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}
}
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… 

Figures from this paper

On change-point estimation under Sobolev sparsity

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

  • Mengjia YuXiaohui Chen
  • 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

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

A Nonparametric Approach for Multiple Change Point Analysis of Multivariate Data

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.

Non-Parametric Change-Point Estimation using String Matching Algorithms

This work introduces a novel non-parametric estimator, CRECHE (CRossings Enumeration CHange Estimator), which requires no assumptions about the form of the source distribution, and avoids the need to estimate its probabilities.

References

SHOWING 1-10 OF 20 REFERENCES

On almost sure behavior of change-point estimators

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

Exponential and polynomial tailbounds for change-point estimators

The change-point problem for dependent observations

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

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.

The effect of long-range dependence on change-point estimators

Time Series: Theory and Methods

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.

Change-point in the mean of dependent observations