## Bernoulli

- Bernoulli
- Volume 24, Number 1 (2018), 526-564.

### A MOSUM procedure for the estimation of multiple random change points

Birte Eichinger and Claudia Kirch

#### Abstract

In this work, we investigate statistical properties of change point estimators based on moving sum statistics. We extend results for testing in a classical situation with multiple deterministic change points by allowing for random exogenous change points that arise in Hidden Markov or regime switching models among others. To this end, we consider a multiple mean change model with possible time series errors and prove that the number and location of change points are estimated consistently by this procedure. Additionally, we derive rates of convergence for the estimation of the location of the change points and show that these rates are strict by deriving the limit distribution of properly scaled estimators. Because the small sample behavior depends crucially on how the asymptotic (long-run) variance of the error sequence is estimated, we propose to use moving sum type estimators for the (long-run) variance and derive their asymptotic properties. While they do not estimate the variance consistently at every point in time, they can still be used to consistently estimate the number and location of the changes. In fact, this inconsistency can even lead to more precise estimators for the change points. Finally, some simulations illustrate the behavior of the estimators in small samples showing that its performance is very good compared to existing methods.

#### Article information

**Source**

Bernoulli, Volume 24, Number 1 (2018), 526-564.

**Dates**

Received: August 2015

Revised: July 2016

First available in Project Euclid: 27 July 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1501142454

**Digital Object Identifier**

doi:10.3150/16-BEJ887

**Mathematical Reviews number (MathSciNet)**

MR3706768

**Zentralblatt MATH identifier**

06778339

**Keywords**

binary segmentation change point hidden Markov model moving sum statistics regime switching model

#### Citation

Eichinger, Birte; Kirch, Claudia. A MOSUM procedure for the estimation of multiple random change points. Bernoulli 24 (2018), no. 1, 526--564. doi:10.3150/16-BEJ887. https://projecteuclid.org/euclid.bj/1501142454

#### Supplemental materials

- Additional simulation results. In this supplement, we give some additional simulations illustrating the performance of the above change point estimators in small samples with an emphasis on how the variance estimator and the bandwidth influence the performance.Digital Object Identifier: doi:10.3150/16-BEJ887SUPPSupplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.