Optimal stratification and clustering on the line using the \(L_ 1\)- norm
A random sample of n continuous one-dimensional observations is partitioned into g groups or clusters by minimizing their within group dispersion \(\hat W_ g\) as measured by the \(L_ 1\)-norm. The asymptotic behaviour of the optimal partition and of the class boundaries is investigated (consistency results, asymptotic normality) by using a representation for the quantile process with the help of a Brownian bridge [\textit{M. Csörgö} and \textit{P. Révész}, Ann. Stat. 6, 882- 894 (1978; Zbl 0378.62050)] and some consistency results by \textit{D. Pollard} [Ann. Probab. 10, 919-926 (1982; Zbl 0502.62055). An asymptotic multivariate normal distribution of the ratios \(\hat R{}_ g:=(\min \hat W_{g+1})/(\min \hat W_ g)\) of criterion values for \(g+1\) resp. g clusters (for \(g=1,2,...,G)\) is derived which leads to a confidence interval for its population analogues \(R_ g\). Some simple examples suggest that in the case of a 'natural' clustering, the true class number g may be indicated by a levelling out and a pronounced widening of these confidence intervals.
scientific article; zbMATH DE number 4007493