An one-parameter compounding discrete distribution

Area of Science:

• Statistics
• Probability Theory
• Mathematical Modeling

Background:

• Discrete distributions are fundamental in statistical modeling.
• Existing distributions may not capture all complex count data patterns.
• The Poisson and xgamma distributions are well-established but have limitations.

Purpose of the Study:

• To propose a new one-parameter discrete distribution by compounding Poisson and xgamma distributions.
• To investigate the statistical properties of this novel distribution.
• To introduce count regression and integer-valued autoregressive models based on the new distribution.

Main Methods:

• Compounding the Poisson and xgamma distributions.
• Derivation of statistical properties: moments, probability generating functions, and moment generating functions.
• Parameter estimation using Maximum Likelihood Estimation (MLE) and Method of Moments (MOM).
• Development of count regression and integer-valued autoregressive (IVAR) models.

Main Results:

• A new one-parameter discrete distribution was successfully derived.
• Key statistical properties, including generating functions, were determined.
• The proposed distribution was shown to be estimable using MLE and MOM.
• Count regression and IVAR models were formulated using the new distribution.

Conclusions:

• The new compound discrete distribution offers a flexible alternative for modeling count data.
• The derived statistical properties and estimation methods provide a foundation for its application.
• The introduced count regression and IVAR models extend its utility to time series and regression analyses.