A weighted negative binomial Lindley distribution with applications to dispersed data.

Area of Science:

• Statistics
• Probability Theory
• Statistical Distributions

Background:

• Discrete distributions are fundamental in statistical modeling.
• Existing models like the negative binomial and Poisson distributions have limitations in capturing complex count data characteristics.
• The discrete Lindley distribution provides a basis for developing more flexible models.

Purpose of the Study:

• Introduce a new discrete distribution with enhanced properties.
• Investigate the mathematical and statistical characteristics of the proposed distribution.
• Evaluate the performance of the new distribution against established models for count data.

Main Methods:

• Developed a new discrete distribution as a weighted version of the two-parameter discrete Lindley distribution.
• Incorporated sub-models such as the negative binomial and size-biased negative binomial distributions.
• Analyzed properties including hazard rate, skewness, dispersion, self-decomposability, and infinite divisibility.
• Employed method of moments and maximum likelihood estimation for parameter estimation.
• Conducted simulation studies to assess estimator performance.
• Validated the distribution using four real-world datasets.

Main Results:

• The proposed distribution exhibits a bathtub-shaped hazard function and can display increasing or decreasing hazard rates.
• It demonstrates properties like positive skewness, symmetry, and the ability to model over- and under-dispersion.
• The distribution is self-decomposable and infinitely divisible, suitable for count data.
• Parameter estimation via method of moments and maximum likelihood is feasible.
• Simulation studies indicate good performance of the estimators.
• The proposed distribution showed superior fit compared to negative binomial, Poisson, and generalized Poisson distributions on real datasets.

Conclusions:

• The newly proposed discrete distribution offers a flexible and robust alternative for modeling various types of count data.
• Its rich mathematical properties and demonstrated performance make it a valuable addition to the statistical toolkit.
• The distribution's ability to outperform standard models highlights its practical utility in applied statistics.