New discrete heavy tailed distributions as models for insurance data

Area of Science:

• Statistics
• Probability Theory
• Data Science

Background:

• Discrete heavy tailed distributions are scarce in statistical literature.
• Real-world data, such as insurance claims, often exhibit discrete and heavy tailed properties.
• Existing continuous models may not adequately represent such data.

Purpose of the Study:

• To address the scarcity of discrete heavy tailed distributions.
• To propose and characterize new discrete heavy tailed distributions.
• To compare the performance of discrete versus continuous models for heavy tailed data.

Main Methods:

• Discussed thirteen known discrete heavy tailed distributions.
• Proposed nine novel discrete heavy tailed distributions.
• Derived key statistical properties including PMF, CDF, hazard rates, moments, entropy, and quantile functions.
• Analyzed tail behavior and asymmetry for distribution comparison.
• Utilized probability plots for model fit assessment on three datasets.
• Conducted a simulation study for maximum likelihood estimator performance.

Main Results:

• Introduced nine new discrete heavy tailed distributions with derived properties.
• Demonstrated superior data fitting of discrete heavy tailed distributions over continuous ones.
• Provided a comprehensive comparison of known and new distributions based on tail behavior and asymmetry.
• Assessed the finite sample performance of maximum likelihood estimators.

Conclusions:

• The proposed discrete heavy tailed distributions offer valuable additions to statistical modeling.
• Discrete heavy tailed distributions provide better fits for discrete, heavy tailed data than continuous counterparts.
• The study enhances the toolkit for analyzing heavy tailed data in various scientific fields.