Bernoulli

  • Bernoulli
  • Volume 20, Number 2 (2014), 747-774.

Asymptotically efficient estimation of a scale parameter in Gaussian time series and closed-form expressions for the Fisher information

Till Sabel and Johannes Schmidt-Hieber

Full-text: Open access

Abstract

Mimicking the maximum likelihood estimator, we construct first order Cramer–Rao efficient and explicitly computable estimators for the scale parameter $\sigma^{2}$ in the model $Z_{i,n}=\sigma n^{-\beta}X_{i}+Y_{i}$, $i=1,\ldots,n$, $\beta>0$ with independent, stationary Gaussian processes $(X_{i})_{i\in\mathbb{N}}$, $(Y_{i})_{i\in\mathbb{N}}$, and $(X_{i})_{i\in\mathbb{N}}$ exhibits possibly long-range dependence. In a second part, closed-form expressions for the asymptotic behavior of the corresponding Fisher information are derived. Our main finding is that depending on the behavior of the spectral densities at zero, the Fisher information has asymptotically two different scaling regimes, which are separated by a sharp phase transition. The most prominent example included in our analysis is the Fisher information for the scaling factor of a high-frequency sample of fractional Brownian motion under additive noise.

Article information

Source
Bernoulli, Volume 20, Number 2 (2014), 747-774.

Dates
First available in Project Euclid: 28 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.bj/1393594005

Digital Object Identifier
doi:10.3150/12-BEJ505

Mathematical Reviews number (MathSciNet)
MR3178517

Zentralblatt MATH identifier
06291821

Keywords
efficient estimation fractional Brownian motion Fisher information regular variation slowly varying function spectral density

Citation

Sabel, Till; Schmidt-Hieber, Johannes. Asymptotically efficient estimation of a scale parameter in Gaussian time series and closed-form expressions for the Fisher information. Bernoulli 20 (2014), no. 2, 747--774. doi:10.3150/12-BEJ505. https://projecteuclid.org/euclid.bj/1393594005


Export citation

References

  • [1] Barndorff-Nielsen, O.E., Hansen, P.R., Lunde, A. and Shephard, N. (2008). Designing realized kernels to measure the ex post variation of equity prices in the presence of noise. Econometrica 76 1481–1536.
  • [2] Belov, A.S. (2002). Remarks on the convergence (boundedness) in the mean of partial sums of a trigonometric series. Mat. Zametki 71 807–817.
  • [3] Bingham, N.H., Goldie, C.M. and Teugels, J.L. (1987). Regular Variation. Encyclopedia of Mathematics and Its Applications 27. Cambridge: Cambridge Univ. Press.
  • [4] Bojanić, R. and Seneta, E. (1971). Slowly varying functions and asymptotic relations. J. Math. Anal. Appl. 34 302–315.
  • [5] Britanak, V., Yip, P.C. and Rao, K.R. (2007). Discrete Cosine and Sine Transforms: General Properties, Fast Algorithms and Integer Approximations. Amsterdam: Elsevier.
  • [6] Cai, T.T., Munk, A. and Schmidt-Hieber, J. (2010). Sharp minimax estimation of the variance of Brownian motion corrupted with Gaussian noise. Statist. Sinica 20 1011–1024.
  • [7] Dahlhaus, R. (1989). Efficient parameter estimation for self-similar processes. Ann. Statist. 17 1749–1766.
  • [8] Davies, R.B. (1973). Asymptotic inference in stationary Gaussian time-series. Adv. in Appl. Probab. 5 469–497.
  • [9] Dzhaparidze, K. (1986). Parameter Estimation and Hypothesis Testing in Spectral Analysis of Stationary Time Series. Springer Series in Statistics. New York: Springer.
  • [10] Fox, R. and Taqqu, M.S. (1986). Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series. Ann. Statist. 14 517–532.
  • [11] Giraitis, L. and Surgailis, D. (1990). A central limit theorem for quadratic forms in strongly dependent linear variables and its application to asymptotical normality of Whittle’s estimate. Probab. Theory Related Fields 86 87–104.
  • [12] Gloter, A. and Hoffmann, M. (2004). Stochastic volatility and fractional Brownian motion. Stochastic Process. Appl. 113 143–172.
  • [13] Gloter, A. and Hoffmann, M. (2007). Estimation of the Hurst parameter from discrete noisy data. Ann. Statist. 35 1947–1974.
  • [14] Gloter, A. and Jacod, J. (2001). Diffusions with measurement errors. I. Local asymptotic normality. ESAIM Probab. Stat. 5 225–242 (electronic).
  • [15] Gloter, A. and Jacod, J. (2001). Diffusions with measurement errors. II. Optimal estimators. ESAIM Probab. Stat. 5 243–260 (electronic).
  • [16] Jacod, J., Li, Y., Mykland, P.A., Podolskij, M. and Vetter, M. (2009). Microstructure noise in the continuous case: The pre-averaging approach. Stochastic Process. Appl. 119 2249–2276.
  • [17] Johnstone, I.M. (1999). Wavelet shrinkage for correlated data and inverse problems: Adaptivity results. Statist. Sinica 9 51–83.
  • [18] Podolskij, M. and Vetter, M. (2009). Estimation of volatility functionals in the simultaneous presence of microstructure noise and jumps. Bernoulli 15 634–658.
  • [19] Porat, B. and Friedlander, B. (1986). Computation of the exact information matrix of Gaussian time series with stationary random components. IEEE Trans. Acoust. Speech Signal Process. 34 118–130.
  • [20] Reiß, M. (2011). Asymptotic equivalence for inference on the volatility from noisy observations. Ann. Statist. 39 772–802.
  • [21] Rohde, A. and Dümbgen, L. (2013). Statistical inference for the optimal approximating model. Probab. Theory Related Fields 155 839–865.
  • [22] Sabel, T. and Schmidt-Hieber, J. (2014). Supplement to “Asymptotically efficient estimation of a scale parameter in Gaussian time series and closed-form expressions for the Fisher information.” DOI:10.3150/12-BEJ505SUPP.
  • [23] Stein, M.L. (1987). Minimum norm quadratic estimation of spatial variograms. J. Amer. Statist. Assoc. 82 765–772.
  • [24] Tikhonov, S. (2007). Trigonometric series with general monotone coefficients. J. Math. Anal. Appl. 326 721–735.
  • [25] Zhang, L. (2006). Efficient estimation of stochastic volatility using noisy observations: A multi-scale approach. Bernoulli 12 1019–1043.

Supplemental materials

  • Supplementary material: Supplement to “Asymptotically efficient estimation of a scale parameter in Gaussian time series and closed-form expressions for the Fisher information”. In the supplement, we provide the proofs of Theorem 3 and Corollary 1 along with some technical propositions and lemmas denoted by B.1, B.2, …, C.1, C.2, ….