Permutation Variation and Alternative Hyper-Sphere Decomposition

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.