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

By Samir Ben Hariz, J. Wylie, Qiang Zhang. 2007

49 citations recorded by the source corpus.

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 processes including long-range dependent sequences and even nonstationary ones. The approach unifies, generalizes and improves on the existing results for both parametric and nonparametric change-point estimation, applied to independent, short-range dependent and as well long-range dependent sequences.

Subfields: Finance, Management Science and Operations Research, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/a29981bc63af0e9a805338604532faafe4a1b96a

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Citations in the corpus

A Nonparametric Approach for Multiple Change Point Analysis of Multivariate Data (2013, background): “For example, given that there are k change points, Hawkins (2001) estimates change point locations by maximizing a likelihood function.”
Inference for single and multiple change‐points in time series (2013, background): “In addition, Hariz et al. (2007) considered the case where the difference before and after the change-point converged to zero.”
Finite sample change point inference and identification for high‐dimensional mean vectors (2017, methodology): “Change point testing and estimation have a long history [45, 46, 41, 16, 31, 49, 10, 20, 33, 6, 23, 13, 37, 29, 27, 22, 24, 40, 3, 4, 7, 51, 30] and there are two major lines of change point tests in literature: sequential tests and fixed sample size tests [34].”
Testing relevant hypotheses in functional time series via self‐normalization (2018, background): “in S= Rd. First, we note that there exists a general and well-developed machinery for establishing convergence rates of change point estimators [see, for example Carlstein (1988), Dum bgen (1991) or Hariz et al. (2007), among others], and we assume (4.8) ^= 0 + o P(n 1=2); throughout this section, ”