Minimax estimation of linear functionals over nonconvex parameter spaces
By Mark G. Low, Tommaso Cai. 2004
52 citations recorded by the source corpus.
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.
Subfields: Control and Systems Engineering
View Paper: https://www.semanticscholar.org/paper/dbc86f0505b6c3f2d15656802148d769378515bf
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In Control and Systems Engineering
- Zijian Guo
- Tommaso Cai
- Mark G. Low
- Anatoli B. Juditsky
- Itai Dattner
- Yihong Wu
- Pengkun Yang
- Michael J. Kolesar
Citations in the corpus
- Confidence intervals for high-dimensional linear regression: Minimax rates and adaptivity (2015, methodology): “For example, such a construction of least favorable set has been used under the Gaussian sequence model in [1] for signal detection and in [7, 8] for estimation and confidence intervals for linear functionals.”
- Nonquadratic estimators of a quadratic functional (2005, background): “On the other hand it was shown in [5] that for estimating linear functionals the maximum risk of linear procedures over a parameter space is equal to the maximum risk over the convex hull of the parameter space.”
- Testing composite hypotheses, Hermite polynomials and optimal estimation of a nonsmooth functional (2011, methodology): “The proof of Theorem 7 is analogous to that of Theorem 7 in Cai and Low (2004), and we omit it here for reasons of space.”
- An adaptation theory for nonparametric confidence intervals (2004, methodology): “It is shown in Cai and Low (2004) that the affine estimator T̂ satisfies sup f∈F |ET̂ − Tf |= sup f∈C.Hull(F) |ET̂ − Tf |.”
- ON DECONVOLUTION OF DISTRIBUTION FUNCTIONS (2010, background): “Note that in the case of nonnested sets an adaptive estimator can be constructed following the ideas of [7].”