Empirical spectral processes for locally stationary time series

Rainer Dahlhaus and Wolfgang Polonik

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A time-varying empirical spectral process indexed by classes of functions is defined for locally stationary time series. We derive weak convergence in a function space, and prove a maximal exponential inequality and a Glivenko–Cantelli-type convergence result. The results use conditions based on the metric entropy of the index class. In contrast to related earlier work, no Gaussian assumption is made. As applications, quasi-likelihood estimation, goodness-of-fit testing and inference under model misspecification are discussed. In an extended application, uniform rates of convergence are derived for local Whittle estimates of the parameter curves of locally stationary time series models.

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Bernoulli Volume 15, Number 1 (2009), 1-39.

First available in Project Euclid: 3 February 2009

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asymptotic normality empirical spectral process locally stationary processes non-stationary time series quadratic forms


Dahlhaus, Rainer; Polonik, Wolfgang. Empirical spectral processes for locally stationary time series. Bernoulli 15 (2009), no. 1, 1--39. doi:10.3150/08-BEJ137.

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  • Alexander, K.S. (1984). Probability inqualities for empirical processes and the law of the iterated logarithm., Ann. Probab. 12 1041–1067. Correction, Ann. Probab. 15 428–430.
  • Brillinger, D.R. (1981)., Time Series: Data Analysis and Theory. San Francisco: Holden Day.
  • Dahlhaus, R. (1988). Empirical spectral processes and their applications to time series analysis., Stochastic Process. Appl. 30 69–83.
  • Dahlhaus, R. (1996). On the Kullback–Leibler information divergence of locally stationary processes., Stochastic Process. Appl. 62 139–168.
  • Dahlhaus, R. (1997). Fitting time series models to nonstationary processes., Ann. Statist. 25 1–37.
  • Dahlhaus, R. (2000). A likelihood approximation for locally stationary processes., Ann. Statist. 28 1762–1794.
  • Dahlhaus, R. and Neumann, M.H. (2001). Locally adaptive fitting of semiparametric models to nonstationary time series., Stochastic Process. Appl. 91 277–308.
  • Dahlhaus, R. and Polonik, W. (2002). Empirical spectral processes and nonparametric maximum likelihood estimation for time series. In, Empirical Process Techniques for Dependent Data (H. Dehling, T. Mikosch and M. Sørensen, eds.) 275–298. Boston: Birkhäuser.
  • Dahlhaus, R. and Polonik, W. (2006). Nonparametric quasi maximum likelihood estimation for Gaussian locally stationary processes., Ann. Statist. 34 2790–2824.
  • Davis, R.A., Lee, T. and Rodriguez-Yam, G. (2005). Structural break estimation for nonstationary time series models., J. Amer. Statist. Assoc. 101 223–239.
  • Fay, G. and Soulier, P. (2001). The periodogram of an i.i.d. sequence., Stochastic Process. Appl. 92 315–343.
  • Fryzlewicz, P., Sapatinas, T. and Subba Rao, S. (2006). A Haar–Fisz technique for locally stationary volatility estimation., Biometrika 93 687–704.
  • Householder, A.S. (1964)., The Theory of Matrices in Numerical Analysis. New York: Blaisdell.
  • Künsch, H.R. (1995). A note on causal solutions for locally stationary AR processes. Preprint, ETH, Zürich.
  • Nason, G. P., von Sachs, R. and Kroisandt, G. (2000). Wavelet processes and adaptive estimation of evolutionary wavelet spectra., J. Roy. Statist. Soc. Ser. B 62 271–292.
  • Mikosch, T. and Norvaisa, R. (1997). Uniform convergence of the empirical spectral distribution function., Stochastic Process. Appl. 70 85–114.
  • Moulines, E., Priouret, P. and Roueff, F. (2005). On recursive estimation for locally stationary time varying autoregressive processes., Ann. Statist. 33 2610–2654.
  • Neumann, M.H. and von Sachs, R. (1997). Wavelet thresholding in anisotropic function classes and applications to adaptive estimation of evolutionary spectra., Ann. Statist. 25 38–76.
  • Priestley, M.B. (1965). Evolutionary spectra and non-stationary processes (with discussion)., J. Roy. Statist. Soc. Ser. B 27 204–237.
  • Sakiyama, K. and Taniguchi, M. (2004). Discriminant analysis for locally stationary processes., J. Multivariate Anal. 90 282–300.
  • Van Bellegem, S. and Dahlhaus, R. (2006). Semiparametric estimation by model selection for locally stationary processes., J. Roy. Statist. Soc. Ser. B 68 721–746.
  • van der Vaart, A.W. and Wellner, J.A. (1996)., Weak Convergence and Empirical Processes. New York: Springer.
  • Whittle, P. (1953). Estimation and information in stationary time series., Ark. Mat. 2 423–434.