### 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 |
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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 - Jan 1989 |

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### Keywords

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

### 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