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Simulating Univariate and Multivariate Nonnormal Distributions through the Method of Percentiles
Jennifer Koran1, Todd C Headrick1, Tzu Chun Kuo1
1a Section on Statistics and Measurement, Southern Illinois University , Carbondale.
Multivariate Behavioral Research
|November 27, 2015
Summary
This study introduces a percentile-based power method for statistical modeling, offering a superior alternative to traditional moment-based estimators when data is limited. This approach simplifies calculations and enhances accuracy in simulations and distribution fitting.
Area of Science:
- Statistics
- Computational Statistics
- Data Science
Background:
- Traditional statistical methods often rely on moment-based estimators (skewness, kurtosis) which may be unknown or difficult to compute.
- Limited data availability, such as only having percentile information, poses challenges for conventional distribution fitting and simulation.
Purpose of the Study:
- To develop a novel power method polynomial transformation utilizing percentiles for Monte Carlo simulations and distribution fitting.
- To provide a method that functions effectively even when conventional moment estimators are unknown or data is unavailable.
Main Methods:
- Derivation of a standard normal-based power method polynomial transformation using the method of percentiles.
- Development of a procedure for simulating power method distributions with specified statistical properties (median, inter-decile range, skew, kurtosis, correlations).
- Modification of the percentile power method for generating non-normal distributions with specified Pearson correlations.
Main Results:
- The percentile-based power method provides closed-form solutions for polynomial coefficients, eliminating the need for numerical equation solving.
- Monte Carlo simulations demonstrate that percentile-based estimators exhibit substantially lower relative bias compared to conventional product-moment estimators.
- The method successfully generates non-normal distributions with specified Pearson correlations and is illustrated using educational assessment data.
Conclusions:
- The percentile power method offers a robust and computationally efficient alternative for statistical modeling, particularly in scenarios with limited data or unknown moments.
- This method enhances the accuracy and applicability of Monte Carlo simulations and distribution fitting.
- The technique is broadly applicable, as demonstrated by its use with real-world educational assessment statistics.
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