Unsupervised Self-Normalized Change-Point Testing for Time Series

ABSTRACT We propose a new self-normalized method for testing change points in the time series setting. Self-normalization has been celebrated for its ability to avoid direct estimation of the nuisance asymptotic variance and its flexibility of being generalized to handle quantities other than the mean. However, it was developed and mainly studied for constructing confidence intervals for quantities associated with a stationary time series, and its adaptation to change-point testing can be nontrivial as direct implementation can lead to tests with nonmonotonic power. Compared with existing results on using self-normalization in this direction, the current article proposes a new self-normalized change-point test that does not require prespecifying the number of total change points and is thus unsupervised. In addition, we propose a new contrast-based approach in generalizing self-normalized statistics to handle quantities other than the mean, which is specifically tailored for change-point testing. Monte Carlo simulations are presented to illustrate the finite-sample performance of the proposed method. Supplementary materials for this article are available online.

[1]  E. J. Hannan,et al.  The central limit theorem for time series regression , 1979 .

[2]  N. Herrndorf.  A Functional Central Limit Theorem for Weakly Dependent Sequences of Random Variables , 1984 .

[3]  W. Newey,et al.  A Simple, Positive Semi-Definite, Heteroskedasticity and Autocorrelationconsistent Covariance Matrix , 1986 .

[4]  Yi-Ching Yao.  Estimating the number of change-points via Schwarz' criterion , 1988 .

[5]  H. Künsch.  The Jackknife and the Bootstrap for General Stationary Observations , 1989 .

[6]  D. Andrews.  Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation , 1991 .

[7]  P. Perron,et al.  Estimating and testing linear models with multiple structural changes , 1995 .

[8]  Jon A. Wellner,et al.  Weak Convergence and Empirical Processes: With Applications to Statistics , 1996 .

[9]  Bing-Yi Jing,et al.  On Sample Reuse Methods for Dependent Data , 1996 .

[10]  T. Vogelsang.  Sources of nonmonotonic power when testing for a shift in mean of a dynamic time series , 1999 .

[11]  Ignacio N. Lobato.  Testing That a Dependent Process Is Uncorrelated , 2001 .

[12]  Alan David Hutson,et al.  Resampling Methods for Dependent Data , 2004, Technometrics.

[13]  Richard A. Davis,et al.  Structural Break Estimation for Nonstationary Time Series Models , 2006 .

[14]  David O Siegmund,et al.  A Modified Bayes Information Criterion with Applications to the Analysis of Comparative Genomic Hybridization Data , 2007, Biometrics.

[15]  Wei Biao Wu,et al.  STRONG INVARIANCE PRINCIPLES FOR DEPENDENT RANDOM VARIABLES , 2007, 0711.3674.

[16]  J. Elsner,et al.  The increasing intensity of the strongest tropical cyclones , 2008, Nature.

[17]  Weidong Liu,et al.  ASYMPTOTICS OF SPECTRAL DENSITY ESTIMATES , 2009, Econometric Theory.

[18]  X. Shao,et al.  The Dependent Wild Bootstrap , 2010 .

[19]  Zhou Zhou,et al.  NONPARAMETRIC INFERENCE OF QUANTILE CURVES FOR NONSTATIONARY TIME SERIES , 2010, 1010.3891.

[20]  X. Shao,et al.  A self‐normalized approach to confidence interval construction in time series , 2010, 1005.2137.

[21]  X. Shao,et al.  Testing for Change Points in Time Series , 2010 .

[22]  D. Politis.  HIGHER-ORDER ACCURATE, POSITIVE SEMIDEFINITE ESTIMATION OF LARGE-SAMPLE COVARIANCE AND SPECTRAL DENSITY MATRICES , 2005, Econometric Theory.

[23]  Wei Biao Wu,et al.  Testing parametric assumptions of trends of a nonstationary time series , 2011 .

[24]  Zhou Zhou,et al.  Heteroscedasticity and Autocorrelation Robust Structural Change Detection , 2013 .

[25]  X. Shao,et al.  A general approach to the joint asymptotic analysis of statistics from sub-samples , 2013, 1305.5618.

[26]  Block sampling under strong dependence , 2013, 1312.5807.

[27]  X. Shao,et al.  Inference for linear models with dependent errors , 2013 .

[28]  Weidong Liu,et al.  Komlós–Major–Tusnády approximation under dependence , 2014, 1402.6517.

[29]  X. Shao,et al.  ON SELF‐NORMALIZATION FOR CENSORED DEPENDENT DATA , 2015 .

[30]  X. Shao,et al.  Self-Normalization for Time Series: A Review of Recent Developments , 2015 .

[31]  Alessandro Casini,et al.  Structural Breaks in Time Series , 2018, Oxford Research Encyclopedia of Economics and Finance.