Minimax Estimation Of Linear Functionals Over Nonconvex Parameter Spaces
Tony Cai and Mark Low
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Abstract:
The minimax theory for estimating linear functionals is extended to
the case of a finite union of convex parameter spaces. Upper and lower
bounds for the
minimax risk can still be described in terms of a modulus of
continuity. However in contrast to the theory for convex parameter
spaces rate optimal procedures are often required to be nonlinear.
A construction of such nonlinear procedures is given. The results
developed in this paper have important applications to the theory of
adaptation.
- Paper:
pdf file.
- Other related papers:
Cai, T. & Low, M. (2004).
An adaptation theory for nonparametric confidence intervals.
The Annals of Statistics 32, 1805-1840.Cai, T. & Low, M. (2005).
Adaptive estimation of linear functionals under different performance measures.
Bernoulli 11, 1-18.Cai, T. & Low, M. (2005).
On adaptive estimation of linear functionals.
The Annals of Statistics 33, 2311-2343.