Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-26T06:03:50.275Z Has data issue: false hasContentIssue false
  • English
  • Français

Efficient estimation of functionals of the spectraldensity of stationary Gaussian fields

Published online by Cambridge University Press:  15 August 2002

Carenne Ludeña
Carenne Ludeña*
Affiliation:
Departamento de Matemáticas, IVIC, Caracas, Venezuela; [email protected].
Get access

Abstract

Minimax bounds for the risk function of estimators of functionals ofthe spectral density of Gaussianfields 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 aregiven for all dimensions. For non-linear integral functionals, theseestimators are constructed for the two and three dimensional problems.

Keywords

Efficient estimationGaussian fieldsperiodogramtapered periodogramspectral densityToeplitz matrices.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 3 , 1999 , pp. 23 - 47
Copyright
© EDP Sciences, SMAI, 1999

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Avram, F., On bilinear forms in Gaussian random variables and Toeplitz matrices. Prob. Th. Rel. Fields 79 (1988) 37-45. CrossRef
R. Azencott and D. Dacunha-Castelle, Series of Irregular Observations. Forecasting and Model Building, Masson Éditions, Paris (1984).
Birgé, L. and Massart, P., Estimation of integral functionals of a density. Ann. Statist. 23 (1995) 11-29. CrossRef
Bouaziz, M., Inégalités de trace pour des matrices de Toeplitz et applications à des vraisemblances gaussiennes. Probability Math. Statistics 13 (1992) 253-267.
Dahlhaus, R., Spectral analysis with tapered data. J. Time Ser. Anal. 4 (1983) 163-175. CrossRef
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 Statistics 26, Springer Verlag (1983).
Dahlhaus, R., Efficient parameter estimation for self similar processes. Ann. Statist. 17 (1989) 1749-1766. CrossRef
Dahlhaus, R. and Künsch, H., Edge effects and efficient parameter estimation for stationary random fields. Biometrika 74 (1987) 877-882. CrossRef
Davies, R., Asymptotic inference in stationary Gaussian time series. Adv. Appl. Prob. 5 (1973) 469-497. CrossRef
Donoho, D. and Liu, R., Geometrizing rates of convergence II. Ann. Statist. 19 (1991) 633-667. CrossRef
Doukhan, P., León, J. and Soulier, P., Central and non-central limit theorems for quadratic forms of a strongly dependent stationary Gaussian field. Rebrape 10 (1996) 205-223.
Efroimovich, S. Yu., Local asymptotic normality for dependent observations. Translated from Problemy Pederachi Informatsii 14 (1978) 73-84.
Guyon, X., Parameter estimation for a stationary process on a d dimensional lattice. Biometrika 69 (1982) 95-105. CrossRef
J. Hajek, Local asymptotic minimax and admissibility in estimation. Sixth Berkeley Symposium (1972) 175-194.
Khas'minskii, R.Z. and Ibragimov, I.A., Asymptotically efficient nonparametric estimation of functionals of a spectral density function. Prob. Th. Rel. Fields 73 (1986) 447-461. CrossRef
Yu.A. Koshevnik, B.Ya. Levit, On a nonparametric analogue of the information matrix. Theor. Prob. Appl. 21 (1976) 738-759. CrossRef
Le Cam, L., On the assumptions used to prove the asymptotic normality of maximum likelihood estimates. Ann. Math. Statist. 41 (1970) 802-828. CrossRef
L. Le Cam and G. Lo Yang, Asymptotics in Statistics Springer Series in Statistics (1990).
Levit, B., Infinite dimensional information lower bounds. Theor. Prob. Appl. 23 (1978) 388-394.
Lude, C. na, Estimación eficiente de funcionales de la densidad espectral de procesos gaussianos multiparamétricos. Proceedings IV CLAPEM 4 (1990) 140-153.
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).
Millar, P.W., Nonparametric applications of an infinite dimensional convolution theorem. Z. Wahrsch. verw. Gebiete 68 (1985) 545-556. CrossRef
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).