Abstract Statistical Estimation and Modern Harmonic Analysis

David L. Donoho
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David L. Donoho: Stanford University, Department of Statistics

A chapter in Proceedings of the International Congress of Mathematicians, 1995, pp 997-1005 from Springer

Abstract: Abstract Suppose that t i are equispaced points in the unit interval t i = i/n, and we observe 1 $${y_i} = f\left( {{t_i}} \right) + s{z_i}, i = 1,...,n$$ , where the z i are i.i.d. N(0,1). Our goal is to recover the object f from these noisy observations. In order to do so, we must know something about f (otherwise we have n observations and 2n unknowns). In the branch of statistics called nonparametric regression, it is traditional to assume quantitative smoothness information about f, often of the form f ∈ F, where F is a ball in a functional class, for example an L2-Sobolev ball $$\left\{ {f:{{\left\| {{f^{\left( m \right)}}} \right\|}_{{L^2}}} \leqslant C} \right\}$$ . Performance is then measured by considering the minimax risk 2 $$M\left( {n,F} \right) = \mathop {min}\limits_{\hat f\left( \cdot \right)} \mathop {max}\limits_{f \in F} E\left\| {\hat f\left( {{y^{\left( n \right)}}} \right) - f} \right\|_{{L^2}{{\left( T \right)}^ \cdot }}^2$$ .

Date: 1995
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DOI: 10.1007/978-3-0348-9078-6_92

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