Efficient estimation of functionals of the spectral density of stationary Gaussian fields

Carenne Ludeña

ESAIM: Probability and Statistics (2010)

  • Volume: 3, page 23-47
  • ISSN: 1292-8100

Abstract

top
Minimax bounds for the risk function of estimators of functionals of the spectral density of Gaussian fields are obtained. This result is a generalization of a previous result of Khas'minskii and Ibragimov on Gaussian processes. Efficient estimators are then constructed for these functionals. In the case of linear functionals these estimators are given for all dimensions. For non-linear integral functionals, these estimators are constructed for the two and three dimensional problems.

How to cite

top

Ludeña, Carenne. "Efficient estimation of functionals of the spectral density of stationary Gaussian fields." ESAIM: Probability and Statistics 3 (2010): 23-47. <http://eudml.org/doc/197742>.

@article{Ludeña2010,
abstract = { Minimax bounds for the risk function of estimators of functionals of the spectral density of Gaussian fields are obtained. This result is a generalization of a previous result of Khas'minskii and Ibragimov on Gaussian processes. Efficient estimators are then constructed for these functionals. In the case of linear functionals these estimators are given for all dimensions. For non-linear integral functionals, these estimators are constructed for the two and three dimensional problems. },
author = {Ludeña, Carenne},
journal = {ESAIM: Probability and Statistics},
keywords = { Efficient estimation; Gaussian fields; periodogram; tapered periodogram; spectral density; Toeplitz matrices.; lower bounds; linear functionals},
language = {eng},
month = {3},
pages = {23-47},
publisher = {EDP Sciences},
title = {Efficient estimation of functionals of the spectral density of stationary Gaussian fields},
url = {http://eudml.org/doc/197742},
volume = {3},
year = {2010},
}

TY - JOUR
AU - Ludeña, Carenne
TI - Efficient estimation of functionals of the spectral density of stationary Gaussian fields
JO - ESAIM: Probability and Statistics
DA - 2010/3//
PB - EDP Sciences
VL - 3
SP - 23
EP - 47
AB - Minimax bounds for the risk function of estimators of functionals of the spectral density of Gaussian fields are obtained. This result is a generalization of a previous result of Khas'minskii and Ibragimov on Gaussian processes. Efficient estimators are then constructed for these functionals. In the case of linear functionals these estimators are given for all dimensions. For non-linear integral functionals, these estimators are constructed for the two and three dimensional problems.
LA - eng
KW - Efficient estimation; Gaussian fields; periodogram; tapered periodogram; spectral density; Toeplitz matrices.; lower bounds; linear functionals
UR - http://eudml.org/doc/197742
ER -

References

top
  1. F. Avram, On bilinear forms in Gaussian random variables and Toeplitz matrices. Prob. Th. Rel. Fields79 (1988) 37-45.  Zbl0648.60043
  2. R. Azencott and D. Dacunha-Castelle, Series of Irregular Observations. Forecasting and Model Building, Masson Éditions, Paris (1984).  Zbl0593.62088
  3. L. Birgé and P. Massart, Estimation of integral functionals of a density. Ann. Statist.23 (1995) 11-29.  Zbl0848.62022
  4. M. Bouaziz, Inégalités de trace pour des matrices de Toeplitz et applications à des vraisemblances gaussiennes. Probability Math. Statistics13 (1992) 253-267.  Zbl0781.60019
  5. R. Dahlhaus, Spectral analysis with tapered data. J. Time Ser. Anal.4 (1983) 163-175.  Zbl0552.62068
  6. R. Dahlhaus, Parameter estimation of stationary processes with spectra containing strong peaks. Robust and nonlinear time series analysis. J. Franke, W. Härdle and D. Martin Eds. Lecture Notes in Statistics26, Springer Verlag (1983).  
  7. R. Dahlhaus, Efficient parameter estimation for self similar processes. Ann. Statist.17 (1989) 1749-1766.  Zbl0703.62091
  8. R. Dahlhaus and H. Künsch, Edge effects and efficient parameter estimation for stationary random fields. Biometrika74 (1987) 877-882.  Zbl0633.62094
  9. R. Davies, Asymptotic inference in stationary Gaussian time series. Adv. Appl. Prob.5 (1973) 469-497.  Zbl0276.62078
  10. D. Donoho and R. Liu, Geometrizing rates of convergence II. Ann. Statist.19 (1991) 633-667.  Zbl0754.62028
  11. P. Doukhan, J. León and P. Soulier, Central and non-central limit theorems for quadratic forms of a strongly dependent stationary Gaussian field. Rebrape10 (1996) 205-223.  Zbl0881.60023
  12. S. Yu. Efroimovich, Local asymptotic normality for dependent observations. Translated from Problemy Pederachi Informatsii14 (1978) 73-84.  
  13. X. Guyon, Parameter estimation for a stationary process on a d dimensional lattice. Biometrika69 (1982) 95-105.  Zbl0485.62107
  14. J. Hajek, Local asymptotic minimax and admissibility in estimation. Sixth Berkeley Symposium (1972) 175-194.  Zbl0281.62010
  15. R.Z. Khas'minskii and I.A. Ibragimov, Asymptotically efficient nonparametric estimation of functionals of a spectral density function. Prob. Th. Rel. Fields73 (1986) 447-461.  
  16. Yu.A. Koshevnik and B.Ya. Levit, On a nonparametric analogue of the information matrix. Theor. Prob. Appl.21 (1976) 738-759.  
  17. L. Le Cam, On the assumptions used to prove the asymptotic normality of maximum likelihood estimates. Ann. Math. Statist.41 (1970) 802-828.  Zbl0246.62039
  18. L. Le Cam and G. Lo Yang, Asymptotics in Statistics Springer Series in Statistics (1990).  
  19. B. Levit, Infinite dimensional information lower bounds. Theor. Prob. Appl.23 (1978) 388-394.  Zbl0404.62053
  20. C. Lude na, Estimación eficiente de funcionales de la densidad espectral de procesos gaussianos multiparamétricos. Proceedings IV CLAPEM4 (1990) 140-153.  
  21. C. Lude na, Estimation des fonctionnelles de la densité spectrale des processus gaussiens dans différents cadres de dépendance. Thèse (Docteur en Sciences), Université de Paris Sud, Centre d'Orsay, France (1996).  
  22. P.W. Millar, Nonparametric applications of an infinite dimensional convolution theorem. Z. Wahrsch. verw. Gebiete68 (1985) 545-556.  Zbl0571.62031
  23. Y.F. Yao, Méthodes bayésiennes en segmentation d'image et estimation par rabotage des modèles spatiaux. Thèse (Docteur en Sciences), Université de Paris Sud, Centre d'Orsay, France (1990).  

NotesEmbed ?

top

You must be logged in to post comments.