The Annals of Statistics

Cross validation for locally stationary processes

Stefan Richter and Rainer Dahlhaus

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Abstract

We propose an adaptive bandwidth selector via cross validation for local M-estimators in locally stationary processes. We prove asymptotic optimality of the procedure under mild conditions on the underlying parameter curves. The results are applicable to a wide range of locally stationary processes such linear and nonlinear processes. A simulation study shows that the method works fairly well also in misspecified situations.

Article information

Source
Ann. Statist., Volume 47, Number 4 (2019), 2145-2173.

Dates
Received: May 2017
Revised: February 2018
First available in Project Euclid: 22 May 2019

Permanent link to this document
https://projecteuclid.org/euclid.aos/1558512018

Digital Object Identifier
doi:10.1214/18-AOS1743

Mathematical Reviews number (MathSciNet)
MR3953447

Zentralblatt MATH identifier
07082282

Subjects
Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]
Secondary: 62G20: Asymptotic properties

Keywords
Locally stationary processes cross validation adaptive bandwidth selection asymptotic optimality

Citation

Richter, Stefan; Dahlhaus, Rainer. Cross validation for locally stationary processes. Ann. Statist. 47 (2019), no. 4, 2145--2173. doi:10.1214/18-AOS1743. https://projecteuclid.org/euclid.aos/1558512018


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References

  • [1] Arkoun, O. (2011). Sequential adaptive estimators in nonparametric autoregressive models. Sequential Anal. 30 229–247.
  • [2] Arkoun, O. and Pergamenchtchikov, S. (2016). Sequential robust estimation for nonparametric autoregressive models. Sequential Anal. 35 489–515.
  • [3] Chiu, S.-T. (1991). Bandwidth selection for kernel density estimation. Ann. Statist. 19 1883–1905.
  • [4] Dahlhaus, R. (2012). 13—Locally stationary processes. In Time Series Analysis: Methods and Applications (T. S. Rao, S. S. Rao and C. R. Rao, eds.). Handbook of Statistics 30 351–413. Elsevier, Amsterdam. DOI:10.1016/B978-0-444-53858-1.00013-2.
  • [5] Dahlhaus, R. and Polonik, W. (2009). Empirical spectral processes for locally stationary time series. Bernoulli 15 1–39.
  • [6] Dahlhaus, R., Richter, S. and Wu, W. B. (2018). Towards a general theory for non-linear locally stationary processes. Bernoulli. To appear.
  • [7] Dahlhaus, R. and Subba Rao, S. (2006). Statistical inference for time-varying ARCH processes. Ann. Statist. 34 1075–1114.
  • [8] Fryzlewicz, P., Sapatinas, T. and Subba Rao, S. (2008). Normalized least-squares estimation in time-varying ARCH models. Ann. Statist. 36 742–786.
  • [9] Giraud, C., Roueff, F. and Sanchez-Perez, A. (2015). Aggregation of predictors for nonstationary sub-linear processes and online adaptive forecasting of time varying autoregressive processes. Ann. Statist. 43 2412–2450.
  • [10] Hall, P., Marron, J. S. and Park, B. U. (1992). Smoothed cross-validation. Probab. Theory Related Fields 92 1–20.
  • [11] Härdle, W., Hall, P. and Marron, J. S. (1988). How far are automatically chosen regression smoothing parameters from their optimum? J. Amer. Statist. Assoc. 83 86–101.
  • [12] Härdle, W. and Marron, J. S. (1985). Optimal bandwidth selection in nonparametric regression function estimation. Ann. Statist. 13 1465–1481.
  • [13] Karmakar, S., Richter, S. and Wu, W. B. (2018). Bahadur representation and simultaneous inference for time-varying models. Technical report.
  • [14] Mallat, S., Papanicolaou, G. and Zhang, Z. (1998). Adaptive covariance estimation of locally stationary processes. Ann. Statist. 26 1–47.
  • [15] Richter, S. and Dahlhaus, R. (2019). Supplement to “Cross validation for locally stationary processes.” DOI:10.1214/18-AOS1743SUPP.
  • [16] Subba Rao, S. (2006). On some nonstationary, nonlinear random processes and their stationary approximations. Adv. in Appl. Probab. 38 1155–1172.
  • [17] Wu, W. B. (2005). Nonlinear system theory: Another look at dependence. Proc. Natl. Acad. Sci. USA 102 14150–14154.
  • [18] Zhou, Z. and Wu, W. B. (2009). Local linear quantile estimation for nonstationary time series. Ann. Statist. 37 2696–2729.

Supplemental materials

  • Supplement: Technical proofs. This material contains some details of the proofs in the paper as well as the proofs of the examples.