A Simple Simulation Technique for Nonnormal Data with Prespecified Skewness, Kurtosis, and Covariance Matrix

Area of Science:

• Statistics
• Data Simulation
• Quantitative Psychology

Background:

• Nonnormal data poses challenges in statistical analyses, particularly in covariance structure analysis.
• Existing methods like the Vale-Maurelli (VM) transform may not fully capture complex nonnormal dependencies.

Purpose of the Study:

• To introduce and investigate a novel method for generating nonnormal data with controlled skewness, kurtosis, and covariance.
• To compare the properties of the new method against the widely used VM transform.

Main Methods:

• The proposed method utilizes linear combinations of independent generator (IG) variables.
• Data simulation with prespecified univariate skewness and kurtosis, and a given covariance matrix.
• Analytical derivation of asymptotic robustness conditions for the IG distribution.

Main Results:

• The generated data exhibit a non-Gaussian copula, unlike data from the VM transform.
• Empirical results show that popular test statistics in covariance analysis exhibit higher rejection rates for true models under the IG transform.
• This suggests that the IG transform provides a more rigorous test environment for nonnormal data.

Conclusions:

• The independent generator (IG) transform is a valuable tool for simulating nonnormal data with specific characteristics.
• The IG transform can temper overly optimistic evaluations of estimators and fit statistics in covariance structure analysis.
• An R implementation of the IG transform is provided for practical use.