Computing the real solutions of Fleishman's equations for simulating non-normal data.

Area of Science:

• Statistics
• Computational Statistics

Background:

• Fleishman's power method is widely used for simulating non-normal data with specified skewness and kurtosis.
• Users often overlook that Fleishman's equations can have multiple solutions for given moments.
• A lack of methods for exploring these multiple solutions leads to reliance on single, potentially unrepresentative, solutions.

Purpose of the Study:

• To develop novel methods for identifying all real-valued solutions to Fleishman's equations.
• To characterize the differences between these solutions, particularly concerning higher-order moments.
• To investigate the impact of multiple solutions on non-normal data simulation and statistical analysis.

Main Methods:

• Development of new algorithms to find all real-valued solutions of Fleishman's nonlinear equations.
• Theoretical analysis to understand the properties and distinctions of these solutions.
• Simulation studies to demonstrate the effects of different solutions on data distribution and sampling properties.

Main Results:

• Fleishman's equations typically possess multiple real-valued solutions for common skewness and kurtosis combinations.
• These distinct solutions often exhibit significant differences in higher-order moments.
• The choice of solution can markedly alter the shape of simulated non-normal distributions and affect statistical inference.

Conclusions:

• The existence of multiple solutions in Fleishman's method is a critical, often unacknowledged, factor.
• Novel methods enable comprehensive exploration of these solutions.
• Understanding and reporting the chosen solution are essential for accurate non-normal data simulation and robust statistical practices.