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

Nathaniel E Helwig1,2

  • 1Department of Psychology, University of Minnesota, Minneapolis, Minnesota, USA.

The British Journal of Mathematical and Statistical Psychology
|November 15, 2021
PubMed
Summary
This summary is machine-generated.

Fleishman's power method can yield multiple solutions for simulating non-normal data. This study introduces methods to find all solutions, revealing significant differences in higher-order moments and distribution shapes.

Keywords:
Monte Carlokurtosispower methodsimulationskewness

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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.