A change-point problem and inference for segment signals

TL;DR

This paper investigates the fundamental limits of detecting and estimating change-points in a mean signal, analyzing how their location and separation affect statistical detection and estimation in noisy data.

Contribution

It provides a comprehensive analysis of the minimal segment size needed for detection and compares it with minimax estimation rates, offering insights into change-point and convex body detection.

Findings

Minimal segment size for detection depends on change-point location and separation.

Detection and estimation rates are characterized in various boundary and separation scenarios.

Insights applicable to high-dimensional convex body estimation.

Abstract

We address the problem of detection and estimation of one or two change-points in the mean of a series of random variables. We use the formalism of set estimation in regression: To each point of a design is attached a binary label that indicates whether that point belongs to an unknown segment and this label is contaminated with noise. The endpoints of the unknown segment are the change-points. We study the minimal size of the segment which allows statistical detection in different scenarios, including when the endpoints are separated from the boundary of the domain of the design, or when they are separated from one another. We compare this minimal size with the minimax rates of convergence for estimation of the segment under the same scenarios. The aim of this extensive study of a simple yet fundamental version of the change-point problem is twofold: Understanding the impact of the location and the separation of the change points on detection and estimation and bringing insights about the estimation and detection of convex bodies in higher dimensions.