Parameter-free testing of the shape of a probability distribution

Area of Science:

• Statistics
• Probability Theory
• Data Analysis

Background:

• The Kolmogorov-Smirnov test assesses data consistency with probability distributions.
• Estimating unknown distribution parameters complicates standard Kolmogorov-Smirnov tests.
• Existing methods are often distribution-specific and parameter-dependent.

Purpose of the Study:

• To develop a universal method for testing data consistency with a given distribution's functional form, irrespective of unknown parameters.
• To provide a direct approach applicable to a wide range of probability distributions.
• To enable standard Kolmogorov-Smirnov testing on transformed data.

Main Methods:

• A data transformation technique is employed to create a parameter-free empirical distribution.
• The transformed data is then subjected to the standard Kolmogorov-Smirnov test.
• Analytical results are derived for specific distributions within the considered class.

Main Results:

• A direct method is presented for assessing data consistency with specified distribution functional forms.
• The transformation successfully removes unknown parameters, allowing standard statistical testing.
• Significance levels and test power were evaluated through simulations.

Conclusions:

• The developed method offers a versatile approach to goodness-of-fit testing for various distributions.
• It simplifies hypothesis testing by eliminating the need to estimate distribution parameters.
• The method has potential applications in fields like biological data analysis.