The Annals of Statistics

Asymptotic equivalence for regression under fractional noise

Johannes Schmidt-Hieber

Full-text: Open access

Abstract

Consider estimation of the regression function based on a model with equidistant design and measurement errors generated from a fractional Gaussian noise process. In previous literature, this model has been heuristically linked to an experiment, where the anti-derivative of the regression function is continuously observed under additive perturbation by a fractional Brownian motion. Based on a reformulation of the problem using reproducing kernel Hilbert spaces, we derive abstract approximation conditions on function spaces under which asymptotic equivalence between these models can be established and show that the conditions are satisfied for certain Sobolev balls exceeding some minimal smoothness. Furthermore, we construct a sequence space representation and provide necessary conditions for asymptotic equivalence to hold.

Article information

Source
Ann. Statist., Volume 42, Number 6 (2014), 2557-2585.

Dates
First available in Project Euclid: 12 November 2014

Permanent link to this document
https://projecteuclid.org/euclid.aos/1415801783

Digital Object Identifier
doi:10.1214/14-AOS1262

Mathematical Reviews number (MathSciNet)
MR3277671

Zentralblatt MATH identifier
1302.62097

Subjects
Primary: 62G08: Nonparametric regression
Secondary: 62G20: Asymptotic properties

Keywords
Asymptotic equivalence long memory fractional Brownian motion fractional Gaussian noise fractional calculus inverse problems nonharmonic Fourier series reproducing kernel Hilbert space (RKHS) stationarity

Citation

Schmidt-Hieber, Johannes. Asymptotic equivalence for regression under fractional noise. Ann. Statist. 42 (2014), no. 6, 2557--2585. doi:10.1214/14-AOS1262. https://projecteuclid.org/euclid.aos/1415801783


Export citation

References

  • [1] Brown, L. D., Cai, T. T., Low, M. G. and Zhang, C.-H. (2002). Asymptotic equivalence theory for nonparametric regression with random design. Ann. Statist. 30 688–707.
  • [2] Brown, L. D. and Low, M. G. (1996). Asymptotic equivalence of nonparametric regression and white noise. Ann. Statist. 24 2384–2398.
  • [3] Carter, A. V. (2006). A continuous Gaussian approximation to a nonparametric regression in two dimensions. Bernoulli 12 143–156.
  • [4] Carter, A. V. (2009). Asymptotically sufficient statistics in nonparametric regression experiments with correlated noise. J. Probab. Stat. Art. ID 275308, 19.
  • [5] Cavalier, L. (2008). Nonparametric statistical inverse problems. Inverse Problems 24 034004, 19.
  • [6] de Branges, L. (1968). Hilbert Spaces of Entire Functions. Prentice Hall, Englewood Cliffs, NJ.
  • [7] Donoho, D. L. (1995). Nonlinear solution of linear inverse problems by wavelet–vaguelette decomposition. Appl. Comput. Harmon. Anal. 2 101–126.
  • [8] Doob, J. L. (1953). Stochastic Processes. Wiley, New York.
  • [9] Dym, H. and McKean, H. (1976). Gaussian Processes, Function Theory, and the Inverse Spectral Measure. Academic Press, San Diego, CA.
  • [10] Dzhaparidze, K. and van Zanten, H. (2005). Krein’s spectral theory and the Paley–Wiener expansion for fractional Brownian motion. Ann. Probab. 33 620–644.
  • [11] Dzhaparidze, K., van Zanten, H. and Zareba, P. (2005). Representations of fractional Brownian motion using vibrating strings. Stochastic Process. Appl. 115 1928–1953.
  • [12] Engl, H. W., Hanke, M. and Neubauer, A. (1996). Regularization of Inverse Problems. Kluwer Academic, Dordrecht.
  • [13] Gloter, A. and Hoffmann, M. (2004). Stochastic volatility and fractional Brownian motion. Stochastic Process. Appl. 113 143–172.
  • [14] Golubev, G. K., Nussbaum, M. and Zhou, H. H. (2010). Asymptotic equivalence of spectral density estimation and Gaussian white noise. Ann. Statist. 38 181–214.
  • [15] Gradshteyn, I. S. and Ryzhik, I. M. (1994). Table of Integrals, Series, and Products, Russian ed. Academic Press, Boston, MA.
  • [16] Grama, I. and Nussbaum, M. (2002). Asymptotic equivalence for nonparametric regression. Math. Methods Statist. 11 1–36.
  • [17] Grenander, U. (1981). Abstract Inference. Wiley, New York.
  • [18] Hall, P. and Hart, J. D. (1990). Nonparametric regression with long-range dependence. Stochastic Process. Appl. 36 339–351.
  • [19] Johnstone, I. M. (1999). Wavelet shrinkage for correlated data and inverse problems: Adaptivity results. Statist. Sinica 9 51–83.
  • [20] Johnstone, I. M. and Silverman, B. W. (1997). Wavelet threshold estimators for data with correlated noise. J. Roy. Statist. Soc. Ser. B 59 319–351.
  • [21] Kreĭn, M. G. (1953). On some cases of effective determination of the density of an inhomogeneous cord from its spectral function. Doklady Akad. Nauk SSSR (N.S.) 93 617–620.
  • [22] Lebedev, N. N. (1972). Special Functions and Their Applications. Dover, New York.
  • [23] Luke, Y. L. (1962). Integrals of Bessel Functions. McGraw-Hill, New York.
  • [24] Meyer, Y., Sellan, F. and Taqqu, M. S. (1999). Wavelets, generalized white noise and fractional integration: The synthesis of fractional Brownian motion. J. Fourier Anal. Appl. 5 465–494.
  • [25] Nussbaum, M. (1996). Asymptotic equivalence of density estimation and Gaussian white noise. Ann. Statist. 24 2399–2430.
  • [26] Picard, J. (2011). Representation formulae for the fractional Brownian motion. In Séminaire de Probabilités XLIII. Lecture Notes in Math. 2006 3–70. Springer, Berlin.
  • [27] Pipiras, V. and Taqqu, M. S. (2000). Integration questions related to fractional Brownian motion. Probab. Theory Related Fields 118 251–291.
  • [28] Reiß, M. (2008). Asymptotic equivalence for nonparametric regression with multivariate and random design. Ann. Statist. 36 1957–1982.
  • [29] Reiß, M. (2011). Asymptotic equivalence for inference on the volatility from noisy observations. Ann. Statist. 39 772–802.
  • [30] Rohde, A. (2004). On the asymptotic equivalence and rate of convergence of nonparametric regression and Gaussian white noise. Statist. Decisions 22 235–243.
  • [31] Samko, S. G., Kilbas, A. A. and Marichev, O. I. (1993). Fractional Integrals and Derivatives. Gordon and Breach, New York.
  • [32] Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman & Hall, New York.
  • [33] Schmidt-Hieber, J. (2014). Supplement to “Asymptotic equivalence for regression under fractional noise.” DOI:10.1214/14-AOS1262SUPP.
  • [34] Sinaĭ, Ja. G. (1976). Self-similar probability distributions. Theory Probab. Appl. 21 64–80.
  • [35] Tautenhahn, U. and Gorenflo, R. (1999). On optimal regularization methods for fractional differentiation. Z. Anal. Anwend. 18 449–467.
  • [36] Tsybakov, A. B. (2009). Introduction to Nonparametric Estimation. Springer, New York.
  • [37] van der Vaart, A. W. and van Zanten, J. H. (2008). Reproducing kernel Hilbert spaces of Gaussian priors. In Pushing the Limits of Contemporary Statistics: Contributions in Honor of Jayanta K. Ghosh. Inst. Math. Stat. Collect. 3 200–222. IMS, Beachwood, OH.
  • [38] Wang, Y. (1996). Function estimation via wavelet shrinkage for long-memory data. Ann. Statist. 24 466–484.
  • [39] Watson, G. N. (1995). A Treatise on the Theory of Bessel Functions. Cambridge Univ. Press, Cambridge. Reprint of the second (1944) edition.
  • [40] Yaglom, A. M. (1987). Correlation Theory of Stationary and Related Random Functions. Vol. I. Springer, New York.
  • [41] Young, R. M. (1980). An Introduction to Nonharmonic Fourier Series. Academic Press, New York.
  • [42] Zareba, P. (2007). Representations of Gaussian processes with stationary increments. Ph.D. thesis, Vrije Univ. Amsterdam.

Supplemental materials

  • Supplementary material: Asymptotic equivalence for regression under fractional noise. The supplement contains proofs for Section 4, some technical results and a brief summary of the Le Cam distance.