Parameter estimation for stable distributions and their mixture

In this paper, we consider estimating the parameters of univariate α-stable distributions and their mixtures. First, using a Gaussian kernel density distribution estimator, we propose an estimation method based on the characteristic function. The optimal bandwidth parameter was selected using a plug...

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Bibliographic Details
Published in:Journal of applied statistics Vol. 52; no. 8; pp. 1594 - 1627
Main Authors: Hajjaji, Omar, Manou-Abi, Solym Mawaki, Slaoui, Yousri
Format: Journal Article
Language:English
Published: England Taylor & Francis 11.06.2025
Taylor & Francis (Routledge)
Subjects:
62-08
bisection algorithm
EM algorithm
Gibbs sampling algorithm
Mathematics
Metropolis-Hastings algorithm
mixture model
Newton-Raphson algorithm
parametric estimation
Stable distribution
Statistics
62P10
Metropolis–Hastings algorithm
bisection algorithm
62-08
mixture model
parametric estimation
Gibbs sampling algorithm
Stable distribution
Newton–Raphson algorithm
62G30
97K80
62C05
EM algorithm
Mixture model
EM
Parametric estimation
MCMC method
ISSN:0266-4763, 1360-0532
Online Access:Get full text
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Summary:In this paper, we consider estimating the parameters of univariate α-stable distributions and their mixtures. First, using a Gaussian kernel density distribution estimator, we propose an estimation method based on the characteristic function. The optimal bandwidth parameter was selected using a plug-in method. We highlight another estimation procedure for the Maximum Likelihood framework based on the False position algorithm to find a numerical root of the log-likelihood through the score functions. For mixtures of α-stable distributions, the EM algorithm and the Bayesian estimation method have been modified to propose an efficient and valuable tool for parameter estimation. The proposed methods can be generalised to multiple mixtures, although we have limited the mixture study to two components. A simulation study is carried out to evaluate the performance of our methods, which are then applied to real data. Our results appear to accurately estimate mixtures of α-stable distributions. Applications concern the estimation of the number of replicates in the Mayotte COVID-19 dataset and the distribution of the N-acetyltransferase activity of the Bechtel et al. data for a urinary caffeine metabolite implicated in carcinogens. We compare the proposed methods, together with a detailed discussion. We conclude with the limitations of this study, together with other forthcoming work and a future implementation of an R package or Python library for the proposed methods in data modelling.
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ISSN:0266-4763
1360-0532
DOI:10.1080/02664763.2024.2434627