### Abstract

A set of k points that optimally summarize a distribution is called a set of k-principal points, which is a generalization of the mean from one point to multiple points and is useful especially for multivariate distributions. This paper discusses the estimation of principal points of multivariate distributions. First, an optimal estimator of principal points is derived for multivariate distributions of location-scale families. In particular, an optimal principal points estimator of a multivariate normal distribution is shown to be obtained by using principal points of a scaled multivariate t-distribution. We also study the case of multivariate location-scale-rotation families. Numerical examples are presented to compare the optimal estimators with maximum likelihood estimators.

Original language | English |
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Pages (from-to) | 1-15 |

Number of pages | 15 |

Journal | Statistical Papers |

DOIs | |

Publication status | Accepted/In press - 2018 Mar 20 |

### Fingerprint

### Keywords

- Location-scale family
- Location-scale-rotation family
- Multivariate normal distribution
- Multivariate t-distribution
- Principal points

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

**Optimal principal points estimators of multivariate distributions of location-scale and location-scale-rotation families.** / Matsuura, Shun; Tarpey, Thaddeus.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Optimal principal points estimators of multivariate distributions of location-scale and location-scale-rotation families

AU - Matsuura, Shun

AU - Tarpey, Thaddeus

PY - 2018/3/20

Y1 - 2018/3/20

N2 - A set of k points that optimally summarize a distribution is called a set of k-principal points, which is a generalization of the mean from one point to multiple points and is useful especially for multivariate distributions. This paper discusses the estimation of principal points of multivariate distributions. First, an optimal estimator of principal points is derived for multivariate distributions of location-scale families. In particular, an optimal principal points estimator of a multivariate normal distribution is shown to be obtained by using principal points of a scaled multivariate t-distribution. We also study the case of multivariate location-scale-rotation families. Numerical examples are presented to compare the optimal estimators with maximum likelihood estimators.

AB - A set of k points that optimally summarize a distribution is called a set of k-principal points, which is a generalization of the mean from one point to multiple points and is useful especially for multivariate distributions. This paper discusses the estimation of principal points of multivariate distributions. First, an optimal estimator of principal points is derived for multivariate distributions of location-scale families. In particular, an optimal principal points estimator of a multivariate normal distribution is shown to be obtained by using principal points of a scaled multivariate t-distribution. We also study the case of multivariate location-scale-rotation families. Numerical examples are presented to compare the optimal estimators with maximum likelihood estimators.

KW - Location-scale family

KW - Location-scale-rotation family

KW - Multivariate normal distribution

KW - Multivariate t-distribution

KW - Principal points

UR - http://www.scopus.com/inward/record.url?scp=85046029562&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85046029562&partnerID=8YFLogxK

U2 - 10.1007/s00362-018-0995-z

DO - 10.1007/s00362-018-0995-z

M3 - Article

AN - SCOPUS:85046029562

SP - 1

EP - 15

JO - Statistical Papers

JF - Statistical Papers

SN - 0932-5026

ER -