Electronic Journal of Statistics

Non-parametric estimation of time varying AR(1)–processes with local stationarity and periodicity

Jean-Marc Bardet and Paul Doukhan

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Extending the ideas of [7], this paper aims at providing a kernel based non-parametric estimation of a new class of time varying AR(1) processes $(X_{t})$, with local stationarity and periodic features (with a known period $T$), inducing the definition $X_{t}=a_{t}(t/nT)X_{t-1}+\xi_{t}$ for $t\in \mathbb{N}$ and with $a_{t+T}\equiv a_{t}$. Central limit theorems are established for kernel estimators $\widehat{a}_{s}(u)$ reaching classical minimax rates and only requiring low order moment conditions of the white noise $(\xi_{t})_{t}$ up to the second order.

Article information

Electron. J. Statist., Volume 12, Number 2 (2018), 2323-2354.

Received: February 2018
First available in Project Euclid: 25 July 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]
Secondary: 60F05: Central limit and other weak theorems

Local stationarity nonparametric estimation central limit theorem

Creative Commons Attribution 4.0 International License.


Bardet, Jean-Marc; Doukhan, Paul. Non-parametric estimation of time varying AR(1)–processes with local stationarity and periodicity. Electron. J. Statist. 12 (2018), no. 2, 2323--2354. doi:10.1214/18-EJS1459. https://projecteuclid.org/euclid.ejs/1532484332

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  • [1] Andrews, D. Laws of large numbers for dependent non-identically distributed random variables., Econometric Theory 4, 3 (1988), 458–467.
  • [2] Azrak, R. and Mélard, G. Asymptotic properties of quasi-maximum likelihood estimators for ARMA models with time-dependent coefficients., Statistical Inference for Stochastic Processes 9 (2006), 279–330.
  • [3] Bibi, A. and Francq, C. Consistent and asymptotically normal estimators for cyclically time-dependent linear models., Annals of the Institute of Statistical Mathematics 55 (2003), 41–68.
  • [4] Dacunha-Castelle, D., Huong Hoang, H. T. and Parey, S. Modeling of air temperatures: preprocessing and trends, reduced stationary process, extremes, simulation., Journal de la Société Française de Statistique 156, 2 (2015), 138–168.
  • [5] Dahlhaus, R. On the Kullback-Leibler information divergence of locally stationary processes., Stochastic Processes and Applications 62 (1996), 139–168.
  • [6] Dahlhaus, R. Fitting time series models to nonstationary processes., Annals of Statistics 25 (1997), 1–37.
  • [7] Dahlhaus, R., Locally Stationary Processes, vol. 30. Time Series Analysis: Methods and Applications, Elsevier, 2012.
  • [8] Dahlhaus, R. and Polonik, W. Empirical spectral processes for locally stationary time series., Bernoulli 15 (2009), 1–39.
  • [9] Doukhan, P., Grublyté, I. and Surgailis, D. A nonlinear model for long memory conditional heteroscedasticity., Lithuanian Journal of Mathematic 56 (2016), 164–188.
  • [10] Fokianos, K. and Tjøstheim, D. Log-linear Poisson autoregression., Journal of Multivariate Analysis 102 (2011), 563–578.
  • [11] Francq, C. and Gautier, E. Large sample properties of parameter least squares estimates for time-varying ARMA models., Journal of Time Series Analysis 25 (2004), 765–783.
  • [12] Grenier, E. ARMA models with time-dependent coefficients: estimators and applications., Traitement du Signal 3 (1986), 219–233.
  • [13] Hall, P. and Heyde, C. C., Martingale Limit Theory and Its Application. Academic Press, 1980.
  • [14] Lund, R., Hurd, H., Bloomfield, P. and Smith, R. Climatological time series with periodic correlation., Journal of Climat 8 (1995), 2787–2809.
  • [15] Major, P. Central limit theorems for martingales., http://www.math-inst.hu/~major/probability/martingale.pdf (2016), 1–22.
  • [16] Maligranda, L. Orlicz spaces and interpolation. In, Seminars in Math. 5, B. Campinas Sao Paulo, Ed., vol. 5 of IMS Lecture Notes-Monograph Series. 1989, pp. 127–140.
  • [17] Moulines, E., Priouret, P. and Roueff, F. On recursive estimation for locally stationary time varying autoregressive processes., Annals of Statistics 33 (2005), 2610–2654.
  • [18] Nadaraya, E. On estimating regression., Theory of Probability and Applications 9 (1964), 1411-1412.
  • [19] Paraschakisa, K. and Dahlhaus, R. Frequency and phase estimation in time series with quasi periodic components., Journal of Time Series Analysis 33 (2012), 13–31.
  • [20] Priestley, M. E., and Chao, M. T. Nonparametric function fitting., Journal of The Royal Statistical Society, Series B 34 (1972), 385–392.
  • [21] Rosenblatt, M., Stochastic curve estimation, vol. 3. NSF-CBMS Regional Conference Series in Probability and Statistics, 1991.
  • [22] Watson, G. Smooth regression analysis., Sankhyä Ser. A 26 (1964), 359–372.