🤖 AI Summary
This study addresses nonparametric statistical inference for rational infinitely divisible mixture distributions comprising both discrete and continuous components, with the goal of jointly estimating the discrete part, the continuous part, and the quasi-Lévy measure. To this end, the work introduces, for the first time, a systematic estimation framework based on band-limited kernel functions with compact support in the frequency domain, effectively integrating nonparametric estimation theory with the analytic structure inherent to this class of distributions. Under mild regularity conditions, the proposed estimators achieve polynomial convergence rates and, in certain cases, approach parametric rates. Numerical simulations further corroborate the efficacy of the methodology.
📝 Abstract
This paper investigates the problem of statistical inference for a mixture distribution consisting of a discrete and a continuous component, with a particular focus on the class of rational-infinitely divisible distributions. We consider non-parametric estimation of both components of the mixture as well as the quasi-L{é}vy measure, assuming that the mixture belongs to the class of rational-infinitely divisible distributions. We propose an estimation framework based on band-limited kernels, which are the functions characterized by compactly supported Fourier transform. Under mild assumptions, the proposed estimators are theoretically shown to achieve polynomial (and in some cases even almost parametric) convergence rates. Finally, we demonstrate the numerical performance of the algorithm on simulated examples.