Optimal clustering on the real line
A method is proposed for assessing the number of groups (clusters) in a random sample (presented by order statistics) drawn from a continuous population distributed on (a,b)\(\subseteq (-\infty,\infty)\). A sample quantile function is defined to be a piecewise linear interpolation of these order statistics. The method is based on the calculation of an asymptotic nonparametric confidence interval for the fractional reduction of the within-group error due to \((g+1)\)-grouping over g-grouping. The confidence interval theory (as well as the central limit theory) is developed for both universally optimal and bounded locally optimal groupings. The derivation makes use of a Donsker-type theorem for the quantile process that is also proven. The consistency of the estimators used in the proofs of the theorem is shown.
piecewise linear interpolation of order statistics
asymptotic nonparametric confidence interval
scientific article; zbMATH DE number 4090609