Asymptotics of a Clustering Criterion for Smooth Distributions

TL;DR

This paper introduces an asymptotic analysis of a clustering criterion for smooth distributions, enabling interval estimation and tests for bimodality and clustering presence.

Contribution

It proposes a new clustering criterion based on order statistics and derives its asymptotic properties for better cluster detection.

Findings

Asymptotic behavior of the splitting point is characterized.

Method for constructing interval estimates of the split point.

Tests for bimodality and clustering are developed.

Abstract

We develop a clustering framework for observations from a population with a smooth probability distribution function and derive its asymptotic properties. A clustering criterion based on a linear combination of order statistics is proposed. The asymptotic behavior of the point at which the observations are split into two clusters is examined. The results obtained can then be utilized to construct an interval estimate of the point which splits the data and develop tests for bimodality and presence of clusters.