Change in the mean in the domain of attraction of the normal law via Darling-Erd\H{o}s theorems

TL;DR

This paper develops a new statistical test based on Darling-Erd"H{o}s theorems to detect changes in the mean of independent observations, applicable even when the variance is infinite, with asymptotic Gumbel distribution under no change.

Contribution

It introduces a novel self-normalized maximal deviation test for mean change detection within the domain of attraction of the normal law, extending applicability to infinite variance cases.

Findings

Test statistic converges to Gumbel distribution under null hypothesis.

Method applicable to variables with potentially infinite variance.

Provides theoretical validation under broad conditions.

Abstract

This paper studies the problem of testing the null assumption of no-change in the mean of chronologically ordered independent observations on a random variable $X$ {\it versus} the at most one change in the mean alternative hypothesis. The approach taken is via a Darling-Erd\H{o}s type self-normalized maximal deviation between sample means before and sample means after possible times of a change in the expected values of the observations of a random sample. Asymptotically, the thus formulated maximal deviations are shown to have a standard Gumbel distribution under the null assumption of no change in the mean. A first such result is proved under the condition that $EX^2\log\log (|X|+1)<\infty$, while in the case of a second one, $X$ is assumed to be in a specific class of the domain of attraction of the normal law, possibly with infinite variance.