# Optimal rate of convergence for nonparametric change-point estimators for nonstationary sequences

@article{Hariz2007OptimalRO,
title={Optimal rate of convergence for nonparametric change-point estimators for nonstationary sequences},
author={Samir Ben Hariz and Jonathan J. Wylie and Qiang Zhang},
journal={Annals of Statistics},
year={2007},
volume={35},
pages={1802-1826}
}
• Published 1 August 2007
• Mathematics
• Annals of Statistics
Let (X i ) ι =1 be a possibly nonstationary sequence such that L(X i ) = P n if i ≤ n6 and L(X i ) = Q n if > nθ, where 0 < θ < 1 is the location of the change-point to be estimated. We construct a class of estimators based on the empirical measures and a seminorm on the space of measures defined through a family of functions F. We prove the consistency of the estimator and give rates of convergence under very general conditions. In particular, the 1/n rate is achieved for a wide class of…
42 Citations

## Figures from this paper

### On change-point estimation under Sobolev sparsity

• Computer Science, Mathematics
Electronic Journal of Statistics
• 2020
An adaptive dimension reduction procedure based on Lepski’s method is proposed and it is proved that there is a bound on the separation of the regimes above which there exists an optimal choice of dimension reduction, leading to the fast rate of estimation.

### Finite sample change point inference and identification for high‐dimensional mean vectors

• Computer Science
Journal of the Royal Statistical Society: Series B (Statistical Methodology)
• 2020
The proposed bootstrap CUSUM test is fully data dependent and it has strong theoretical guarantees under arbitrary dependence structures and mild moment conditions and it achieves the minimax separation rate under the sparse alternatives when the dimension p can be larger than the sample size n.

### Posterior Convergence and Model Estimation in Bayesian Change-point Problems

• H. Lian
• Mathematics, Computer Science
• 2008
It is argued that the point-wise posterior convergence property as demonstrated might have bad finite sample performance in that consistent posterior for model selection necessarily implies the maximal squared risk will be asymptotically larger than the optimal $O(1/\sqrt{n})$ rate.

### Statistical inference for the slope parameter in functional linear regression

• Mathematics
Electronic Journal of Statistics
• 2022
In this paper we consider the linear regression model Y = SX + ε with functional regressors and responses. We develop new inference tools to quantify deviations of the true slope S from a

### On Some Unsupervised Learning Problems for Highly Dependent Time Series. (Sur quelques problèmes non-supervisés impliquant des séries temporelles hautement dèpendantes)

This thesis is devoted to the theoretical analysis of unsupervised learning problems involving highly dependent time-series, and constructively demonstrates that natural formulations of the considered problems exist which admit consistent solutions in this general framework.

### A Nonparametric Approach for Multiple Change Point Analysis of Multivariate Data

• Mathematics
• 2013
The divisive method is shown to provide consistent estimates of both the number and the location of change points under standard regularity assumptions, and methods from cluster analysis are applied to assess performance and to allow simple comparisons of location estimates, even when the estimated number differs.

## References

SHOWING 1-10 OF 20 REFERENCES

### On almost sure behavior of change-point estimators

• Mathematics
• 1994
We propose a natural setting for the simple change-point problem which is particularly useful for studying almost sure convergence properties of changepoint estimators. It is shown that each member

### Limit Theorems for Nonlinear Functionals of a Stationary Gaussian Sequence of Vectors

Limit theorems for functions of stationary mean-zero Gaussian sequences of vectors satisfying long range dependence conditions are considered. Depending on the rate of decay of the coefficients, the

### Weak Convergence and Empirical Processes: With Applications to Statistics

• Mathematics
• 1996
This chapter discusses Convergence: Weak, Almost Uniform, and in Probability, which focuses on the part of Convergence of the Donsker Property which is concerned with Uniformity and Metrization.

### Time Series: Theory and Methods

• Mathematics
• 1991
This paper presents a meta-modelling framework for estimating the mean and the Autocovariance Function of Stationary Time Series using ARMA Models and State-Space Models and Kalman Recursions.

### Boundary estimation based on set-indexed empirical processes

This work induces a partition from which it is shown that the number of misclassified data is stochastically bounded as the sample sizes increase to infinity, and estimates the common topological boundary of the two regions.