The asymptotic distribution of a cluster-index for i.i.d. normal random variables

TL;DR

This paper establishes that the asymptotic distribution of a cluster index used in normal sample cluster detection converges to the Gumbel distribution, aiding in statistical inference.

Contribution

It proves that the cluster index's asymptotic distribution for i.i.d. normal variables is Gumbel, providing theoretical insight into its behavior.

Findings

Asymptotic distribution of the cluster index is Gumbel.

Supports the use of the index in cluster detection for normal data.

Enhances understanding of sample variance decomposition in clustering.

Abstract

In a sample variance decomposition, with components functions of the sample's spacings, the largest component $\tilde{I}_n$ is used in cluster detection. It is shown for normal samples that the asymptotic distribution of $\tilde{I}_n$ is the Gumbel distribution.