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Poisson XLindley Distribution for Count Data: Statistical and Reliability Properties with Estimation Techniques and

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A novel count distribution, combining Poisson and XLindley distributions, is introduced for modeling skewed and dispersed data. Its statistical properties and parameter estimation methods are explored, demonstrating its flexibility in real-world applications.

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Area of Science:

  • Statistics
  • Probability Theory
  • Mathematical Modeling

Background:

  • Existing count distributions may not adequately capture complex data characteristics.
  • There is a need for flexible discrete distributions to model various data phenomena.
  • Combining established distributions can yield novel models with enhanced properties.

Purpose of the Study:

  • To propose a new one-parameter count distribution by integrating Poisson and XLindley distributions.
  • To investigate the comprehensive statistical and reliability properties of the proposed distribution.
  • To assess the performance of various parameter estimation techniques and demonstrate the model's applicability.

Main Methods:

  • Analytical derivation of statistical properties: order statistics, hazard rates, moments, generating functions, entropy, and dispersion measures.
  • Parameter estimation using six distinct approaches.
  • Monte Carlo simulations to evaluate estimation method behavior.
  • Real-data case studies for practical illustration.

Main Results:

  • The new distribution exhibits a flexible probability mass function suitable for positively skewed, leptokurtic data.
  • It effectively models equi- and over-dispersed data with an increasing hazard rate.
  • Explicit forms for all investigated statistical and reliability properties were derived.
  • Parameter estimation methods showed varying behaviors, analyzed via simulation.

Conclusions:

  • The proposed combined distribution offers a versatile and mathematically tractable tool for count data analysis.
  • It addresses limitations of existing models for skewed, dispersed, and time-to-event data.
  • The study validates its practical utility through real-world applications.