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Parameter estimation for stable distributions and their mixture.

Omar Hajjaji1, Solym Mawaki Manou-Abi1,2, Yousri Slaoui1

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This study introduces novel methods for estimating parameters in alpha-stable distributions and their mixtures, crucial for modeling heavy-tailed data. The techniques demonstrate accurate estimation for complex datasets, including COVID-19 and carcinogen metabolite data.

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62-0862C0562G3062P1097K80EM algorithmGibbs sampling algorithmMetropolis–Hastings algorithmNewton–Raphson algorithmStable distributionbisection algorithmmixture modelparametric estimation

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Area of Science:

  • Statistics
  • Probability Theory
  • Data Modeling

Background:

  • Alpha-stable distributions are essential for modeling data with heavy tails and asymmetry.
  • Parameter estimation for these distributions and their mixtures presents significant statistical challenges.
  • Existing methods may lack efficiency or applicability to complex mixture scenarios.

Purpose of the Study:

  • To develop and evaluate novel methods for estimating parameters of univariate alpha-stable distributions and their mixtures.
  • To provide efficient and accurate tools for analyzing complex, non-Gaussian data.
  • To apply these methods to real-world datasets in epidemiology and toxicology.

Main Methods:

  • Gaussian kernel density estimation with characteristic function-based approach.
  • Maximum Likelihood estimation utilizing the False Position algorithm.
  • Modified Expectation-Maximization (EM) and Bayesian methods for mixture models.
  • Simulation studies for performance evaluation.

Main Results:

  • Proposed methods accurately estimate parameters for univariate alpha-stable distributions and their mixtures.
  • The techniques are robust and perform well on simulated and real-world data.
  • Successful application to COVID-19 replicate estimation and N-acetyltransferase activity data.

Conclusions:

  • The developed methods offer valuable tools for parameter estimation in alpha-stable mixture models.
  • The study validates the accuracy and applicability of the proposed techniques.
  • Future work includes generalization to multiple mixtures and development of R/Python packages.