Permutation Variation and Alternative Hyper-Sphere Decomposition

Qingze Li,Jianxin Pan

doi:10.3390/math10040562

Abstract

Current covariance modeling methods work well in longitudinal data analysis. In the analysis of data with no nature order, a common covariance modeling method would be inadequate. In this paper, a study is implemented to investigate the effects of permutations of data on the estimation of covariance matrix Σ. Based on the Hyper-sphere decomposition method (HPC), this study suggests that the change of data’s permutation breaks the consistency of covariance estimation. An alternative Hyper-sphere decomposition method with permutation invariant is introduced later in this paper. The alternative method’s consistency and asymptotic normality are studied when the observations follow a normal distribution. These results are tested using some example studies. Furthermore, a real data analysis is conducted for illustration purposes.

Highlights

The covariance matrix is a simple and popular method to describe the variation and correlation between random variables in multivariate statistics
Even worse, when using γ1/50 to create an estimate R1/50, we find R1/50 is a poor estimator of R1/50 with a large sum of the absolute difference (SAD), the Frobenius norm
We addressed the permutation variation of Hyper-sphere decomposition method (HPC) through its geometrical interpretations

Summary

Introduction

The covariance matrix is a simple and popular method to describe the variation and correlation between random variables in multivariate statistics. A valid covariance matrix must be symmetric and semi-positive definite. Like social sciences, finance, economics and geology, people are more interested in studying the covariance matrices than mean models. An appropriate working covariance matrix could increase the estimator’s efficiency. Pourahmadi (2013) [1] suggests a good estimate of the covariance matrix can lead to an accurate statistical inference and test results.

Objectives

Methods

Conclusion