Summary
Given a collection of nested closed, convex symmetric sets and a linear functional, we find estimates which are within a logarithm term of being simultaneously asymptotically minimax. Moreover, these estimates can be constructed so that the loss of this logarithm term only occurs on a small subset of functions. These estimates are quasi-optimal since there do not exist estimators which do not lose a logarithm term on some part of the parameter spaces.
References
Brown, L.D., Low, M.G.: Superefficiency and lack of adaptability in nonparametric functional estimation. Technical Report (1992)
Donoho, D.L., Johnstone, I.M.: Adapting to unknown smoothness at a point via wavelet shrinkage. Technical Report (1992)
Donoho, D.L., Liu, R.C.: Geometrizing rates of convergence, III. Ann. Stat.19, 668–701 (1991)
Efromovich, S., Pinsker, M.S.: Learning algorithm for nonparametric filtering. Autom. Remote Control11, 1434–1440 (1984)
Ibragimov, I.A., Khasminskii, R.Z.: Statistical estimation: Asymptotic theory. Berlin Heidelberg New York: Springer 1981
Ibragimov, I.A., Khasminskii, R.Z.: On nonparametric estimation of the value of a linear functional in gaussian white noise. Theory Probab. Appl.29, 18–32 (1984)
Lepskii, O.V.: On one problem of adaptive estimation in Gaussian white noise. Theory Probab. Appl.35, 459–470 (1990)
Lepskii, O.V.: On problem of adaptive estimation in white Gaussian noise. Adv. Sov. Math.12, 87–106 (1992)
Lepskii, O.V.: Asymptotically minimax adaptive estimation 1: Upper bounds. Optimally adaptive estimates. Theory Probab. Appl.36, 682–697 (1991)
Low, M.G.: Bias-variance tradeoffs in functional estimation problems. Technical Report (1992)
Pinsker, M.S.: Optimal filtration of square-integrable signals in Gaussian noise. Probl. Inf. Transm.16, 52–68 (1980)
Author information
Authors and Affiliations
Additional information
This author was partially supported by an NSF Grant DMS-9123956
This author was supported by an NSF Postdoctoral Research Fellowship
Rights and permissions
About this article
Cite this article
Efromovich, S., Low, M.G. Adaptive estimates of linear functionals. Probab. Th. Rel. Fields 98, 261–275 (1994). https://doi.org/10.1007/BF01192517
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01192517
Mathematics Subject Classification (1991)
- 62C05
- 62E20
- 62J02
- 62G05
- 62M99