Exploration of distributions of ratio of partial sum of sample eigenvalues when all population eigenvalues are the same

Moonseong Heo

Research output: Contribution to journalArticle


This paper explores empirically the first two moments of ratio of the partial sum of the first two sample eigenvalues to the sum of all eigenvalues when the population eigenvalues of a covariance matrix are all the same. Estimation of the first two moments can be practically crucial in assessing non-randomness of observed patterns on planar graphical displays based on lower rank approximations of data matrices. For derivation of the moments, exact and large sample asymptotic distributions of the sample ratios are reviewed but neither can be applicable to derivation of the moments. Therefore, I rely on simulations, where data matrices X with order nxm element-wise independent normal distribution with mean 0 and variance σ2 are assumed, that is, X ∼ N(0, σ2Inm), and then derive formulas for estimates of means and standard deviations of the sample ratios within a range of order of the data matrix. The derivations are based on the biplot graphical diagnostic methods proposed by Bradu and Gabriel (1976).

Original languageEnglish (US)
Pages (from-to)420-427
Number of pages8
JournalJournal of Modern Applied Statistical Methods
Issue number2
Publication statusPublished - Jan 1 2002



  • Bias
  • Biplot
  • Eigenvalues
  • Multivariate Gaussian; Schönemann-Lingoes- Gower coefficient

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Cite this