We address the problem of detection and estimation of one or two change-points in the mean of a series of random variables. We use the formalism of set estimation in regression: to each point of a design is attached a binary label that indicates whether that point belongs to an unknown segment and this label is contaminated with noise. The endpoints of the unknown segment are the change-points. We study the minimal size of the segment which allows statistical detection in different scenarios, including when the endpoints are separated from the boundary of the domain of the design, or when they are separated from one another. We compare this minimal size with the minimax rates of convergence for estimation of the segment under the same scenarios. The aim of this extensive study of a simple yet fundamental version of the change-point problem is two-fold: understanding the impact of the location and the separation of the change points on detection and estimation and bringing insights about the estimation and detection of convex bodies in higher dimensions.
DOI : 10.1051/ps/2018014
Keywords: Change-point, detection, hypothesis testing, minimax, separation rate, set estimation
Brunel, Victor-Emmanuel 1
@article{PS_2018__22__210_0,
author = {Brunel, Victor-Emmanuel},
title = {A change-point problem and inference for segment signals},
journal = {ESAIM: Probability and Statistics},
pages = {210--235},
year = {2018},
publisher = {EDP Sciences},
volume = {22},
doi = {10.1051/ps/2018014},
mrnumber = {3891756},
zbl = {1409.62165},
language = {en},
url = {https://www.numdam.org/articles/10.1051/ps/2018014/}
}
TY - JOUR AU - Brunel, Victor-Emmanuel TI - A change-point problem and inference for segment signals JO - ESAIM: Probability and Statistics PY - 2018 SP - 210 EP - 235 VL - 22 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/ps/2018014/ DO - 10.1051/ps/2018014 LA - en ID - PS_2018__22__210_0 ER -
Brunel, Victor-Emmanuel. A change-point problem and inference for segment signals. ESAIM: Probability and Statistics, Tome 22 (2018), pp. 210-235. doi: 10.1051/ps/2018014
[1] and , Bayesian-type estimators of change points. Prague Workshop on Perspectives in Modern Statistical Inference: Parametrics, Semi-parametrics, Non-parametrics 1998. J. Stat. Plan. Inference 91 (2000) 195–208. | MR | Zbl | DOI
[2] , Testing for change-points in long-range dependent time series by means of a self-normalized Wilcoxon test. J. Time Ser. Anal. 37 (2016) 785–809. | MR | Zbl | DOI
[3] and , Asymptotically optimal estimates in the smooth change-point problem. Math. Method. Stat. 13 (2004) 1–24. | MR | Zbl
[4] , Adaptive estimation of convex polytopes and convex sets from noisy data. Electron. J. Stat. 7 (2013) 1301–1327. | MR | Zbl
[5] and , Statistical inference of covariance change points in gaussian model. Statistics 38 (2004) 17–28. | MR | Zbl | DOI
[6] and , Detection with the scan and the average likelihood ratio. Stat. Sin. 23 (2013) 409–428. | MR | Zbl
[7] and , Rate-optimal detection of very short signal segments. Technical report (2014).
[8] , and , Multiscale change point inference. J. R. Stat. Soc. Ser. B. Stat. Methodol. 76 (2014) 495–580. With 32 discussions by 47 authors and a rejoinder by the authors. | MR | Zbl | DOI
[9] , Convergence rates for pointwise curve estimation with a degenerate design. Math. Method. Stat. 14 (2005) 1–27. | MR
[10] , Minimax hypothesis testing about the density support. Bernoulli 7 (2001) 507–525. | MR | Zbl | DOI
[11] and , On the estimation of a changepoint in a tail index. Liet. Mat. Rink. 45 (2005) 333–348. | MR | Zbl
[12] and , Estimation of the parameter of a discontinuous signal in Gaussian white noise. Prob. Peredači Inform. 11 (1975) 31–43. | MR | Zbl
[13] and , Statistical Estimation: Asymptotic Theory. Springer Science & Business, New York, (2013).
[14] , On minimax estimation of a discontinuous signal (translation of a paper in russian, published in 1986). Theory Probab. Appl. 32 (2006) 727–730. | Zbl | DOI
[15] and , Asymptotically minimax image reconstruction problems, in Topics in Nonparametric Estimation. Vol. 12 of Advances in Soviet Mathematics. American Mathematical Society, Providence, RI (1992) 45–86. | MR | Zbl | DOI
[16] and , Minimax theory of image reconstruction. Vol. 82 of Lecture Notes in Statistics. Springer-Verlag, New York (1993). | MR | Zbl | DOI
[17] , Detecting multiple change-points in the mean of gaussian process by model selection. Signal Process. 85 (2005) 717–736. | Zbl | DOI
[18] , , and , The cusum test for parameter change in time series models. Scand. J. Stat. 30 (2003) 781–796. | MR | Zbl | DOI
[19] and , Asymptotically exact nonparametric hypothesis testing in sup-norm and at a fixed point. Probab. Theory Rel. Fields 117 (2000) 17–48. | MR | Zbl | DOI
[20] , Information criterion for Gaussian change-point model. Stat. Probab. Lett. 72 (2005) 237–247. | MR | Zbl | DOI
[21] and , Estimation of a regression function with a sharp change point using boundary wavelets. Stat. Probab. Lett. 66 (2004) 435–448. | MR | Zbl | DOI
[22] , Minimax estimation of sharp change points. Ann. Stat. 26 (1998) 1379–1397. | MR | Zbl | DOI
[23] and , Bayesian single change point detection in a sequence of multivariate normal observations. Statistics 39 (2005) 373–387. | MR | Zbl | DOI
[24] and , Testing for change points in time series. J. Am. Stat. Assoc. 105 (2010) 1228–1240. | MR | Zbl | DOI
[25] , Introduction to nonparametric estimation. Revised and extended from the 2004 French original, Translated by Vladimir Zaiats. Springer Series in Statistics. Springer, New York (2009). | MR | Zbl | DOI
[26] , Inference for change-point and post-change mean with possible change in variance. Seq. Anal. 24 (2005) 279–302. | MR | Zbl | DOI
Cité par Sources :
