Optimal clustering on the real line (Q1116592)

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.