Renormalizing Experiments for Nonlinear Functionals

David L. Donoho
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David L. Donoho: Stanford and University of California at Berkeley

Chapter 11 in Festschrift for Lucien Le Cam, 1997, pp 167-181 from Springer

Abstract: Abstract Let f = f(t), t ∈ R d be an unknown “object” (real-valued function), and suppose we are interested in recovering the nonlinear functional T(f). We know a priori that f ∈F, a certain convex class of functions (e.g. a class of smooth functions). For various types of measurements Yn=(yl, y2,…, yn), problems of this form arise in statistical settings, such as nonparametric density estimation and nonparametric regression estimation; but they also arise in signal recovery and image processing. In such problems, there generally exists an “optimal rate of convergence”: the minimax risk from n observations, $$ R\left( n \right) = \mathop{{\inf }}\limits_{{\hat{T}}} \mathop{{\sup }}\limits_{{f \in F}} E{{\left( {\hat{T}\left( {{{Y}_{n}}} \right) - T\left( f \right)} \right)}^{2}}$$ tends to zero as. $$ R\left( n \right) \asymp {{n}^{{ - r}}}$$ There is ariety of functionals T, function classes.F, and types of observation Yn; the literature is really too extensive to list here, although we mention Ibragimov & Has’minskii (1981), Sacks & Ylvisaker (1981), and Stone (1980). Lucien Le Cam (1973) has contributed directly to this literature, in his typical abstract and profound way; his ideas have stimulated the work of others in the field, e.g. Donoho & Liu (1991a).

Keywords: Nonparametric Regression; Unimodal Function; Nonparametric Density Estimation; Minimax Risk; White Noise Model (search for similar items in EconPapers)
Date: 1997
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DOI: 10.1007/978-1-4612-1880-7_11

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