# Assessing One-Dimensional Cluster Stability by Extreme-Point Trimming

## TL;DR

This paper introduces a probabilistic method for evaluating the stability of one-dimensional data clusters by analyzing how their range contracts when extreme points are trimmed, providing theoretical insights and practical tools.

## Contribution

It develops a new shrinkage-based test for cluster stability, with analytical expressions and a clustering validation pipeline, outperforming classical methods in small-sample and noisy conditions.

## Key findings

- The method accurately distinguishes distribution types using shrinkage curves.
- The proposed test outperforms likelihood-ratio tests in small samples.
- It effectively validates one-dimensional clusters within a clustering pipeline.

## Abstract

We develop a probabilistic method for assessing the tail behavior and geometric stability of one-dimensional n i.i.d. samples by tracking how their span contracts when the most extreme points are trimmed. Central to our approach is the diameter-shrinkage ratio, that quantifies the relative reduction in data range as extreme points are successively removed. We derive analytical expressions, including finite-sample corrections, for the expected shrinkage under both the uniform and Gaussian hypotheses, and establish that these curves remain distinct even for moderate number of removal. We construct an elementary decision rule that assigns a sample to whichever theoretical shrinkage profile it most closely follows. This test achieves higher classification accuracy than the classical likelihood-ratio test in small-sample or noisy regimes, while preserving asymptotic consistency for large n. We further integrate our criterion into a clustering pipeline (e.g. DBSCAN), demonstrating its ability to validate one-dimensional clusters without any density estimation or parameter tuning. This work thus provides both theoretical insight and practical tools for robust distributional inference and cluster stability analysis.