Open Access
February 2021 Convergence of covariance and spectral density estimates for high-dimensional locally stationary processes
Danna Zhang, Wei Biao Wu
Ann. Statist. 49(1): 233-254 (February 2021). DOI: 10.1214/20-AOS1954
Abstract

Covariances and spectral density functions play a fundamental role in the theory of time series. There is a well-developed asymptotic theory for their estimates for low-dimensional stationary processes. For high-dimensional nonstationary processes, however, many important problems on their asymptotic behaviors are still unanswered. This paper presents a systematic asymptotic theory for the estimates of time-varying second-order statistics for a general class of high-dimensional locally stationary processes. Using the framework of functional dependence measure, we derive convergence rates of the estimates which depend on the sample size $T$, the dimension $p$, the moment condition and the dependence of the underlying processes.

References

1.

Adak, S. (1998). Time-dependent spectral analysis of nonstationary time series. J. Amer. Statist. Assoc. 93 1488–1501. 1064.62565 10.1080/01621459.1998.10473808Adak, S. (1998). Time-dependent spectral analysis of nonstationary time series. J. Amer. Statist. Assoc. 93 1488–1501. 1064.62565 10.1080/01621459.1998.10473808

2.

Alexopoulos, C. and Goldsman, D. (2004). To batch or not to batch? ACM Trans. Model. Comput. Simul. 14 76–114. 1390.65018 10.1145/974734.974738Alexopoulos, C. and Goldsman, D. (2004). To batch or not to batch? ACM Trans. Model. Comput. Simul. 14 76–114. 1390.65018 10.1145/974734.974738

3.

An, H. Z., Chen, Z. G. and Hannan, E. J. (1982). Autocorrelation, autoregression and autoregressive approximation. Ann. Statist. 10 926–936. 0512.62087 10.1214/aos/1176345882 euclid.aos/1176345882An, H. Z., Chen, Z. G. and Hannan, E. J. (1982). Autocorrelation, autoregression and autoregressive approximation. Ann. Statist. 10 926–936. 0512.62087 10.1214/aos/1176345882 euclid.aos/1176345882

4.

Anderson, T. W. (1971). The Statistical Analysis of Time Series. Wiley, New York. MR0283939 0225.62108Anderson, T. W. (1971). The Statistical Analysis of Time Series. Wiley, New York. MR0283939 0225.62108

5.

Baccalá, L. A. and Sameshima, K. (2001). Partial directed coherence: A new concept in neural structure determination. Biol. Cybernet. 84 463–474. 1160.92306 10.1007/PL00007990Baccalá, L. A. and Sameshima, K. (2001). Partial directed coherence: A new concept in neural structure determination. Biol. Cybernet. 84 463–474. 1160.92306 10.1007/PL00007990

6.

Bach, F. R. and Jordan, M. I. (2004). Learning graphical models for stationary time series. IEEE Trans. Signal Process. 52 2189–2199. 1369.62219 10.1109/TSP.2004.831032Bach, F. R. and Jordan, M. I. (2004). Learning graphical models for stationary time series. IEEE Trans. Signal Process. 52 2189–2199. 1369.62219 10.1109/TSP.2004.831032

7.

Barigozzi, M., Hallin, M., Soccorsi, S. and Von Sachs, R. (2019). Time-varying general dynamic factor models and the measurement of financial connectedness. Available at SSRN 3329445.Barigozzi, M., Hallin, M., Soccorsi, S. and Von Sachs, R. (2019). Time-varying general dynamic factor models and the measurement of financial connectedness. Available at SSRN 3329445.

8.

Bercu, B., Gamboa, F. and Rouault, A. (1997). Large deviations for quadratic forms of stationary Gaussian processes. Stochastic Process. Appl. 71 75–90. 0941.60050 10.1016/S0304-4149(97)00071-9Bercu, B., Gamboa, F. and Rouault, A. (1997). Large deviations for quadratic forms of stationary Gaussian processes. Stochastic Process. Appl. 71 75–90. 0941.60050 10.1016/S0304-4149(97)00071-9

9.

Bickel, P. J. and Levina, E. (2008). Covariance regularization by thresholding. Ann. Statist. 36 2577–2604. 1196.62062 10.1214/08-AOS600 euclid.aos/1231165180Bickel, P. J. and Levina, E. (2008). Covariance regularization by thresholding. Ann. Statist. 36 2577–2604. 1196.62062 10.1214/08-AOS600 euclid.aos/1231165180

10.

Bickel, P. J., Ritov, Y. and Tsybakov, A. B. (2009). Simultaneous analysis of lasso and Dantzig selector. Ann. Statist. 37 1705–1732. 1173.62022 10.1214/08-AOS620 euclid.aos/1245332830Bickel, P. J., Ritov, Y. and Tsybakov, A. B. (2009). Simultaneous analysis of lasso and Dantzig selector. Ann. Statist. 37 1705–1732. 1173.62022 10.1214/08-AOS620 euclid.aos/1245332830

11.

Blinowska, K. J. (2011). Review of the methods of determination of directed connectivity from multichannel data. Med. Biol. Eng. Comput. 49 521–529.Blinowska, K. J. (2011). Review of the methods of determination of directed connectivity from multichannel data. Med. Biol. Eng. Comput. 49 521–529.

12.

Brillinger, D. R. (1975). Time Series: Data Analysis and Theory. Holt, Rinehart, and Winston, New York. 0321.62004Brillinger, D. R. (1975). Time Series: Data Analysis and Theory. Holt, Rinehart, and Winston, New York. 0321.62004

13.

Brillinger, D. R. (1996). Remarks concerning graphical models for time series and point processes. Revista de Econometria 16 23.Brillinger, D. R. (1996). Remarks concerning graphical models for time series and point processes. Revista de Econometria 16 23.

14.

Brockwell, P. J. and Davis, R. A. (1991). Time Series: Theory and Methods, 2nd ed. Springer Series in Statistics. Springer, New York. 0709.62080Brockwell, P. J. and Davis, R. A. (1991). Time Series: Theory and Methods, 2nd ed. Springer Series in Statistics. Springer, New York. 0709.62080

15.

Bryc, W. and Dembo, A. (1997). Large deviations for quadratic functionals of Gaussian processes. J. Theoret. Probab. 10 307–332. 0894.60026 10.1023/A:1022656331883Bryc, W. and Dembo, A. (1997). Large deviations for quadratic functionals of Gaussian processes. J. Theoret. Probab. 10 307–332. 0894.60026 10.1023/A:1022656331883

16.

Bühlmann, P. (2002). Bootstraps for time series. Statist. Sci. 17 52–72.Bühlmann, P. (2002). Bootstraps for time series. Statist. Sci. 17 52–72.

17.

Cai, T., Liu, W. and Luo, X. (2011). A constrained $\ell_{1}$ minimization approach to sparse precision matrix estimation. J. Amer. Statist. Assoc. 106 594–607. 1232.62087 10.1198/jasa.2011.tm10155Cai, T., Liu, W. and Luo, X. (2011). A constrained $\ell_{1}$ minimization approach to sparse precision matrix estimation. J. Amer. Statist. Assoc. 106 594–607. 1232.62087 10.1198/jasa.2011.tm10155

18.

Candes, E. and Tao, T. (2007). The Dantzig selector: Statistical estimation when $p$ is much larger than $n$. Ann. Statist. 35 2313–2351. 1139.62019 euclid.aos/1201012958Candes, E. and Tao, T. (2007). The Dantzig selector: Statistical estimation when $p$ is much larger than $n$. Ann. Statist. 35 2313–2351. 1139.62019 euclid.aos/1201012958

19.

Chen, X., Xu, M. and Wu, W. B. (2013). Covariance and precision matrix estimation for high-dimensional time series. Ann. Statist. 41 2994–3021. 1294.62123 10.1214/13-AOS1182 euclid.aos/1388545676Chen, X., Xu, M. and Wu, W. B. (2013). Covariance and precision matrix estimation for high-dimensional time series. Ann. Statist. 41 2994–3021. 1294.62123 10.1214/13-AOS1182 euclid.aos/1388545676

20.

Dahlhaus, R. (1997). Fitting time series models to nonstationary processes. Ann. Statist. 25 1–37. 0871.62080 10.1214/aos/1034276620 euclid.aos/1034276620Dahlhaus, R. (1997). Fitting time series models to nonstationary processes. Ann. Statist. 25 1–37. 0871.62080 10.1214/aos/1034276620 euclid.aos/1034276620

21.

Dahlhaus, R. (2000a). A likelihood approximation for locally stationary processes. Ann. Statist. 28 1762–1794. 1010.62078 10.1214/aos/1015957480 euclid.aos/1015957480Dahlhaus, R. (2000a). A likelihood approximation for locally stationary processes. Ann. Statist. 28 1762–1794. 1010.62078 10.1214/aos/1015957480 euclid.aos/1015957480

22.

Dahlhaus, R. (2000b). Graphical interaction models for multivariate time series. Metrika 51 157–172. 1093.62571 10.1007/s001840000055Dahlhaus, R. (2000b). Graphical interaction models for multivariate time series. Metrika 51 157–172. 1093.62571 10.1007/s001840000055

23.

Dahlhaus, R. (2012). Locally stationary processes. Handb. Statist. 30 351–412.Dahlhaus, R. (2012). Locally stationary processes. Handb. Statist. 30 351–412.

24.

Dahlhaus, R. and Polonik, W. (2009). Empirical spectral processes for locally stationary time series. Bernoulli 15 1–39. 1204.62156 10.3150/08-BEJ137 euclid.bj/1233669881Dahlhaus, R. and Polonik, W. (2009). Empirical spectral processes for locally stationary time series. Bernoulli 15 1–39. 1204.62156 10.3150/08-BEJ137 euclid.bj/1233669881

25.

Dahlhaus, R., Richter, S. and Wu, W. B. (2019). Towards a general theory for nonlinear locally stationary processes. Bernoulli 25 1013–1044. 1427.60057 10.3150/17-BEJ1011 euclid.bj/1551862842Dahlhaus, R., Richter, S. and Wu, W. B. (2019). Towards a general theory for nonlinear locally stationary processes. Bernoulli 25 1013–1044. 1427.60057 10.3150/17-BEJ1011 euclid.bj/1551862842

26.

Dahlhaus, R. and Subba Rao, S. (2006). Statistical inference for time-varying ARCH processes. Ann. Statist. 34 1075–1114. 1113.62099 10.1214/009053606000000227 euclid.aos/1152540743Dahlhaus, R. and Subba Rao, S. (2006). Statistical inference for time-varying ARCH processes. Ann. Statist. 34 1075–1114. 1113.62099 10.1214/009053606000000227 euclid.aos/1152540743

27.

Dahlhaus, R. and Subba Rao, S. (2007). A recursive online algorithm for the estimation of time-varying ARCH parameters. Bernoulli 13 389–422. 1127.62078 10.3150/07-BEJ5009 euclid.bj/1179498754Dahlhaus, R. and Subba Rao, S. (2007). A recursive online algorithm for the estimation of time-varying ARCH parameters. Bernoulli 13 389–422. 1127.62078 10.3150/07-BEJ5009 euclid.bj/1179498754

28.

Eichler, M. (2007). Granger causality and path diagrams for multivariate time series. J. Econometrics 137 334–353. 1360.62455 10.1016/j.jeconom.2005.06.032Eichler, M. (2007). Granger causality and path diagrams for multivariate time series. J. Econometrics 137 334–353. 1360.62455 10.1016/j.jeconom.2005.06.032

29.

Eichler, M. (2012). Graphical modelling of multivariate time series. Probab. Theory Related Fields 153 233–268. 1316.60049 10.1007/s00440-011-0345-8Eichler, M. (2012). Graphical modelling of multivariate time series. Probab. Theory Related Fields 153 233–268. 1316.60049 10.1007/s00440-011-0345-8

30.

Eichler, M., Dahlhaus, R. and Sandkühler, J. (2003). Partial correlation analysis for the identification of synaptic connections. Biol. Cybernet. 89 289–302. 1105.92311 10.1007/s00422-003-0400-3Eichler, M., Dahlhaus, R. and Sandkühler, J. (2003). Partial correlation analysis for the identification of synaptic connections. Biol. Cybernet. 89 289–302. 1105.92311 10.1007/s00422-003-0400-3

31.

Fiecas, M., Leng, C., Liu, W. and Yu, Y. (2019). Spectral analysis of high-dimensional time series. Electron. J. Stat. 13 4079–4101. 1431.62369Fiecas, M., Leng, C., Liu, W. and Yu, Y. (2019). Spectral analysis of high-dimensional time series. Electron. J. Stat. 13 4079–4101. 1431.62369

32.

Forni, M., Hallin, M., Lippi, M. and Reichlin, L. (2000). The generalized dynamic-factor model: Identification and estimation. Rev. Econ. Stat. 82 540–554. 1117.62334 10.1198/016214504000002050Forni, M., Hallin, M., Lippi, M. and Reichlin, L. (2000). The generalized dynamic-factor model: Identification and estimation. Rev. Econ. Stat. 82 540–554. 1117.62334 10.1198/016214504000002050

33.

Fried, R. and Didelez, V. (2003). Decomposability and selection of graphical models for multivariate time series. Biometrika 90 251–267. 1036.62072 10.1093/biomet/90.2.251Fried, R. and Didelez, V. (2003). Decomposability and selection of graphical models for multivariate time series. Biometrika 90 251–267. 1036.62072 10.1093/biomet/90.2.251

34.

Fryzlewicz, P. and Subba Rao, S. (2011). Mixing properties of ARCH and time-varying ARCH processes. Bernoulli 17 320–346. 1284.62550 10.3150/10-BEJ270 euclid.bj/1297173845Fryzlewicz, P. and Subba Rao, S. (2011). Mixing properties of ARCH and time-varying ARCH processes. Bernoulli 17 320–346. 1284.62550 10.3150/10-BEJ270 euclid.bj/1297173845

35.

Gather, U., Imhoff, M. and Fried, R. (2002). Graphical models for multivariate time series from intensive care monitoring. Stat. Med. 21 2685–2701.Gather, U., Imhoff, M. and Fried, R. (2002). Graphical models for multivariate time series from intensive care monitoring. Stat. Med. 21 2685–2701.

36.

Giurcanu, M. and Spokoiny, V. (2004). Confidence estimation of the covariance function of stationary and locally stationary processes. Statist. Decisions 22 283–300. MR2158265 1063.62118 10.1524/stnd.22.4.283.64315Giurcanu, M. and Spokoiny, V. (2004). Confidence estimation of the covariance function of stationary and locally stationary processes. Statist. Decisions 22 283–300. MR2158265 1063.62118 10.1524/stnd.22.4.283.64315

37.

Grenier, Y. (1983). Time-dependent ARMA modeling of nonstationary signals. IEEE Trans. Acoust. Speech Signal Process. 31 899–911.Grenier, Y. (1983). Time-dependent ARMA modeling of nonstationary signals. IEEE Trans. Acoust. Speech Signal Process. 31 899–911.

38.

Hafner, C. M. and Linton, O. (2010). Efficient estimation of a multivariate multiplicative volatility model. J. Econometrics 159 55–73. 1431.62381 10.1016/j.jeconom.2010.04.007Hafner, C. M. and Linton, O. (2010). Efficient estimation of a multivariate multiplicative volatility model. J. Econometrics 159 55–73. 1431.62381 10.1016/j.jeconom.2010.04.007

39.

Hannan, E. J. (1974). The uniform convergence of autocovariances. Ann. Statist. 2 803–806. 0284.62061 10.1214/aos/1176342767 euclid.aos/1176342767Hannan, E. J. (1974). The uniform convergence of autocovariances. Ann. Statist. 2 803–806. 0284.62061 10.1214/aos/1176342767 euclid.aos/1176342767

40.

Hannan, E. J. and Deistler, M. (1988). The Statistical Theory of Linear Systems. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. Wiley, New York. MR940698 0641.93002Hannan, E. J. and Deistler, M. (1988). The Statistical Theory of Linear Systems. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. Wiley, New York. MR940698 0641.93002

41.

Hanson, D. L. and Wright, F. T. (1971). A bound on tail probabilities for quadratic forms in independent random variables. Ann. Math. Stat. 42 1079–1083. 0216.22203 euclid.aoms/1177693335Hanson, D. L. and Wright, F. T. (1971). A bound on tail probabilities for quadratic forms in independent random variables. Ann. Math. Stat. 42 1079–1083. 0216.22203 euclid.aoms/1177693335

42.

Hooten, M. B. and Wikle, C. K. (2008). A hierarchical Bayesian non-linear spatio-temporal model for the spread of invasive species with application to the Eurasian collared-dove. Environ. Ecol. Stat. 15 59–70.Hooten, M. B. and Wikle, C. K. (2008). A hierarchical Bayesian non-linear spatio-temporal model for the spread of invasive species with application to the Eurasian collared-dove. Environ. Ecol. Stat. 15 59–70.

43.

Jacquier, E., Polson, N. G. and Rossi, P. E. (2004). Bayesian analysis of stochastic volatility models with fat-tails and correlated errors. J. Econometrics 122 185–212. 1328.91254 10.1016/j.jeconom.2003.09.001Jacquier, E., Polson, N. G. and Rossi, P. E. (2004). Bayesian analysis of stochastic volatility models with fat-tails and correlated errors. J. Econometrics 122 185–212. 1328.91254 10.1016/j.jeconom.2003.09.001

44.

Jirak, M. (2011). On the maximum of covariance estimators. J. Multivariate Anal. 102 1032–1046. 1274.60068 10.1016/j.jmva.2011.02.003Jirak, M. (2011). On the maximum of covariance estimators. J. Multivariate Anal. 102 1032–1046. 1274.60068 10.1016/j.jmva.2011.02.003

45.

Kakizawa, Y. (2007). Moderate deviations for quadratic forms in Gaussian stationary processes. J. Multivariate Anal. 98 992–1017. 1118.60020 10.1016/j.jmva.2006.07.004Kakizawa, Y. (2007). Moderate deviations for quadratic forms in Gaussian stationary processes. J. Multivariate Anal. 98 992–1017. 1118.60020 10.1016/j.jmva.2006.07.004

46.

Kondrashov, D., Kravtsov, S., Robertson, A. W. and Ghil, M. (2005). A hierarchy of data-based ENSO models. J. Climate 18 4425–4444.Kondrashov, D., Kravtsov, S., Robertson, A. W. and Ghil, M. (2005). A hierarchy of data-based ENSO models. J. Climate 18 4425–4444.

47.

Lahiri, S. N. (2003). Resampling Methods for Dependent Data. Springer Series in Statistics. Springer, New York. 1028.62002Lahiri, S. N. (2003). Resampling Methods for Dependent Data. Springer Series in Statistics. Springer, New York. 1028.62002

48.

Lennartz, C., Schiefer, J., Rotter, S., Hennig, J. and LeVan, P. (2018). Sparse estimation of resting-state effective connectivity from fMRI cross-spectra. Front. Neurosci. 12 287.Lennartz, C., Schiefer, J., Rotter, S., Hennig, J. and LeVan, P. (2018). Sparse estimation of resting-state effective connectivity from fMRI cross-spectra. Front. Neurosci. 12 287.

49.

Lindquist, M. A., Xu, Y., Nebel, M. B. and Caffo, B. S. (2014). Evaluating dynamic bivariate correlations in resting-state fMRI: A comparison study and a new approach. NeuroImage 101 531–546.Lindquist, M. A., Xu, Y., Nebel, M. B. and Caffo, B. S. (2014). Evaluating dynamic bivariate correlations in resting-state fMRI: A comparison study and a new approach. NeuroImage 101 531–546.

50.

Liu, C., Gaetz, W. and Zhu, H. (2010). Estimation of time-varying coherence and its application in understanding brain functional connectivity. EURASIP Journal on Advances in Signal Processing 2010.Liu, C., Gaetz, W. and Zhu, H. (2010). Estimation of time-varying coherence and its application in understanding brain functional connectivity. EURASIP Journal on Advances in Signal Processing 2010.

51.

Majda, A. J., Abramov, R. V. and Grote, M. J. (2005). Information Theory and Stochastics for Multiscale Nonlinear Systems. CRM Monograph Series 25. Amer. Math. Soc., Providence, RI. 1082.60002Majda, A. J., Abramov, R. V. and Grote, M. J. (2005). Information Theory and Stochastics for Multiscale Nonlinear Systems. CRM Monograph Series 25. Amer. Math. Soc., Providence, RI. 1082.60002

52.

Mallat, S., Papanicolaou, G. and Zhang, Z. (1998). Adaptive covariance estimation of locally stationary processes. Ann. Statist. 26 1–47. 0949.62082 10.1214/aos/1030563977 euclid.aos/1030563977Mallat, S., Papanicolaou, G. and Zhang, Z. (1998). Adaptive covariance estimation of locally stationary processes. Ann. Statist. 26 1–47. 0949.62082 10.1214/aos/1030563977 euclid.aos/1030563977

53.

Medkour, T., Walden, A. T. and Burgess, A. (2009). Graphical modelling for brain connectivity via partial coherence. J. Neurosci. Methods 180 374–383.Medkour, T., Walden, A. T. and Burgess, A. (2009). Graphical modelling for brain connectivity via partial coherence. J. Neurosci. Methods 180 374–383.

54.

Moulines, E., Priouret, P. and Roueff, F. (2005). On recursive estimation for time varying autoregressive processes. Ann. Statist. 33 2610–2654. 1084.62089 10.1214/009053605000000624 euclid.aos/1140191668Moulines, E., Priouret, P. and Roueff, F. (2005). On recursive estimation for time varying autoregressive processes. Ann. Statist. 33 2610–2654. 1084.62089 10.1214/009053605000000624 euclid.aos/1140191668

55.

Nason, G. P., von Sachs, R. and Kroisandt, G. (2000). Wavelet processes and adaptive estimation of the evolutionary wavelet spectrum. J. R. Stat. Soc. Ser. B. Stat. Methodol. 62 271–292.Nason, G. P., von Sachs, R. and Kroisandt, G. (2000). Wavelet processes and adaptive estimation of the evolutionary wavelet spectrum. J. R. Stat. Soc. Ser. B. Stat. Methodol. 62 271–292.

56.

Newey, W. K. and West, K. D. (1987). A simple, positive semidefinite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica 55 703–708. 0658.62139 10.2307/1913610Newey, W. K. and West, K. D. (1987). A simple, positive semidefinite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica 55 703–708. 0658.62139 10.2307/1913610

57.

Ombao, H. and Van Bellegem, S. (2008). Evolutionary coherence of nonstationary signals. IEEE Trans. Signal Process. 56 2259–2266. 1390.94344 10.1109/TSP.2007.914341Ombao, H. and Van Bellegem, S. (2008). Evolutionary coherence of nonstationary signals. IEEE Trans. Signal Process. 56 2259–2266. 1390.94344 10.1109/TSP.2007.914341

58.

Ombao, H., von Sachs, R. and Guo, W. (2005). SLEX analysis of multivariate nonstationary time series. J. Amer. Statist. Assoc. 100 519–531. 1117.62407 10.1198/016214504000001448Ombao, H., von Sachs, R. and Guo, W. (2005). SLEX analysis of multivariate nonstationary time series. J. Amer. Statist. Assoc. 100 519–531. 1117.62407 10.1198/016214504000001448

59.

Ombao, H. C., Raz, J. A., von Sachs, R. and Malow, B. A. (2001). Automatic statistical analysis of bivariate nonstationary time series. J. Amer. Statist. Assoc. 96 543–560. 1018.62080 10.1198/016214501753168244Ombao, H. C., Raz, J. A., von Sachs, R. and Malow, B. A. (2001). Automatic statistical analysis of bivariate nonstationary time series. J. Amer. Statist. Assoc. 96 543–560. 1018.62080 10.1198/016214501753168244

60.

Paparoditis, E. and Politis, D. N. (2012). Nonlinear spectral density estimation: Thresholding the correlogram. J. Time Series Anal. 33 386–397. 1301.62041Paparoditis, E. and Politis, D. N. (2012). Nonlinear spectral density estimation: Thresholding the correlogram. J. Time Series Anal. 33 386–397. 1301.62041

61.

Park, T., Eckley, I. A. and Ombao, H. C. (2014). Estimating time-evolving partial coherence between signals via multivariate locally stationary wavelet processes. IEEE Trans. Signal Process. 62 5240–5250. 1394.94446 10.1109/TSP.2014.2343937Park, T., Eckley, I. A. and Ombao, H. C. (2014). Estimating time-evolving partial coherence between signals via multivariate locally stationary wavelet processes. IEEE Trans. Signal Process. 62 5240–5250. 1394.94446 10.1109/TSP.2014.2343937

62.

Politis, D. N. (2011). Higher-order accurate, positive semidefinite estimation of large-sample covariance and spectral density matrices. Econometric Theory 27 703–744. 1219.62144 10.1017/S0266466610000484Politis, D. N. (2011). Higher-order accurate, positive semidefinite estimation of large-sample covariance and spectral density matrices. Econometric Theory 27 703–744. 1219.62144 10.1017/S0266466610000484

63.

Politis, D. N. and Romano, J. P. (1995). Bias-corrected nonparametric spectral estimation. J. Time Series Anal. 16 67–103. 0811.62088 10.1111/j.1467-9892.1995.tb00223.xPolitis, D. N. and Romano, J. P. (1995). Bias-corrected nonparametric spectral estimation. J. Time Series Anal. 16 67–103. 0811.62088 10.1111/j.1467-9892.1995.tb00223.x

64.

Politis, D. N. and Romano, J. P. (1999). Multivariate density estimation with general flat-top kernels of infinite order. J. Multivariate Anal. 68 1–25. 0954.62042 10.1006/jmva.1998.1774Politis, D. N. and Romano, J. P. (1999). Multivariate density estimation with general flat-top kernels of infinite order. J. Multivariate Anal. 68 1–25. 0954.62042 10.1006/jmva.1998.1774

65.

Politis, D. N., Romano, J. P. and Wolf, M. (1999). Subsampling. Springer Series in Statistics. Springer, New York.Politis, D. N., Romano, J. P. and Wolf, M. (1999). Subsampling. Springer Series in Statistics. Springer, New York.

66.

Prado, R., West, M. and Krystal, A. D. (2001). Multichannel electroencephalographic analyses via dynamic regression models with time-varying lag–lead structure. J. R. Stat. Soc. Ser. C. Appl. Stat. 50 95–109.Prado, R., West, M. and Krystal, A. D. (2001). Multichannel electroencephalographic analyses via dynamic regression models with time-varying lag–lead structure. J. R. Stat. Soc. Ser. C. Appl. Stat. 50 95–109.

67.

Priestley, M. B. (1981). Spectral Analysis and Time Series. Vol. 1. Academic Press, London. 0537.62075Priestley, M. B. (1981). Spectral Analysis and Time Series. Vol. 1. Academic Press, London. 0537.62075

68.

Priestley, M. B. (1988). Nonlinear and Nonstationary Time Series Analysis. Academic Press, London.Priestley, M. B. (1988). Nonlinear and Nonstationary Time Series Analysis. Academic Press, London.

69.

Priestley, M. B. and Tong, H. (1973). On the analysis of bivariate non-stationary processes. J. Roy. Statist. Soc. Ser. B 35 153–166, 179–188. 0269.62077 10.1111/j.2517-6161.1973.tb00949.xPriestley, M. B. and Tong, H. (1973). On the analysis of bivariate non-stationary processes. J. Roy. Statist. Soc. Ser. B 35 153–166, 179–188. 0269.62077 10.1111/j.2517-6161.1973.tb00949.x

70.

Rosenblatt, M. (1971). Markov Processes. Structure and Asymptotic Behavior. Springer, New York. 0236.60002Rosenblatt, M. (1971). Markov Processes. Structure and Asymptotic Behavior. Springer, New York. 0236.60002

71.

Rosenblatt, M. (1985). Stationary Sequences and Random Fields. Birkhäuser, Inc., Boston, MA.Rosenblatt, M. (1985). Stationary Sequences and Random Fields. Birkhäuser, Inc., Boston, MA.

72.

Rudelson, M. and Vershynin, R. (2013). Hanson–Wright inequality and sub-Gaussian concentration. Electron. Commun. Probab. 18 no. 82, 9. 1329.60056 10.1214/ECP.v18-2865Rudelson, M. and Vershynin, R. (2013). Hanson–Wright inequality and sub-Gaussian concentration. Electron. Commun. Probab. 18 no. 82, 9. 1329.60056 10.1214/ECP.v18-2865

73.

Salvador, R., Suckling, J., Schwarzbauer, C. and Bullmore, E. (2005). Undirected graphs of frequency-dependent functional connectivity in whole brain networks. Philos. Trans. R. Soc. Lond. B, Biol. Sci. 360 937–946.Salvador, R., Suckling, J., Schwarzbauer, C. and Bullmore, E. (2005). Undirected graphs of frequency-dependent functional connectivity in whole brain networks. Philos. Trans. R. Soc. Lond. B, Biol. Sci. 360 937–946.

74.

Sanderson, J., Fryzlewicz, P. and Jones, M. W. (2010). Estimating linear dependence between nonstationary time series using the locally stationary wavelet model. Biometrika 97 435–446. 1406.62100 10.1093/biomet/asq007Sanderson, J., Fryzlewicz, P. and Jones, M. W. (2010). Estimating linear dependence between nonstationary time series using the locally stationary wavelet model. Biometrika 97 435–446. 1406.62100 10.1093/biomet/asq007

75.

Simpson, S. L., Bowman, F. D. and Laurienti, P. J. (2013). Analyzing complex functional brain networks: Fusing statistics and network science to understand the brain. Stat. Surv. 7 1–36. 1279.92021 10.1214/13-SS103Simpson, S. L., Bowman, F. D. and Laurienti, P. J. (2013). Analyzing complex functional brain networks: Fusing statistics and network science to understand the brain. Stat. Surv. 7 1–36. 1279.92021 10.1214/13-SS103

76.

Subba Rao, T. (1970). The fitting of non-stationary time-series models with time-dependent parameters. J. Roy. Statist. Soc. Ser. B 32 312–322. 0225.62109 10.1111/j.2517-6161.1970.tb00844.xSubba Rao, T. (1970). The fitting of non-stationary time-series models with time-dependent parameters. J. Roy. Statist. Soc. Ser. B 32 312–322. 0225.62109 10.1111/j.2517-6161.1970.tb00844.x

77.

Sun, Y., Li, Y., Kuceyeski, A. and Basu, S. (2018). Large spectral density matrix estimation by thresholding. Preprint. Available at  arXiv:1812.005321812.00532Sun, Y., Li, Y., Kuceyeski, A. and Basu, S. (2018). Large spectral density matrix estimation by thresholding. Preprint. Available at  arXiv:1812.005321812.00532

78.

Timmer, J., Lauk, M., Häußler, S., Radt, V., Köster, B., Hellwig, B., Guschlbauer, B., Lücking, C. H., Eichler, M. et al. (2000). Cross-spectral analysis of tremor time series. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 10 2595–2610. 0967.92010 10.1142/S0218127400001663Timmer, J., Lauk, M., Häußler, S., Radt, V., Köster, B., Hellwig, B., Guschlbauer, B., Lücking, C. H., Eichler, M. et al. (2000). Cross-spectral analysis of tremor time series. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 10 2595–2610. 0967.92010 10.1142/S0218127400001663

79.

Tong, H. (1990). Nonlinear Time Series: A Dynamical System Approach. Oxford Statistical Science Series 6. The Clarendon Press, Oxford University Press, New York. 0716.62085Tong, H. (1990). Nonlinear Time Series: A Dynamical System Approach. Oxford Statistical Science Series 6. The Clarendon Press, Oxford University Press, New York. 0716.62085

80.

Tsay, R. S. (2005). Analysis of Financial Time Series, 2nd ed. Wiley Series in Probability and Statistics. Wiley Interscience, Hoboken, NJ.Tsay, R. S. (2005). Analysis of Financial Time Series, 2nd ed. Wiley Series in Probability and Statistics. Wiley Interscience, Hoboken, NJ.

81.

Vogt, M. (2012). Nonparametric regression for locally stationary time series. Ann. Statist. 40 2601–2633. 1373.62459 10.1214/12-AOS1043 euclid.aos/1359987532Vogt, M. (2012). Nonparametric regression for locally stationary time series. Ann. Statist. 40 2601–2633. 1373.62459 10.1214/12-AOS1043 euclid.aos/1359987532

82.

Wiener, N. (1958). Nonlinear Problems in Random Theory. Technology Press Research Monographs. Wiley, New York. 0121.12302Wiener, N. (1958). Nonlinear Problems in Random Theory. Technology Press Research Monographs. Wiley, New York. 0121.12302

83.

Wikle, C. K. and Hooten, M. B. (2010). A general science-based framework for dynamical spatio-temporal models. TEST 19 417–451. 1203.37141 10.1007/s11749-010-0209-zWikle, C. K. and Hooten, M. B. (2010). A general science-based framework for dynamical spatio-temporal models. TEST 19 417–451. 1203.37141 10.1007/s11749-010-0209-z

84.

Wright, F. T. (1973). A bound on tail probabilities for quadratic forms in independent random variables whose distributions are not necessarily symmetric. Ann. Probab. 1 1068–1070. 0271.60033 10.1214/aop/1176996815 euclid.aop/1176996815Wright, F. T. (1973). A bound on tail probabilities for quadratic forms in independent random variables whose distributions are not necessarily symmetric. Ann. Probab. 1 1068–1070. 0271.60033 10.1214/aop/1176996815 euclid.aop/1176996815

85.

Wu, W. B. (2005). Nonlinear system theory: Another look at dependence. Proc. Natl. Acad. Sci. USA 102 14150–14154. 1135.62075 10.1073/pnas.0506715102Wu, W. B. (2005). Nonlinear system theory: Another look at dependence. Proc. Natl. Acad. Sci. USA 102 14150–14154. 1135.62075 10.1073/pnas.0506715102

86.

Wu, W. B. and Shao, X. (2004). Limit theorems for iterated random functions. J. Appl. Probab. 41 425–436. 1046.60024 10.1239/jap/1082999076 euclid.jap/1082999076Wu, W. B. and Shao, X. (2004). Limit theorems for iterated random functions. J. Appl. Probab. 41 425–436. 1046.60024 10.1239/jap/1082999076 euclid.jap/1082999076

87.

Xiao, H. and Wu, W. B. (2012). Covariance matrix estimation for stationary time series. Ann. Statist. 40 466–493. 1246.62191 10.1214/11-AOS967 euclid.aos/1334581750Xiao, H. and Wu, W. B. (2012). Covariance matrix estimation for stationary time series. Ann. Statist. 40 466–493. 1246.62191 10.1214/11-AOS967 euclid.aos/1334581750

88.

Xiao, H. and Wu, W. B. (2014). Portmanteau test and simultaneous inference for serial covariances. Statist. Sinica 24 577–599. 06290111Xiao, H. and Wu, W. B. (2014). Portmanteau test and simultaneous inference for serial covariances. Statist. Sinica 24 577–599. 06290111

89.

Zani, M. (2002). Large deviations for quadratic forms of locally stationary processes. J. Multivariate Anal. 81 205–228. 1130.60307 10.1006/jmva.2001.2003Zani, M. (2002). Large deviations for quadratic forms of locally stationary processes. J. Multivariate Anal. 81 205–228. 1130.60307 10.1006/jmva.2001.2003

90.

Zhang, D. and Wu, W. B. (2017). Gaussian approximation for high dimensional time series. Ann. Statist. 45 1895–1919. 1381.62254 10.1214/16-AOS1512 euclid.aos/1509436822Zhang, D. and Wu, W. B. (2017). Gaussian approximation for high dimensional time series. Ann. Statist. 45 1895–1919. 1381.62254 10.1214/16-AOS1512 euclid.aos/1509436822

91.

Zhang, D. and Wu, W. B (2021). Supplement to “Convergence of covariance and spectral density estimates for high-dimensional locally stationary processes.”  https://doi.org/10.1214/20-AOS1954SUPP.Zhang, D. and Wu, W. B (2021). Supplement to “Convergence of covariance and spectral density estimates for high-dimensional locally stationary processes.”  https://doi.org/10.1214/20-AOS1954SUPP.

92.

Zhou, Z. (2010). Nonparametric inference of quantile curves for nonstationary time series. Ann. Statist. 38 2187–2217. 1202.62062 10.1214/09-AOS769 euclid.aos/1278861246Zhou, Z. (2010). Nonparametric inference of quantile curves for nonstationary time series. Ann. Statist. 38 2187–2217. 1202.62062 10.1214/09-AOS769 euclid.aos/1278861246

93.

Zhou, Z. and Wu, W. B. (2009). Local linear quantile estimation for nonstationary time series. Ann. Statist. 37 2696–2729. 1173.62066 10.1214/08-AOS636 euclid.aos/1247836666Zhou, Z. and Wu, W. B. (2009). Local linear quantile estimation for nonstationary time series. Ann. Statist. 37 2696–2729. 1173.62066 10.1214/08-AOS636 euclid.aos/1247836666
Copyright © 2021 Institute of Mathematical Statistics
Danna Zhang and Wei Biao Wu "Convergence of covariance and spectral density estimates for high-dimensional locally stationary processes," The Annals of Statistics 49(1), 233-254, (February 2021). https://doi.org/10.1214/20-AOS1954
Received: 1 November 2018; Published: February 2021
Vol.49 • No. 1 • February 2021
Back to Top