Exploring Multivariate Statistics: Unveiling the Power of Eigenvalues in Wishart Distribution Analysis

Abstract

This research examines how degrees of freedom and covariance matrix configurations affect Wishart distributed matrix eigenvalue distributions. We use the energy distance metric to compare eigenvalue distributions in Identity, Diagonal, and Structured covariance matrices. Through extensive simulations, we show that degrees of freedom and covariance matrix architectures greatly affect eigenvalue dispersion and energy distance distributions. Statistical models are more reliable with smaller, more stable distributions from higher degrees of freedom. We analyze the eigenvalue distributions of NextEra Energy (NEE), Enphase Energy (ENPH), First Solar (FSLR), SunPower (SPWR), and Brookfield Renewable Partners (BEP) stocks using this analytical methodology. Daily returns and covariance matrices were calculated using daily closing prices from January 1, 2018, to January 1, 2023. Our findings support simulation studies indicating larger degrees of freedom lead to more stable energy distance distributions.The findings of this study are useful in finance, genetics, and environmental studies, where stability and variability of the covariance structure are important. The requirement for higher-dimensional settings and real-world datasets to validate the theoretical framework are our research’s constraints. This research expands Wishart distribution knowledge and provides a solid analytical foundation for data analysis and model fitting in numerous scientific fields.