New technique for parameter estimation and improved fits to experimental data for a set of compound Poisson distributions

TL;DR

This paper introduces a power-spectrum–based parameter estimation technique for a set of compound Poisson distributions (notably Neyman Type A, Poisson-binomial, and Poisson-Pascal, with geometric Poisson as a related case). By analyzing the power spectrum of the pmf via the characteristic function, the author identifies periodic modal structure and leverages peak locations to estimate distribution parameters, often yielding substantially better fits than traditional method-of-moments approaches across multiple published datasets. The method is demonstrated on Beall, Beall–Rescia, McGuire–Brindley–Bancroft, and Fracker–Brischle data, with improvements attributed to more accurate peak identification and aliasing-aware adjustments. The results suggest that Fourier-based parameter estimation can be a valuable addition to the statistical toolbox for compound Poisson models, though it requires careful peak-detection and acknowledgment of aliasing effects. Overall, the power-spectrum approach provides competitive, sometimes superior, fits and broadens the set of tools available for modeling count data with clustered or overdispersed structure.

Abstract

Compound Poisson distributions have been employed by many authors to fit experimental data, typically via the method of moments or maximum likelihood estimation. We propose a new technique and apply it to several sets of published data. It yields better fits than those obtained by the original authors for a set of widely employed compound Poisson distributions (in some cases, significantly better). The technique employs the power spectrum (the absolute square of the characteristic function). The new idea is suggested as a useful addition to the tools for parameter estimation of compound Poisson distributions.

Figures (18)

• Figure 1: Plots of the scaled pmf $h_n(=P_n/P_0)$ of the Neyman Type A (dash), Poisson-binomial (dots) and Poisson-Pascal (dotdash) distributions, with the respective parameter values $(\lambda,\phi)=(5,25)$, $(\lambda,k,p)=(5,50,0.5)$ and $(\Lambda,k,P)=(5,50,0.5)$. Note that $h_0=1$ in all cases, although this is difficult to observe in the plot.
• Figure 2: Plots of the scaled pmf $h_n(=P_n/P_0)$ of the Neyman Type A (top), Poisson-binomial (middle) and Poisson-Pascal (bottom) distributions, with the respective parameter values $(\lambda,\phi)=(10,100)$, $(\lambda,k,p)=(10,1000,0.1)$ and $(\Lambda,k,P)=(10,200,0.5)$. Similar to Fig. \ref{['fig:neyman_binom_pascal_pmf_lam5']}, but with different parameter values.
• Figure 3: Plots of the power spectra of Neyman Type A (top), Poisson-binomial (middle) and (c) Poisson-Pascal (bottom). The respective parameter values are stated in the text.
• Figure 4: Graph of the power spectrum of the data in Beall Table II column 1 (upper panel), with a magnified view in the lower panel.
• Figure 5: Plot of the data in Beall Table II columns 1 (top) and 2 (bottom). Here $c_n$ denotes the count in each histogram bin. In each panel, the points indicate the data and the curves are the fits using the method of moments (dash) and power spectrum (dotdash).