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Two simple approximations to the distributions of quadratic forms
1University of Notre Dame, Notre Dame, Indiana 46556, USA. kyuan@nd.edu
The British Journal of Mathematical and Statistical Psychology
|October 2, 2009
Summary
This study evaluates approximations for a specific statistical distribution (T), crucial for model evaluation. Findings show adjusted distributions are accurate, but caution is advised when comparing rescaled statistics to chi-squared distributions.
Area of Science:
- Statistics
- Statistical modeling
Background:
- Many statistical test statistics approximate quadratic forms of normal variables.
- These are often represented as T = Σ(i=1 to d) λᵢzᵢ², where zᵢ are independent N(0,1) variables.
Purpose of the Study:
- To systematically assess the accuracy of two common approximations for the distribution of T.
- To examine the influence of λᵢ coefficients and degrees of freedom (d) on approximation quality.
Main Methods:
- Analytical methods.
- Monte Carlo simulations.
Main Results:
- The adjusted distribution for T demonstrates high accuracy, comparable to the exact distribution.
- The rescaled statistic T(R) = dT / (Σ(i=1 to d) λᵢ) is adequate for practical inference when λᵢ variation is low.
- Comparing T(R) against chi-squared distribution inflates Type I errors with large differences in λᵢ and large d.
Conclusions:
- Adjusted distributions offer a reliable alternative to exact distributions for T.
- Careful consideration of λᵢ and d is necessary when using rescaled statistics for model inference to avoid inflated Type I errors.
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