Distinguishing between a power law and a Pareto distribution

Area of Science:

• Statistics
• Probability Theory
• Data Analysis

Background:

• Power law distributions are widely used in various scientific fields.
• Distinguishing between power law and related distributions can be challenging.
• Classical methods may lack sensitivity in differentiating similar models.

Purpose of the Study:

• Introduce the location Pareto distribution as a natural extension of the power law distribution.
• Develop and evaluate a likelihood ratio test for model selection.
• Provide guidance on choosing the appropriate distribution for empirical data.

Main Methods:

• Derivation of the location Pareto distribution.
• Development of a likelihood ratio test statistic.
• Investigation of the statistical properties of the distribution and test.
• Application and validation using real-world datasets.

Main Results:

• The location Pareto distribution offers a flexible alternative to the power law distribution.
• The likelihood ratio test demonstrates high discrimination power.
• The test is particularly effective for data exhibiting linearity on a log-log scale.
• Similar performance of both models for large observation values is explained.

Conclusions:

• The location Pareto distribution is a valuable addition to statistical modeling tools.
• The proposed likelihood ratio test provides a simple and powerful method for model selection.
• The test is recommended for datasets where the complementary cumulative distribution function is linear on a log-log plot.