### Abstract

A survey is given of known proofs of the antitonicity of the inverse matrix function for positive definite matrices w.r.t. the Lowner partial ordering, and of the corresponding result for the Moore-Penrose inverse of nonnegative definite matrices [the theorem of Milliken and Akdeniz (1977)]. A short new proof of the latter result is obtained by employing an extremal representation of a nonnegative definite quadratic form. Another proof of this result involving Schur complements is also given, and is seen to be extendable to the case of symmetric (not necessarily nonnegative definite) matrices. A geometrical interpretation of Milliken and Akdeniz’s theorem is presented. As an application, the relationship between the concepts of greater (maximum) concentration and smaller (minimum) dispersion is considered for a pair (class) of vector-valued statistics with possibly degenerate distributions.

Original language | English |
---|---|

Pages (from-to) | 4471-4489 |

Number of pages | 19 |

Journal | Communications in Statistics - Theory and Methods |

Volume | 18 |

Issue number | 12 |

DOIs | |

Publication status | Published - 1989 |

Externally published | Yes |

### Fingerprint

### Keywords

- comparison of vector-valued statistics
- concentration
- concentration ellipsoid
- ellipsoidal cylinder
- extremal representation of quadratic form
- inertia
- linear support
- maximum
- minimum dispersion
- polar set
- Schur complement

### ASJC Scopus subject areas

- Statistics and Probability

### Cite this

*Communications in Statistics - Theory and Methods*,

*18*(12), 4471-4489. https://doi.org/10.1080/03610928908830167

**Some Further Aspects of the Owner-Ordering Antitonicity of the Moore-Penrose Inverse.** / Nordstrom, Kenneth.

Research output: Contribution to journal › Article

*Communications in Statistics - Theory and Methods*, vol. 18, no. 12, pp. 4471-4489. https://doi.org/10.1080/03610928908830167

}

TY - JOUR

T1 - Some Further Aspects of the Owner-Ordering Antitonicity of the Moore-Penrose Inverse

AU - Nordstrom, Kenneth

PY - 1989

Y1 - 1989

N2 - A survey is given of known proofs of the antitonicity of the inverse matrix function for positive definite matrices w.r.t. the Lowner partial ordering, and of the corresponding result for the Moore-Penrose inverse of nonnegative definite matrices [the theorem of Milliken and Akdeniz (1977)]. A short new proof of the latter result is obtained by employing an extremal representation of a nonnegative definite quadratic form. Another proof of this result involving Schur complements is also given, and is seen to be extendable to the case of symmetric (not necessarily nonnegative definite) matrices. A geometrical interpretation of Milliken and Akdeniz’s theorem is presented. As an application, the relationship between the concepts of greater (maximum) concentration and smaller (minimum) dispersion is considered for a pair (class) of vector-valued statistics with possibly degenerate distributions.

AB - A survey is given of known proofs of the antitonicity of the inverse matrix function for positive definite matrices w.r.t. the Lowner partial ordering, and of the corresponding result for the Moore-Penrose inverse of nonnegative definite matrices [the theorem of Milliken and Akdeniz (1977)]. A short new proof of the latter result is obtained by employing an extremal representation of a nonnegative definite quadratic form. Another proof of this result involving Schur complements is also given, and is seen to be extendable to the case of symmetric (not necessarily nonnegative definite) matrices. A geometrical interpretation of Milliken and Akdeniz’s theorem is presented. As an application, the relationship between the concepts of greater (maximum) concentration and smaller (minimum) dispersion is considered for a pair (class) of vector-valued statistics with possibly degenerate distributions.

KW - comparison of vector-valued statistics

KW - concentration

KW - concentration ellipsoid

KW - ellipsoidal cylinder

KW - extremal representation of quadratic form

KW - inertia

KW - linear support

KW - maximum

KW - minimum dispersion

KW - polar set

KW - Schur complement

UR - http://www.scopus.com/inward/record.url?scp=78651463332&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78651463332&partnerID=8YFLogxK

U2 - 10.1080/03610928908830167

DO - 10.1080/03610928908830167

M3 - Article

VL - 18

SP - 4471

EP - 4489

JO - Communications in Statistics - Theory and Methods

JF - Communications in Statistics - Theory and Methods

SN - 0361-0926

IS - 12

ER -