Electronic Journal of Statistics

Estimation of spectral functionals for Levy-driven continuous-time linear models with tapered data

Mamikon S. Ginovyan and Artur A. Sahakyan

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The paper is concerned with the nonparametric statistical estimation of linear spectral functionals for Lévy-driven continuous-time stationary linear models with tapered data. As an estimator for unknown functional we consider the averaged tapered periodogram. We analyze the bias of the estimator and obtain sufficient conditions assuring the proper rate of convergence of the bias to zero, necessary for asymptotic normality of the estimator. We prove a a central limit theorem for a suitable normalized stochastic process generated by a tapered Toeplitz type quadratic functional of the model. As a consequence of these results we obtain the asymptotic normality of our estimator.

Article information

Electron. J. Statist., Volume 13, Number 1 (2019), 255-283.

Received: December 2017
First available in Project Euclid: 30 January 2019

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Zentralblatt MATH identifier

Primary: 62F12: Asymptotic properties of estimators 62G20: Asymptotic properties
Secondary: 60G10: Stationary processes 60F05: Central limit and other weak theorems

Lévy-driven continuous-time model tapered data smoothed periodogram central limit theorem nonparametric estimation asymptotic normality Toeplitz type quadratic functional

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Ginovyan, Mamikon S.; Sahakyan, Artur A. Estimation of spectral functionals for Levy-driven continuous-time linear models with tapered data. Electron. J. Statist. 13 (2019), no. 1, 255--283. doi:10.1214/18-EJS1525. https://projecteuclid.org/euclid.ejs/1548817591

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  • [1] V. V. Anh, N. N. Leonenko, and L. Sakhno, Minimum contrast estimation of random processes based on information of second and third orders, J. Statist. Planning Inference 137 (2007), 1302–1331.
  • [2] F. Avram, N. N. Leonenko, and L. Sakhno, Harmonic analysis tools for statistical inference in the spectral domain, In: Dependence in Probability and Statistics, P. Doukhan et al. (eds.), Lecture Notes in Statistics, vol. 200, Springer, 2010, 59–70.
  • [3] S. Bai, M. S. Ginovyan, and M. S. Taqqu, Limit theorems for quadratic forms of Levy-driven continuous-time linear processes. Stochast. Process. Appl. 126 (2016), 1036–1065.
  • [4] R. Bentkus, On the error of the estimate of the spectral function of a stationary process, Litovskii Mat. Sb. 12 (1972), 55–71.
  • [5] P. Billingsley, Convergence of Probability Measures, John Wiley & Sons, New York, 1999.
  • [6] V. Bogachev, Measure Theory, vol. I, Springer-Verlag, Berlin, 2007
  • [7] D. R. Brillinger, Time Series: Data Analysis and Theory, Holden Day, San Francisco, 1981.
  • [8] P. J. Brockwell, Recent results in the theory and applications of CARMA processes, Annals of the Institute of Statistical Mathematics 66(4) (2014), 647–685.
  • [9] R. Dahlhaus, Spectral analysis with tapered data, J. Time Ser. Anal. 4 (1983), 163–174.
  • [10] R. Dahlhaus, A functional limit theorem for tapered empirical spectral functions, Stoch. Process. Appl. 19 (1985), 135–149.
  • [11] R. Dahlhaus and H. Künsch, Edge effects and efficient parameter estimation for stationary random fields, Biometrika 74(4) (1987), 877–882.
  • [12] R. Dahlhaus and W. Wefelmeyer, Asymptotically optimal estimation in misspecified time series models, Ann. Statist. 24 (1996), 952–974.
  • [13] M. Eichler, Empirical spectral processes and their applications to stationary point processes, Ann. Appl. Probab. 5(4) (1995), 1161–1176.
  • [14] M. Farré, M. Jolis, and F. Utzet, Multiple Stratonovich integral and Hu–Meyer formula for Lévy processes, The Annals of Probability 38(6) (2010), 2136–2169.
  • [15] M. S. Ginovyan, Asymptotically efficient nonparametric estimation of functionals of a spectral density having zeros, Theory Probab. Appl. 33(2) (1988), 296–303.
  • [16] M. S. Ginovyan, On estimating the value of a linear functional of the spectral density of a Gaussian stationary process, Theory Probab. Appl. 33(4) (1988), 722–726.
  • [17] M. S. Ginovyan, On Toeplitz type quadratic functionals in Gaussian stationary process, Probab. Theory Relat. Fields 100 (1994), 395–406.
  • [18] M. S. Ginovyan, Asymptotic properties of spectrum estimate of stationary Gaussian processes, J. Cont. Math. Anal. 30(1) (1995), 1–16.
  • [19] M. S. Ginovyan, Asymptotically efficient nonparametric estimation of nonlinear spectral functionals, Acta Appl. Math. 78 (2003), 145–154.
  • [20] M. S. Ginovyan, Efficient Estimation of Spectral Functionals for Gaussian Stationary Models, Comm. Stochast. Anal. 5(1) (2011), 211–232.
  • [21] M. S. Ginovyan, Efficient Estimation of Spectral Functionals for Continuous-time Stationary Models, Acta Appl. Math. 115(2) (2011), 233–254.
  • [22] M. S. Ginovyan and A. A. Sahakyan, Limit Theorems for Toeplitz quadratic functionals of continuous-time stationary process, Probab. Theory Relat. Fields 138 (2007), 551–579.
  • [23] M. S. Ginovyan and A. A. Sahakyan, Robust estimation for continuous-time linear models with memory, Probability Theory and Mathematical Statistics, 95 (2016), 75–91.
  • [24] M. S. Ginovyan, A. A. Sahakyan, and M. S. Taqqu, The trace problem for Toeplitz matrices and operators and its impact in probability, Probability Surveys 11 (2014), 393–440.
  • [25] X. Guyon, Random Fields on a Network: Modelling, Statistics and Applications, Springer, New York, 1995.
  • [26] R. Z. Has’minskii and I. A. Ibragimov, Asymptotically efficient nonparametric estimation of functionals of a spectral density function, Probab. Theory Related Fields 73 (1986), 447–461.
  • [27] I. A. Ibragimov and Yu. V. Linnik, Independent and stationary sequences of random variables, Wolters-Noordhoff, 1971.
  • [28] I. A. Ibragimov and R. Z. Khasminskii, Asymptotically normal families of distributions and efficient estimation, Ann. Statist. 19 (1991), 1681–1724.
  • [29] N. N. Leonenko and L. Sakhno, On the Whittle estimators for some classes of continuous-parameter random processes and fields, Stat & Probab. Letters 76 (2006), 781–795.
  • [30] G. Peccati and M. S. Taqqu, Moments, Cumulants and Diagrams: a Survey With Computer Implementation, Springer Verlag, 2011.
  • [31] L. Sakhno, Minimum Contrast Method for Parameter Estimation in the Spectral Domain, In: Modern Stochastics and Applications (V. Korolyuk et al. eds.), Springer Optimization and Its Applications, vol. 90, Springer, 2014, 319–336.
  • [32] M. Taniguchi, Minimum contrast estimation for spectral densities of stationary processes. J. R. Stat. Soc. Ser. B-Stat. Methodol. 49 (1987), 315–325.