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A New F Approximation for the Pillai-Bartlett Trace under H0.
1Department of Biostatistics CB #7400, University of North Carolina, Chapel Hill, NC 27599-7400 ( Keith_Muller@UNC.EDU ).
Journal of Computational and Graphical Statistics : a Joint Publication of American Statistical Association, Institute of Mathematical Statistics, Interface Foundation of North America
|July 30, 2013
Summary
Two new approximations for the Pillai-Bartlett trace statistic offer improved accuracy. The second new approximation provides the best balance of logical properties and numerical precision for statistical analysis.
Area of Science:
- Multivariate statistical analysis
- Statistical inference
Background:
- The Pillai-Bartlett trace statistic is crucial for hypothesis testing in multivariate analysis.
- Existing approximations, particularly the first Pillai approximation, exhibit limitations in accuracy with moderate sample sizes.
Purpose of the Study:
- To develop and evaluate new approximations for the Pillai-Bartlett trace statistic under null hypotheses.
- To enhance the numerical accuracy and logical properties of these approximations.
Main Methods:
- Developing two new approximations based on matching moments of a beta distribution.
- Comparing the accuracy of new approximations against Pillai's existing approximations using the Pearson system.
Main Results:
- The first Pillai approximation, while common, shows insufficient accuracy.
- Pillai's second approximation offers greater accuracy by matching moment ratios.
- The two new approximations match two moments of a beta distribution, achieving high accuracy.
- The second new approximation demonstrates the optimal blend of theoretical soundness and numerical precision.
Conclusions:
- The second new approximation for the Pillai-Bartlett trace statistic is recommended for its superior performance.
- These advancements improve the reliability of statistical inference in multivariate settings.
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