Matrix Healy Plot: A Practical Tool for Visual Assessment of Matrix-Variate Normality

TL;DR

The paper introduces the Matrix Healy (MHealy) plot, a new graphical tool for visually assessing matrix-variate normality that overcomes limitations of existing methods by utilizing matrix-based Mahalanobis distances.

Contribution

It proposes the MHealy plot, extending the Healy plot to matrix data, providing a more reliable and applicable visual assessment of matrix-normality without sample size restrictions.

Findings

MHealy plot outperforms DD plot in small sample scenarios.

It provides a more accurate visual assessment of matrix-normality.

Empirical results confirm its effectiveness and practicality.

Abstract

Matrix-valued data, where each observation is represented as a matrix, frequently arises in various scientific disciplines. Modeling such data often relies on matrix-variate normal distributions, making matrix-variate normality testing crucial for valid statistical inference. Recently, the Distance-Distance (DD) plot has been introduced as a graphical tool for visually assessing matrix-variate normality. However, the Mahalanobis squared distances (MSD) used in the DD plot require vectorizing matrix observations, restricting its applicability to cases where the dimension of the vectorized data does not exceed the sample size. To address this limitation, we propose a novel graphical method called the Matrix Healy (MHealy) plot, an extension of the Healy plot for vector-valued data. This new plot is based on more accurate matrix-based MSD that leverages the inherent structure of matrix data. Consequently, it offers a more reliable visual assessment. Importantly, the MHealy plot eliminates the sample size restriction of the DD plot and hence more applicable to matrix-valued data. Empirical results demonstrate its effectiveness and practicality compared to the DD plot across various scenarios, particularly in cases where the DD plot is not available due to limited sample sizes.