Optimal estimators of principal points for minimizing expected mean squared distance

• Optimal estimators of principal points under expected mean squared distance are derived.
• For univariate normal distributions, optimal estimators are determined by tt-distributions.
• Extensions to location-scale families and multivariate distributions are given.
• An optimal estimator of principal surfaces is also discussed.

kk-Principal points of a random variable are kk points that minimize the mean squared distance (MSD) between the random variable and the nearest of the kk 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 kk-principal points of univariate normal distributions is determined by the kk-principal points of the tt-distribution with n+1n+1 degrees of freedom, where nn is the sample size. Extensions of the results to location-scale families, multivariate distributions, and principal surfaces are also discussed.