The Annals of Statistics

Asymptotic equivalence of spectral density estimation and Gaussian white noise

Georgi K. Golubev, Michael Nussbaum, and Harrison H. Zhou

Full-text: Open access

Enhanced PDF (304 KB)

Abstract

We consider the statistical experiment given by a sample y(1), …, y(n) of a stationary Gaussian process with an unknown smooth spectral density f. Asymptotic equivalence, in the sense of Le Cam’s deficiency Δ-distance, to two Gaussian experiments with simpler structure is established. The first one is given by independent zero mean Gaussians with variance approximately f(ωi), where ωi is a uniform grid of points in (−π, π) (nonparametric Gaussian scale regression). This approximation is closely related to well-known asymptotic independence results for the periodogram and corresponding inference methods. The second asymptotic equivalence is to a Gaussian white noise model where the drift function is the log-spectral density. This represents the step from a Gaussian scale model to a location model, and also has a counterpart in established inference methods, that is, log-periodogram regression. The problem of simple explicit equivalence maps (Markov kernels), allowing to directly carry over inference, appears in this context but is not solved here.

Article information

Source
Ann. Statist., Volume 38, Number 1 (2010), 181-214.

Dates
First available in Project Euclid: 31 December 2009

Permanent link to this document
https://projecteuclid.org/euclid.aos/1262271613

Digital Object Identifier
doi:10.1214/09-AOS705

Mathematical Reviews number (MathSciNet)
MR2589320

Zentralblatt MATH identifier
1181.62152

Subjects
Primary: 62G07: Density estimation 62G20: Asymptotic properties

Keywords
Stationary Gaussian process spectral density Sobolev classes Le Cam distance asymptotic equivalence Whittle likelihood log-periodogram regression nonparametric Gaussian scale model signal in Gaussian white noise

Citation

Golubev, Georgi K.; Nussbaum, Michael; Zhou, Harrison H. Asymptotic equivalence of spectral density estimation and Gaussian white noise. Ann. Statist. 38 (2010), no. 1, 181--214. doi:10.1214/09-AOS705. https://projecteuclid.org/euclid.aos/1262271613


Export citation

References

  • [1] Bickel, P. and Doksum, K. (2001). Mathematical Statistics 1, 2nd ed. Prentice Hall, Upper Saddle River, NJ.
    Zentralblatt MATH: 0403.62001
  • [2] Balakrishnan, A. V. (1976). Applied Functional Analysis. Springer, New York.
    Mathematical Reviews (MathSciNet): MR470699
    Zentralblatt MATH: 0333.93051
  • [3] Brockwell, P. J. and Davis, R. A. (1991). Time Series: Theory and Methods, 2nd ed. Springer, New York.
    Mathematical Reviews (MathSciNet): MR1093459
    Zentralblatt MATH: 0709.62080
  • [4] Brown, L. D. and Low, M. (1996). Asymptotic equivalence of nonparametric regression and white noise. Ann. Statist. 24 2384–2398.
    Mathematical Reviews (MathSciNet): MR1425958
    Zentralblatt MATH: 0867.62022
    Digital Object Identifier: doi:10.1214/aos/1032181159
    Project Euclid: euclid.aos/1032181159
  • [5] Brown, L. D., Low, M. and Zhang, C.-H. (2002). Asymptotic equivalence theory for nonparametric regression with random design. Ann. Statist. 30 688–707.
    Mathematical Reviews (MathSciNet): MR1922538
    Zentralblatt MATH: 1029.62044
    Digital Object Identifier: doi:10.1214/aos/1028674838
    Project Euclid: euclid.aos/1028674838
  • [6] Brown, L. D., Carter, A. V., Low, M. G. and Zhang, C.-H. (2004). Equivalence theory for density estimation, Poisson processes and Gaussian white noise with drift. Ann. Statist. 32 2074–2097.
    Mathematical Reviews (MathSciNet): MR2102503
    Zentralblatt MATH: 1062.62083
    Digital Object Identifier: doi:10.1214/009053604000000454
    Project Euclid: euclid.aos/1098883782
  • [7] Carter, A. V. (2002). Deficiency distance between multinomial and multivariate normal experiments. Ann. Statist. 30 708–730.
    Mathematical Reviews (MathSciNet): MR1922539
    Zentralblatt MATH: 1029.62005
    Digital Object Identifier: doi:10.1214/aos/1028674839
    Project Euclid: euclid.aos/1028674839
  • [8] Choudhouri, N., Ghosal, S. and Roy, A. (2004). Contiguity of the Whittle measure for a Gaussian time series. Biometrika 91 211–218.
    Mathematical Reviews (MathSciNet): MR2050470
    Zentralblatt MATH: 1132.62350
    Digital Object Identifier: doi:10.1093/biomet/91.1.211
  • [9] Coursol, J. and Dacunha-Castelle, D. (1982). Remarks on the approximation of the likelihood function of a stationary Gaussian process. Theory Probab. Appl. 27 162–167.
    Mathematical Reviews (MathSciNet): MR645138
    Zentralblatt MATH: 0537.60030
  • [10] Dahlhaus, R. (1988). Small sample effects in time series analysis: A new asymptotic theory and a new estimate. Ann. Statist. 16 808–841.
    Mathematical Reviews (MathSciNet): MR947580
    Zentralblatt MATH: 0662.62100
    Digital Object Identifier: doi:10.1214/aos/1176350838
    Project Euclid: euclid.aos/1176350838
  • [11] Dahlhaus, R. and Janas, D. (1996). A frequency domain bootstrap for ratio statistics in time series analysis. Ann. Statist. 24 1934–1963.
    Mathematical Reviews (MathSciNet): MR1421155
    Zentralblatt MATH: 0867.62072
    Digital Object Identifier: doi:10.1214/aos/1069362304
    Project Euclid: euclid.aos/1069362304
  • [12] 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. Birkhäuser, Boston.
    Zentralblatt MATH: 1022.62091
  • [13] Davies, R. B. (1973). Asymptotic inference in stationary Gaussian time-series. Adv. in Appl. Probab. 5 469–497.
    Mathematical Reviews (MathSciNet): MR341699
    Zentralblatt MATH: 0276.62078
    Digital Object Identifier: doi:10.2307/1425830
    JSTOR: links.jstor.org
  • [14] Delattre, S. and Hoffmann, M. (2002). Asymptotic equivalence for a null recurrent diffusion. Bernoulli 8 139–174.
    Mathematical Reviews (MathSciNet): MR1895888
    Project Euclid: euclid.bj/1078866865
    Zentralblatt MATH: 1040.60067
  • [15] Dzhaparidze, K. (1986). Parameter Estimation and Hypothesis Testing in Spectral Analysis of Stationary Time Series. Springer, New York.
    Mathematical Reviews (MathSciNet): MR812272
    Zentralblatt MATH: 0584.62157
  • [16] Fan, J. and Gijbels, I. (1996). Local Polynomial Modelling and Its Applications. Chapman and Hall, London.
    Mathematical Reviews (MathSciNet): MR1383587
    Zentralblatt MATH: 0873.62037
  • [17] Golubev, G. K., Nussbaum, M. and Zhou, H. H. (2005). Asymptotic equivalence of spectral density estimation and Gaussian white noise. Technical report. Available at arXiv:0903.1314v1 [math.ST].
    Mathematical Reviews (MathSciNet): MR2589320
    Zentralblatt MATH: 1181.62152
    Digital Object Identifier: doi:10.1214/09-AOS705
    Project Euclid: euclid.aos/1262271613
  • [18] Grama, I. and Nussbaum, M. (1998). Asymptotic equivalence for nonparametric generalized linear models. Probab. Theory Related Fields 111 167–214.
    Mathematical Reviews (MathSciNet): MR1633574
    Zentralblatt MATH: 0953.62039
    Digital Object Identifier: doi:10.1007/s004400050166
  • [19] Ingster, Y. I. (1985). An asymptotic minimax test of nonparametric hypotheses about spectral density. Theory Probab. Appl. 29 846–847.
  • [20] Ingster, Y. I. (1993). Asymptotically minimax hypothesis testing for nonparametric alternatives I–III. Math. Methods Statist. 2 85–114; 3 171–189; 4 249–268.
    Mathematical Reviews (MathSciNet): MR1259685
    Zentralblatt MATH: 0798.62059
  • [21] Lahiri, S. N. (2003). A necessary and sufficient condition for asymptotic independence of discrete Fourier transforms under short- and long-range dependence. Ann. Statist. 31 613–641.
    Mathematical Reviews (MathSciNet): MR1983544
    Zentralblatt MATH: 1039.62087
    Digital Object Identifier: doi:10.1214/aos/1051027883
    Project Euclid: euclid.aos/1051027883
  • [22] Le Cam, L. (1974). On the information contained in additional observations. Ann. Statist. 2 630–649.
    Mathematical Reviews (MathSciNet): MR436400
    Zentralblatt MATH: 0286.62004
    Digital Object Identifier: doi:10.1214/aos/1176342753
    Project Euclid: euclid.aos/1176342753
  • [23] Le Cam, L. and Yang, G. (2000). Asymptotics in Statistics, 2nd ed. Springer, New York.
    Mathematical Reviews (MathSciNet): MR1784901
    Zentralblatt MATH: 0952.62002
  • [24] Low, M. G. and Zhou, H. H. (2007). A complement to Le Cam’s theorem. Ann. Statist. 35 1146–1165.
    Mathematical Reviews (MathSciNet): MR2341701
    Zentralblatt MATH: 05186964
    Digital Object Identifier: doi:10.1214/009053607000000091
    Project Euclid: euclid.aos/1185304001
  • [25] Mammen, E. (1986). The statistical information contained in additional observations. Ann. Statist. 14 665–678.
    Mathematical Reviews (MathSciNet): MR840521
    Zentralblatt MATH: 0633.62006
    Digital Object Identifier: doi:10.1214/aos/1176349945
    Project Euclid: euclid.aos/1176349945
  • [26] Nussbaum, M. (1996). Asymptotic equivalence of density estimation and Gaussian white noise. Ann. Statist. 24 2399–2430.
    Mathematical Reviews (MathSciNet): MR1425959
    Digital Object Identifier: doi:10.1214/aos/1032181160
    Project Euclid: euclid.aos/1032181160
    Zentralblatt MATH: 0867.62035
  • [27] Runst, T. (1986). Mapping properties of nonlinear operators in spaces of Triebel–Lizorkin and Besov type. Anal. Math. 12 313–346.
    Mathematical Reviews (MathSciNet): MR877164
    Zentralblatt MATH: 0644.46022
    Digital Object Identifier: doi:10.1007/BF01909369
  • [28] Shiryaev, A. N. (1996). Probability, 2nd ed. Springer, New York.
    Mathematical Reviews (MathSciNet): MR1368405
    Zentralblatt MATH: 0909.01009
  • [29] Sickel, W. (1996). Composition operators acting on Sobolev spaces of fractional order—a survey on sufficient and necessary conditions. In Function Spaces, Differential Operators and Nonlinear Analysis 159–182. Prometheus, Prague.
    Mathematical Reviews (MathSciNet): MR1480936
    Zentralblatt MATH: 0896.47052
  • [30] Spokoiny, V. G. (1996). Adaptive hypothesis testing using wavelets. Ann. Statist. 24 2477–2498.
    Mathematical Reviews (MathSciNet): MR1425962
    Zentralblatt MATH: 0898.62056
    Digital Object Identifier: doi:10.1214/aos/1032181163
    Project Euclid: euclid.aos/1032181163
  • [31] Strasser, H. (1985). Mathematical Theory of Statistics. Walter de Gruyter, Berlin.
    Mathematical Reviews (MathSciNet): MR812467
    Zentralblatt MATH: 0594.62017
  • [32] van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge Univ. Press, Cambridge.
    Mathematical Reviews (MathSciNet): MR1652247
    Zentralblatt MATH: 0910.62001
  • [33] Zhou, H. H. (2004). Minimax estimation with thresholding and asymptotic equivalence for Gaussian variance regression. Ph.D. thesis, Cornell Univ. Press, Ithaca, NY. Available at http://www.stat.yale.edu/~hz68/.

The Institute of Mathematical Statistics

The Annals of Statistics

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more