Optimal stratification and clustering on the line using the \(L_ 1\)- norm
type: Item
instance of: scholarly article
title: Optimal stratification and clustering on the line using the \(L_ 1\)- norm
zbMATH Open document ID: 0621.62061
DOI: 10.1016/0047-259X(86)90099-0
author: R. W. Butler
published in: Journal of Multivariate Analysis
publication date: 1986-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 62H30
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 62G15
zbMATH DE Number: 4007493
zbMATH Keywords: optimal stratification
zbMATH Keywords: L1-clustering
zbMATH Keywords: central limit theory
zbMATH Keywords: continuous one-dimensional observations
zbMATH Keywords: clusters
zbMATH Keywords: within group dispersion
zbMATH Keywords: asymptotic behaviour
zbMATH Keywords: optimal partition
zbMATH Keywords: class boundaries
zbMATH Keywords: consistency
zbMATH Keywords: asymptotic normality
zbMATH Keywords: quantile process
zbMATH Keywords: Brownian bridge
zbMATH Keywords: confidence interval
reviewed by: H.-H. Bock
MaRDI profile type: Publication
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full work available at URL: 0047-259x(86)90099-0
OpenAlex ID: W1989289856
description: scientific article; zbMATH DE number 4007493