an inverse Fourier approach based on the second derivative of the characteristic exponent.

The estimation procedure proceeds in two main stages:

• Component Separation: The authors estimate the discrete atoms $x_j$ and probabilities $p_j$ by solving optimization problems involving the modulus and argument of the convolved empirical characteristic function over specific frequency intervals. The continuous density is then recovered by subtracting the estimated discrete component from the total empirical characteristic function and applying an inverse Fourier transform with a truncation parameter.
• Quasi-Lévy Measure Recovery: The quasi-Lévy measure $\nu$ is estimated by inverting the second derivative of the estimated characteristic exponent, smoothed by a band-limited kernel to handle noise and restrict the domain.

Key Contributions

• Novel Estimation Framework: The paper introduces a non-parametric estimation approach for mixtures in class $\mathcal{Q}$ using band-limited kernels, a technique previously studied in signal processing (e.g., Whittaker) but not extensively applied to this specific statistical problem.
• Dual Estimation Targets: The work provides simultaneous solutions for:
  1. Estimating the mixing parameters ($\omega$), the discrete support ($x_j$), and the discrete probabilities ($p_j$), as well as the continuous density $g_{ac}$.
  2. Estimating the quasi-Lévy measure $\nu$ and its discrete component $\nu_d$, utilizing the relationship between the characteristic triplet and the mixture structure.
• Theoretical Convergence Rates: The authors establish theoretical convergence rates for the proposed estimators under mild assumptions (e.g., separation of atoms, boundedness of the continuous density and its derivative).
  • For the discrete parameters and mixing weight, the estimators achieve polynomial convergence rates, which become almost parametric (up to a logarithmic factor) under specific choices of tuning parameters.
  • For the continuous density, the rates depend on the smoothness of the density (ordinary smooth vs. supersmooth), achieving polynomial rates for ordinary smooth densities and almost parametric rates for supersmooth densities.
  • For the quasi-Lévy measure, convergence rates are derived in the operator norm of the dual Sobolev space $H^{-1}(\mathbb{R})$, yielding polynomial rates.

Results
The theoretical findings are supported by numerical experiments using a mixture of a Poisson distribution (discrete) and an Exponential distribution (continuous).

• Discrete Component: The simulations demonstrate that the proposed estimators successfully identify the atoms and probabilities of the discrete component, with estimation errors decreasing as the sample size increases.
• Continuous Component: The non-parametric estimates of the continuous density closely match the true density, validating the deconvolution approach.
• Quasi-Lévy Measure: The paper verifies that under specific parameter constraints (ensuring the mixture belongs to $\mathcal{Q}$), the estimator for the Fourier transform of the weighted quasi-Lévy measure converges to the true function as the sample size grows.

Significance and Claims
The authors claim that this work fills a gap in the literature regarding statistical inference for rational-infinitely divisible distributions, a class that is significantly larger than the class of standard infinitely divisible distributions. The significance of the paper lies in:

• Providing a rigorous theoretical foundation for estimating mixtures where the continuous component may not be infinitely divisible, provided the overall mixture is rational-infinitely divisible.
• Demonstrating that the use of band-limited kernels allows for the separation of discrete and continuous components without requiring strong parametric assumptions on the continuous part.
• Offering a uniform upper bound for convergence rates across the class of rational-infinitely divisible distributions, addressing the difficulty of analyzing asymptotic properties of specific characteristic function ratios that plagues other methods.

The paper concludes that the proposed estimators are theoretically sound, achieving polynomial convergence rates, and numerically effective, as evidenced by the simulation studies.