Bernoulli
- Bernoulli
- Volume 19, Number 1 (2013), 205-227.
Inference for modulated stationary processes
Full-text: Open access
Abstract
We study statistical inferences for a class of modulated stationary processes with time-dependent variances. Due to non-stationarity and the large number of unknown parameters, existing methods for stationary, or locally stationary, time series are not applicable. Based on a self-normalization technique, we address several inference problems, including a self-normalized central limit theorem, a self-normalized cumulative sum test for the change-point problem, a long-run variance estimation through blockwise self-normalization, and a self-normalization-based wild bootstrap. Monte Carlo simulation studies show that the proposed self-normalization-based methods outperform stationarity-based alternatives. We demonstrate the proposed methodology using two real data sets: annual mean precipitation rates in Seoul from 1771–2000, and quarterly U.S. Gross National Product growth rates from 1947–2002.
Article information
Source
Bernoulli, Volume 19, Number 1 (2013), 205-227.
Dates
First available in Project Euclid: 18 January 2013
Permanent link to this document
https://projecteuclid.org/euclid.bj/1358531747
Digital Object Identifier
doi:10.3150/11-BEJ399
Mathematical Reviews number (MathSciNet)
MR3019492
Zentralblatt MATH identifier
1259.62077
Keywords
change-point analysis confidence interval long-run variance modulated stationary process self-normalization strong invariance principle wild bootstrap
Citation
Zhao, Zhibiao; Li, Xiaoye. Inference for modulated stationary processes. Bernoulli 19 (2013), no. 1, 205--227. doi:10.3150/11-BEJ399. https://projecteuclid.org/euclid.bj/1358531747
References
- [1] Adak, S. (1998). Time-dependent spectral analysis of nonstationary time series. J. Amer. Statist. Assoc. 93 1488–1501.Mathematical Reviews (MathSciNet): MR1666643
Zentralblatt MATH: 1064.62565
Digital Object Identifier: doi:10.1080/01621459.1998.10473808 - [2] Altman, N.S. (1990). Kernel smoothing of data with correlated errors. J. Amer. Statist. Assoc. 85 749–759.Mathematical Reviews (MathSciNet): MR1138355
Digital Object Identifier: doi:10.1080/01621459.1990.10474936 - [3] Anděl, J., Netuka, I. and Zvźra, K. (1984). On threshold autoregressive processes. Kybernetika (Prague) 20 89–106.
- [4] Andrews, D.W.K. (1993). Tests for parameter instability and structural change with unknown change point. Econometrica 61 821–856.
- [5] Aue, A., Hörmann, S., Horváth, L. and Reimherr, M. (2009). Break detection in the covariance structure of multivariate time series models. Ann. Statist. 37 4046–4087.Mathematical Reviews (MathSciNet): MR2572452
Zentralblatt MATH: 1191.62143
Digital Object Identifier: doi:10.1214/09-AOS707
Project Euclid: euclid.aos/1256303536 - [6] Aue, A., Horváth, L., Hušková, M. and Kokoszka, P. (2008). Testing for changes in polynomial regression. Bernoulli 14 637–660.Mathematical Reviews (MathSciNet): MR2537806
Digital Object Identifier: doi:10.3150/08-BEJ122
Project Euclid: euclid.bj/1219669624 - [7] Aue, A., Horváth, L., Kokoszka, P. and Steinebach, J. (2008). Monitoring shifts in mean: Asymptotic normality of stopping times. TEST 17 515–530.Mathematical Reviews (MathSciNet): MR2470096
Zentralblatt MATH: 05560822
Digital Object Identifier: doi:10.1007/s11749-006-0041-7 - [8] Bai, J. and Perron, P. (1998). Estimating and testing linear models with multiple structural changes. Econometrica 66 47–78.
- [9] Bentkus, V., Bloznelis, M. and Götze, F. (1996). A Berry–Esséen bound for Student’s statistic in the non-i.i.d. case. J. Theoret. Probab. 9 765–796.
- [10] Berkes, I., Gombay, E. and Horváth, L. (2009). Testing for changes in the covariance structure of linear processes. J. Statist. Plann. Inference 139 2044–2063.Mathematical Reviews (MathSciNet): MR2497559
Zentralblatt MATH: 1159.62031
Digital Object Identifier: doi:10.1016/j.jspi.2008.09.004 - [11] Billingsley, P. (1968). Convergence of Probability Measures. New York: Wiley.Mathematical Reviews (MathSciNet): MR233396
- [12] Bühlmann, P. (2002). Bootstraps for time series. Statist. Sci. 17 52–72.Mathematical Reviews (MathSciNet): MR1910074
Digital Object Identifier: doi:10.1214/ss/1023798998
Project Euclid: euclid.ss/1023798998 - [13] Carlstein, E. (1986). The use of subseries values for estimating the variance of a general statistic from a stationary sequence. Ann. Statist. 14 1171–1179.Mathematical Reviews (MathSciNet): MR856813
Zentralblatt MATH: 0602.62029
Digital Object Identifier: doi:10.1214/aos/1176350057
Project Euclid: euclid.aos/1176350057 - [14] Csörgő, M. and Horváth, L. (1997). Limit Theorems in Change-Point Analysis. Wiley Series in Probability and Statistics. Chichester: Wiley.Mathematical Reviews (MathSciNet): MR2743035
- [15] Dahlhaus, R. (1997). Fitting time series models to nonstationary processes. Ann. Statist. 25 1–37.Mathematical Reviews (MathSciNet): MR1429916
Zentralblatt MATH: 0871.62080
Digital Object Identifier: doi:10.1214/aos/1034276620
Project Euclid: euclid.aos/1034276620 - [16] Dahlhaus, R. and Polonik, W. (2009). Empirical spectral processes for locally stationary time series. Bernoulli 15 1–39.Mathematical Reviews (MathSciNet): MR2546797
Digital Object Identifier: doi:10.3150/08-BEJ137
Project Euclid: euclid.bj/1233669881 - [17] Davidson, R. and Flachaire, E. (2008). The wild bootstrap, tamed at last. J. Econometrics 146 162–169.Mathematical Reviews (MathSciNet): MR2459651
Digital Object Identifier: doi:10.1016/j.jeconom.2008.08.003 - [18] de Jong, R.M. and Davidson, J. (2000). Consistency of kernel estimators of heteroscedastic and autocorrelated covariance matrices. Econometrica 68 407–423.
- [19] Efron, B. (1979). Bootstrap methods: Another look at the jackknife. Ann. Statist. 7 1–26.Mathematical Reviews (MathSciNet): MR515681
Zentralblatt MATH: 0406.62024
Digital Object Identifier: doi:10.1214/aos/1176344552
Project Euclid: euclid.aos/1176344552 - [20] Fan, J. and Yao, Q. (2003). Nonlinear Time Series: Nonparametric and Parametric Methods. Springer Series in Statistics. New York: Springer.Mathematical Reviews (MathSciNet): MR1964455
- [21] Götze, F. and Künsch, H.R. (1996). Second-order correctness of the blockwise bootstrap for stationary observations. Ann. Statist. 24 1914–1933.Mathematical Reviews (MathSciNet): MR1421154
Zentralblatt MATH: 0906.62040
Digital Object Identifier: doi:10.1214/aos/1069362303
Project Euclid: euclid.aos/1069362303 - [22] Ha, K.J. and Ha, E. (2006). Climatic change and interannual fluctuations in the long-term record of monthly precipitation for Seoul. Int. J. Climatol. 26 607–618.
- [23] Hansen, B.E. (1995). Regression with nonstationary volatility. Econometrica 63 1113–1132.
- [24] Hansen, B.E. (2000). Testing for structural change in conditional models. J. Econometrics 97 93–115.Mathematical Reviews (MathSciNet): MR1788819
Zentralblatt MATH: 1122.62326
Digital Object Identifier: doi:10.1016/S0304-4076(99)00068-8 - [25] Horváth, L. (1993). The maximum likelihood method for testing changes in the parameters of normal observations. Ann. Statist. 21 671–680.Mathematical Reviews (MathSciNet): MR1232511
Zentralblatt MATH: 0778.62016
Digital Object Identifier: doi:10.1214/aos/1176349143
Project Euclid: euclid.aos/1176349143 - [26] Inclán, C. and Tiao, G.C. (1994). Use of cumulative sums of squares for retrospective detection of changes of variance. J. Amer. Statist. Assoc. 89 913–923.
- [27] Künsch, H.R. (1989). The jackknife and the bootstrap for general stationary observations. Ann. Statist. 17 1217–1241.Mathematical Reviews (MathSciNet): MR1015147
Zentralblatt MATH: 0684.62035
Digital Object Identifier: doi:10.1214/aos/1176347265
Project Euclid: euclid.aos/1176347265 - [28] Kwiatkowski, D., Phillips, P.C.B., Schmidt, P. and Shin, Y. (1992). Testing the null hypothesis of stationarity against the alternative of a unit root. J. Econometrics 54 159–178.
- [29] Lahiri, S.N. (2003). Resampling Methods for Dependent Data. Springer Series in Statistics. New York: Springer.
- [30] Liu, R.Y. (1988). Bootstrap procedures under some non-i.i.d. models. Ann. Statist. 16 1696–1708.Mathematical Reviews (MathSciNet): MR964947
Zentralblatt MATH: 0655.62031
Digital Object Identifier: doi:10.1214/aos/1176351062
Project Euclid: euclid.aos/1176351062 - [31] Müller, U.K. (2007). A theory of robust long-run variance estimation. J. Econometrics 141 1331–1352.Mathematical Reviews (MathSciNet): MR2413504
Digital Object Identifier: doi:10.1016/j.jeconom.2007.01.019 - [32] Pettitt, A.N. (1980). A simple cumulative sum type statistic for the change-point problem with zero–one observations. Biometrika 67 79–84.Mathematical Reviews (MathSciNet): MR570508
Zentralblatt MATH: 0424.62015
Digital Object Identifier: doi:10.1093/biomet/67.1.79 - [33] Phillips, P.C.B., Sun, Y. and Jin, S. (2007). Long run variance estimation and robust regression testing using sharp origin kernels with no truncation. J. Statist. Plann. Inference 137 985–1023.Mathematical Reviews (MathSciNet): MR2301731
Zentralblatt MATH: 1104.62099
Digital Object Identifier: doi:10.1016/j.jspi.2006.06.033 - [34] Politis, D.N. and Romano, J.P. (1994). Large sample confidence regions based on subsamples under minimal assumptions. Ann. Statist. 22 2031–2050.Mathematical Reviews (MathSciNet): MR1329181
Zentralblatt MATH: 0828.62044
Digital Object Identifier: doi:10.1214/aos/1176325770
Project Euclid: euclid.aos/1176325770 - [35] Robbins, M.W., Lund, R.B., Gallagher, C.M. and Lu, Q. (2011). Changepoints in the North Atlantic tropical cyclone record. J. Amer. Statist. Assoc. 106 89–99.Mathematical Reviews (MathSciNet): MR2816704
Digital Object Identifier: doi:10.1198/jasa.2011.ap10023 - [36] Shao, Q.M. (1993). Almost sure invariance principles for mixing sequences of random variables. Stochastic Process. Appl. 48 319–334.Mathematical Reviews (MathSciNet): MR1244549
Zentralblatt MATH: 0793.60038
Digital Object Identifier: doi:10.1016/0304-4149(93)90051-5 - [37] Shao, X. and Zhang, X. (2010). Testing for change points in time series. J. Amer. Statist. Assoc. 105 1228–1240.Mathematical Reviews (MathSciNet): MR2752617
Digital Object Identifier: doi:10.1198/jasa.2010.tm10103 - [38] Shumway, R.H. and Stoffer, D.S. (2006). Time Series Analysis and Its Applications: With R Examples, 2nd ed. Springer Texts in Statistics. New York: Springer.Mathematical Reviews (MathSciNet): MR2228626
- [39] Wu, C.F.J. (1986). Jackknife, bootstrap and other resampling methods in regression analysis. Ann. Statist. 14 1261–1295.Mathematical Reviews (MathSciNet): MR868303
Zentralblatt MATH: 0618.62072
Digital Object Identifier: doi:10.1214/aos/1176350142
Project Euclid: euclid.aos/1176350142 - [40] Wu, W.B. (2007). Strong invariance principles for dependent random variables. Ann. Probab. 35 2294–2320.Mathematical Reviews (MathSciNet): MR2353389
Zentralblatt MATH: 1166.60307
Digital Object Identifier: doi:10.1214/009117907000000060
Project Euclid: euclid.aop/1191860422 - [41] Zhao, Z. (2011). A self-normalized confidence interval for the mean of a class of non-stationary processes. Biometrika 98 81–90.Mathematical Reviews (MathSciNet): MR2804211
Zentralblatt MATH: 1214.62089
Digital Object Identifier: doi:10.1093/biomet/asq076
- You have access to this content.
- You have partial access to this content.
- You do not have access to this content.
More like this
- A hierarchical multivariate spatio-temporal model for clustered climate data with annual cycles
Mastrantonio, Gianluca, Jona Lasinio, Giovanna, Pollice, Alessio, Capotorti, Giulia, Teodonio, Lorenzo, Genova, Giulio, and Blasi, Carlo, The Annals of Applied Statistics, 2019 - Nonparametric specification for non-stationary time series regression
Zhou, Zhou, Bernoulli, 2014 - Spatial and Temporal Variation of Annual Precipitation in a River of the Loess Plateau in China
Shen, Cui and Qiang, Huang, Journal of Applied Mathematics, 2013
- A hierarchical multivariate spatio-temporal model for clustered climate data with annual cycles
Mastrantonio, Gianluca, Jona Lasinio, Giovanna, Pollice, Alessio, Capotorti, Giulia, Teodonio, Lorenzo, Genova, Giulio, and Blasi, Carlo, The Annals of Applied Statistics, 2019 - Nonparametric specification for non-stationary time series regression
Zhou, Zhou, Bernoulli, 2014 - Spatial and Temporal Variation of Annual Precipitation in a River of the Loess Plateau in China
Shen, Cui and Qiang, Huang, Journal of Applied Mathematics, 2013 - Detecting long-range dependence in non-stationary time series
Dette, Holger, Preuss, Philip, and Sen, Kemal, Electronic Journal of Statistics, 2017 - Recursive estimation of time-average variance constants
Wu, Wei Biao, The Annals of Applied Probability, 2009 - Dirichlet Process Hidden Markov Multiple Change-point Model
Ko, Stanley I. M., Chong, Terence T. L., and Ghosh, Pulak, Bayesian Analysis, 2015 - Empirical likelihood methods with weakly dependent processes
Kitamura, Yuichi, The Annals of Statistics, 1997 - A test for stationarity based on empirical processes
Preuß, Philip, Vetter, Mathias, and Dette, Holger, Bernoulli, 2013 - Scalable high-resolution forecasting of sparse spatiotemporal events with kernel methods: A winning solution to the NIJ “Real-Time Crime Forecasting Challenge”
Flaxman, Seth, Chirico, Michael, Pereira, Pau, and Loeffler, Charles, The Annals of Applied Statistics, 2019 - Least tail-trimmed absolute deviation estimation for autoregressions with infinite/finite variance
Wu, Rongning and Cui, Yunwei, Electronic Journal of Statistics, 2018