Discriminating between Light- and Heavy-Tailed Distributions with Limit Theorem.

Area of Science:

• Statistics
• Data Analysis
• Probability Theory

Background:

• Distinguishing between light- and heavy-tailed probability distributions is crucial for accurate data analysis.
• Standard statistical tests often struggle to differentiate between similar distribution types.
• Identifying the correct probability law is essential for modeling complex phenomena.

Purpose of the Study:

• To propose a novel algorithm for distinguishing between light- and heavy-tailed probability laws.
• To provide a visual and easily implementable method for classifying distributions.
• To address limitations of existing tests in differentiating subtle distribution differences.

Main Methods:

• Developed a visual algorithm based on examining the rate of convergence of random datasets.
• The algorithm checks if the underlying law falls within the domain of attraction of Gaussian or non-Gaussian stable distributions.
• Applied the method to differentiate between stable, non-stable, Student's t, and tempered stable distributions.

Main Results:

• The proposed algorithm successfully discriminates between stable and various non-stable distributions.
• The method differentiates distributions that appear identical using the standard Kolmogorov-Smirnov test.
• Successfully distinguished between stable and Student's t laws, and between stable and tempered stable laws.

Conclusions:

• The algorithm offers a robust and practical solution for identifying underlying probability laws in datasets.
• This method enhances the ability to accurately model and interpret complex data, particularly in fields like plasma physics.
• The approach provides a valuable tool for researchers dealing with challenging distribution classifications.