Sequential change point tests based on U-statistics

TL;DR

This paper introduces a flexible framework for sequential change point detection using U-statistics, including robust and early-detection schemes, with theoretical guarantees and empirical validation on temperature data.

Contribution

It develops a unified framework for sequential change point tests based on U-statistics, incorporating various monitoring schemes and providing asymptotic properties and practical performance analysis.

Findings

All proposed tests control the asymptotic type-I-error.

Tests have asymptotic power one.

Simulation shows competitive detection delay and power.

Abstract

We propose a general framework of sequential testing procedures based on $U$-statistics which contains as an example a sequential CUSUM test based on differences in mean but also includes a robust sequential Wilcoxon change point procedure. Within this framework, we consider several monitoring schemes that take different observations into account to make a decision at a given time point. Unlike the originally proposed scheme that takes all observations of the monitoring period into account, we also consider a modified moving-sum-version as well as a version of a Page-monitoring scheme. The latter behave almost as good for early changes while being advantageous for later changes. For all proposed procedures we provide the limit distribution under the null hypothesis which yields the threshold to control the asymptotic type-I-error. Furthermore, we show that the proposed tests have asymptotic power one. In a simulation study we compare the performance of the sequential procedures via their empirical size, power and detection delay, which is further illustrated by means of a temperature data set.