# Bootstrap Confidence Intervals for Multiple Change Points Based on Two-Stage Procedures

## TL;DR

This paper introduces a two-step method to build confidence intervals for multiple change points in regression models using bootstrapping.

## Contribution

A novel two-stage procedure combining variable selection and refinement with a valid bootstrapping method for multiple change points.

## Key findings

- The orthogonal greedy algorithm consistently recovers the number of change points.
- The sup-Wald-type test statistic accurately determines change point locations.
- Bootstrap confidence intervals are asymptotically valid and perform well in simulations and real data.

## Abstract

This paper investigates the construction of confidence intervals for multiple change points in linear regression models. First, we detect multiple change points by performing variable selection on blocks of the input sequence; second, we re-estimate their exact locations in a refinement step. Specifically, we exploit an orthogonal greedy algorithm to recover the number of change points consistently in the cutting stage, and employ the sup-Wald-type test statistic to determine the locations of multiple change points in the refinement stage. Based on a two-stage procedure, we propose bootstrapping the estimated centered error sequence, which can accommodate unknown magnitudes of changes and ensure the asymptotic validity of the proposed bootstrapping method. This enables us to construct confidence intervals using the empirical distribution of the resampled data. The proposed method is illustrated with simulations and real data examples.