### Abstract

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 σ^{2} are assumed, that is, X ∼ N(0, σ^{2}I_{nm}), 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 language | English (US) |
---|---|

Pages (from-to) | 420-427 |

Number of pages | 8 |

Journal | Journal of Modern Applied Statistical Methods |

Volume | 1 |

Issue number | 2 |

State | Published - 2002 |

Externally published | Yes |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Statistics, Probability and Uncertainty
- Statistics and Probability

### Cite this

**Exploration of distributions of ratio of partial sum of sample eigenvalues when all population eigenvalues are the same.** / Heo, Moonseong.

Research output: Contribution to journal › Article

*Journal of Modern Applied Statistical Methods*, vol. 1, no. 2, pp. 420-427.

}

TY - JOUR

T1 - Exploration of distributions of ratio of partial sum of sample eigenvalues when all population eigenvalues are the same

AU - Heo, Moonseong

PY - 2002

Y1 - 2002

N2 - 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).

AB - 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).

KW - Bias

KW - Biplot

KW - Eigenvalues

KW - Multivariate Gaussian; Schönemann-Lingoes- Gower coefficient

UR - http://www.scopus.com/inward/record.url?scp=18444412528&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=18444412528&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:18444412528

VL - 1

SP - 420

EP - 427

JO - Journal of Modern Applied Statistical Methods

JF - Journal of Modern Applied Statistical Methods

SN - 1538-9472

IS - 2

ER -