### Abstract

k-Principal points of a random variable are k points that minimize the mean squared distance (MSD) between the random variable and the nearest of the k points. This paper focuses on finding optimal estimators of principal points in terms of the expected mean squared distance (EMSD) between the random variable and the nearest principal point estimator. These estimators are compared with nonparametric and maximum likelihood estimators. It turns out that a minimum EMSD estimator of k-principal points of univariate normal distributions is determined by the k-principal points of the t-distribution with n+. 1 degrees of freedom, where n is the sample size. Extensions of the results to location-scale families, multivariate distributions, and principal surfaces are also discussed.

Original language | English |
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Pages (from-to) | 102-122 |

Number of pages | 21 |

Journal | Journal of Statistical Planning and Inference |

Volume | 167 |

DOIs | |

Publication status | Published - 2015 Dec 1 |

### Fingerprint

### Keywords

- Elliptical distributions
- K-means clustering
- Location-scale family
- Normal distribution
- Principal curves and surfaces
- Self-consistency
- T-distribution

### ASJC Scopus subject areas

- Statistics, Probability and Uncertainty
- Applied Mathematics
- Statistics and Probability

### Cite this

*Journal of Statistical Planning and Inference*,

*167*, 102-122. https://doi.org/10.1016/j.jspi.2015.05.005

**Optimal estimators of principal points for minimizing expected mean squared distance.** / Matsuura, Shun; Kurata, Hiroshi; Tarpey, Thaddeus.

Research output: Contribution to journal › Article

*Journal of Statistical Planning and Inference*, vol. 167, pp. 102-122. https://doi.org/10.1016/j.jspi.2015.05.005

}

TY - JOUR

T1 - Optimal estimators of principal points for minimizing expected mean squared distance

AU - Matsuura, Shun

AU - Kurata, Hiroshi

AU - Tarpey, Thaddeus

PY - 2015/12/1

Y1 - 2015/12/1

N2 - k-Principal points of a random variable are k points that minimize the mean squared distance (MSD) between the random variable and the nearest of the k points. This paper focuses on finding optimal estimators of principal points in terms of the expected mean squared distance (EMSD) between the random variable and the nearest principal point estimator. These estimators are compared with nonparametric and maximum likelihood estimators. It turns out that a minimum EMSD estimator of k-principal points of univariate normal distributions is determined by the k-principal points of the t-distribution with n+. 1 degrees of freedom, where n is the sample size. Extensions of the results to location-scale families, multivariate distributions, and principal surfaces are also discussed.

AB - k-Principal points of a random variable are k points that minimize the mean squared distance (MSD) between the random variable and the nearest of the k points. This paper focuses on finding optimal estimators of principal points in terms of the expected mean squared distance (EMSD) between the random variable and the nearest principal point estimator. These estimators are compared with nonparametric and maximum likelihood estimators. It turns out that a minimum EMSD estimator of k-principal points of univariate normal distributions is determined by the k-principal points of the t-distribution with n+. 1 degrees of freedom, where n is the sample size. Extensions of the results to location-scale families, multivariate distributions, and principal surfaces are also discussed.

KW - Elliptical distributions

KW - K-means clustering

KW - Location-scale family

KW - Normal distribution

KW - Principal curves and surfaces

KW - Self-consistency

KW - T-distribution

UR - http://www.scopus.com/inward/record.url?scp=84945438926&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84945438926&partnerID=8YFLogxK

U2 - 10.1016/j.jspi.2015.05.005

DO - 10.1016/j.jspi.2015.05.005

M3 - Article

AN - SCOPUS:84945438926

VL - 167

SP - 102

EP - 122

JO - Journal of Statistical Planning and Inference

JF - Journal of Statistical Planning and Inference

SN - 0378-3758

ER -