Cui, Yan; Levine, Michael; Zhou, Zhou Estimation and inference of time-varying auto-covariance under complex trend: a difference-based approach. (English) Zbl 1476.62077 Electron. J. Stat. 15, No. 2, 4264-4294 (2021). Summary: We propose a difference-based nonparametric methodology for the estimation and inference of the time-varying auto-covariance functions of a locally stationary time series when it is contaminated by a complex trend with both abrupt and smooth changes. Simultaneous confidence bands (SCB) with asymptotically correct coverage probabilities are constructed for the auto-covariance functions under complex trend. A simulation-assisted bootstrapping method is proposed for the practical construction of the SCB. Detailed simulation and a real data example round out our presentation. MSC: 62G10 Nonparametric hypothesis testing 62G15 Nonparametric tolerance and confidence regions 62G05 Nonparametric estimation Keywords:change points; Gaussian approximation; local stationarity; simultaneous confidence bands Software:itsmr; ITSM2000; AR1seg PDFBibTeX XMLCite \textit{Y. Cui} et al., Electron. J. Stat. 15, No. 2, 4264--4294 (2021; Zbl 1476.62077) Full Text: DOI arXiv Link References: [1] Brockwell, P. J. and Davis, R. A. (2016). Introduction to Time Series and Forecasting. Springer. · Zbl 1355.62001 [2] Brown, L. D. and Levine, M. (2007). Variance estimation in nonparametric regression via the difference sequence method. The Annals of Statistics 35 2219-2232. · Zbl 1126.62024 [3] Cai, T. T., Levine, M. and Wang, L. (2009). Variance function estimation in multivariate nonparametric regression with fixed design. Journal of Multivariate Analysis 100 126-136. · Zbl 1151.62029 [4] Chakar, S., Lebarbier, E., Lévy-Leduc, C. and Robin, S. (2017). A robust approach for estimating change-points in the mean of an AR(1) process. Bernoulli 23 1408-1447. · Zbl 1378.62059 [5] Craven, P. and Wahba, G. (1978). Smoothing noisy data with spline functions. Numerische mathematik 31 377-403. · Zbl 0377.65007 [6] Dahlhaus, R. (1997). Fitting time series models to nonstationary processes. The Annals of Statistics 25 1-37. · Zbl 0871.62080 [7] Dai, W., Tong, T. and Zhu, L. (2017). On the choice of difference sequence in a unified framework for variance estimation in nonparametric regression. Statistical Science 32 455-468. · Zbl 1442.62081 [8] Dette, H., PreuSS, P. and Vetter, M. (2010). A measure of stationarity in locally stationary processes with applications to testing. Journal of the American Statistical Association 106 1113-1124. · Zbl 1229.62119 [9] Dette, H., Wu, W. and Zhou, Z. (2019). Change Point Analysis of Correlation in Non-stationary Time Series. Statistica Sinica 29 611-643. · Zbl 1433.62257 [10] Dimitris N. Politis, J. P. R. and Wolf, M. (1999). Subsampling. New York: Springer. [11] Ding, X. and Zhou, Z. (2019). Estimation and inference for precision matrices of non-stationary time series. The Annals of Statistics. arXiv:1803.01188. [12] Dwivedi, Y. and Rao, S. S. (2011). A test for second-order stationarity of a time series based on the discrete Fourier transform. Journal of Time Series Analysis 32 68-91. · Zbl 1290.62059 [13] Eubank, R. L. and Spiegelman, C. H. (1990). Testing the goodness of fit of a linear model via nonparametric regression techniques. Journal of the American Statistical Association 85 387-392. · Zbl 0702.62037 [14] Fan, J. and Gijbels, I. (1994). Local Polynomial Modelling and Its Applications. London: Chapman and Hall. · Zbl 0873.62037 [15] Gasser, T., Kneip, A. and Köhler, W. (1991). A flexible and fast method for automatic smoothing. Journal of the American Statistical Association 86 643-652. · Zbl 0733.62047 [16] Hall, P. and Keilegom, I. V. (2003). Using difference-based methods for inference in nonparametric regression with time series errors. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 65 443-456. · Zbl 1065.62067 [17] Hamilton, J. D. (1994). Time Series Analysis. Princeton University Press. · Zbl 0831.62061 [18] Härdle, W. and Tsybakov, A. (1997). Local polynomial estimators of the volatility function in nonparametric autoregression. Journal of Econometrics 81 223-242. · Zbl 0904.62047 [19] Herrmann, E., Gasser, T. and Kneip, A. (1992). Choice of bandwidth for kernel regression when residuals are correlated. Biometrika 79 783-795. · Zbl 0765.62044 [20] Müller, H.-G. and Stadtmüller, U. (1987). Estimation of heteroscedasticity in regression analysis. The Annals of Statistics 15 610-625. · Zbl 0632.62040 [21] Müller, H.-G. and Stadtmüller, U. (1988). Detecting dependencies in smooth regression models. Biometrika 75 639-650. · Zbl 0659.62043 [22] Müller, H.-G. and Stadtmüller, U. (1993). On variance function estimation with quadratic forms. Journal of Statistical Planning and Inference 35 213-231. · Zbl 0769.62029 [23] Nason, G. (2013). A test for second-order stationarity and approximate confidence intervals for localized autocovariances for locally stationary time series. Journal of the Royal Statistical Society. Series B (Statistical Methodology) 75 879-904. · Zbl 1411.62259 [24] Nason, G. P., von Sachs, R. and Kroisandt, G. (2000). Wavelet processes and adaptive estimation of the evolutionary wavelet spectrum. Journal of the Royal Statistical Society. Series B. Statistical Methodology 62 271-292. [25] Paparoditis, E. (2010). Validating stationarity assumptions in time series analysis by rolling local periodograms. Journal of the American Statistical Association 105 839-851. · Zbl 1392.62275 [26] Park, B. U., Lee, Y. K., Kim, T. Y. and Park, C. (2006). A Simple Estimator of Error Correlation in Non-parametric Regression Models. Scandinavian Journal of Statistics 33 451-462. · Zbl 1113.62050 [27] Rao, S. S. (2004). On multiple regression models with nonstationary correlated errors. Biometrika 91 645-659. · Zbl 1108.62093 [28] Rao, T. S. and Tsolaki, E. (2004). Nonstationary time series analysis of monthly global temperature anomalies. In Time Series Analysis and Applications to Geophysical Systems 73-103. Springer. · Zbl 1062.62250 [29] Rice, J. et al. (1984). Bandwidth choice for nonparametric regression. The Annals of Statistics 12 1215-1230. · Zbl 0554.62035 [30] Tecuapetla-Gómez, I. and Munk, A. (2017). Autocovariance Estimation in Regression with a Discontinuous Signal and \(m\)-Dependent Errors: A Difference-Based Approach. Scandinavian Journal of Statistics 44 346-368. · Zbl 1422.62154 [31] Wang, L., Brown, L. D., Cai, T. T. and Levine, M. (2008). Effect of mean on variance function estimation in nonparametric regression. The Annals of Statistics 36 646-664. · Zbl 1133.62033 [32] Wu, W. B. (2005). Nonlinear system theory: another look at dependence. Proceedings of the National Academy of Sciences of the United States of America 102 14150-14154. · Zbl 1135.62075 [33] Wu, W. B. and Zhou, Z. (2011). Gaussian approximations for non-stationary multiple time series. Statistica Sinica 21 1397-1413. · Zbl 1251.60029 [34] Zhou, Y., Cheng, Y., Wang, L. and Tong, T. (2015). Optimal difference-based variance estimation in heteroscedastic nonparametric regression. Statistica Sinica 25 1377-1397. · Zbl 1377.62122 [35] Zhou, Z. (2010). Nonparametric inference of quantile curves for nonstationary time series. The Annals of Statistics 38 2187-2217. · Zbl 1202.62062 [36] Zhou, Z. (2013). Heteroscedasticity and Autocorrelation Robust Structural Change Detection. Journal of the American Statistical Association 108 726-740. · Zbl 06195974 [37] Zhou, Z. and Wu, W. B. (2009). Local linear quantile estimation for nonstationary time series. The Annals of Statistics 37 2696-2729. · Zbl 1173.62066 [38] Zhou, Z. and Wu, W. B. (2010). Simultaneous inference of linear models with time varying coefficients. Journal of the Royal Statistical Society. Series B (Statistical Methodology). 72 513-531. · Zbl 1411.62267 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.