Abstract Statistical Estimation and Modern Harmonic Analysis

Abstract

Suppose that t _i are equispaced points in the unit interval t _i = i/n, and we observe

$${y_i} = f\left( {{t_i}} \right) + s{z_i}, i = 1,...,n$$

(1)

, 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

$$M\left( {n,F} \right) = \mathop {min}\limits_{\hat f\left( \cdot \right)} \mathop {max}\limits_{f \in F} E\left\| {\hat f\left( {{y^{\left( n \right)}}} \right) - f} \right\|_{{L^2}{{\left( T \right)}^ \cdot }}^2$$

(2)

.