A Framework for Estimation of Convex Functions
Tony Cai and Mark Low
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Abstract:
A general non-asymptotic framework, which evaluates the performance of any procedure at individual functions, is introduced in the context of estimating convex functions at a point. This framework, which is significantly different from the conventional minimax theory, is also applicable to other problems in shape constrained inference.
A benchmark is provided for the mean squared error of any estimate for each convex function in the same way that Fisher Information depends on the unknown parameter in a regular parametric model. A local modulus of continuity is introduced and is shown to capture the difficulty of estimating individual convex functions. A fully data-driven estimator is proposed and is shown to perform uniformly within a constant factor of the ideal benchmark for every convex function. Such an estimator is thus adaptive to every unknown function instead of to a collection of function classes as is typical in the nonparametric function estimation literature.
- Paper:
pdf file.
- Supplement: Additional technical arguments are contained in the
supplement.
- Other related paper:
Cai, T. T., Low, M. & Xia, Y. (2013).
Adaptive confidence intervals for regression functions under shape constraints.
The Annals of Statistics 42, 722-750.