0.6 Nonparametric regression with wavelets  (Page 5/5)

where the infimum is taken over all measurable estimators $T_n$ of $m.$ Similarly, we define the linear $L_p-$ minimax risk as

where the infimum is now taken over all linear estimators $T_n^{\text{lin}}.$ Obviously, $R_n^{\text{lin}}(V,p)\ge R_n(V,p).$ We first state some definitions.

In order to fix the idea, we consider only the $L_2-$ risk in the remaining part of this section, thus $p:=2$ .

In [link] , [link] , the authors found that the optimal rate of convergence attainable by an estimator when the underlying function belongs to the Sobolev class $W_q^s$ is $a_n=n^{\frac{-s}{2s+1}}$ , hence $R_n(V,2)=n^{\frac{-2s}{2s+1}}$ . We saw in "Linear smoothing with wavelets" that linear wavelet estimators attain the optimal rate for $s-$ Hölder function in case of the $L_2-$ risk (also called `IMSE'). For a Sobolev class $W_q^s$ , the same result holds provided that $q\ge 2$ . More precisely, we have the two following situations.

1. If $q\ge 2,$ we are in the so-called homogeneous zone. In this zone of spatial homogeneity, linear estimators can attain the optimal rate of convergence $n^{-s/(2s+1)}.$
2. If $q<2,$ we are in the non-homogeneous zone, where linear estimators do not attain the optimal rate of convergence. Instead, we have:
   $$R_n^{\text{lin}}(V,2)/R_n(V,2)\to \infty ,\text{as}n\to \infty .$$

The second result is due to the spatial variability of functions in Sobolev spaces with small index $q$ . Linear estimators are based on the idea of spatial homogeneity of the function and hence do perform poorly in the presence of non-homogeneous functions. In contrast, even if $q<2$ , the SureShrink estimator attains the minimax rate [link] . The same type of results holds for more general Besov spaces, see for example [link] , Chapter 10.

Adaptivity of wavelet estimator

We just saw that a nonlinear wavelet estimator is able to estimate in an optimal way functions ofinhomogeneous regularity. However, it may not be sufficient to know that for $m$ belonging to a given space, the estimator performs well. Indeed, in general we do not know which space the function belongs to. Hence it is ofgreat interest to consider a scale of function classes and to look for an estimator that attains simultaneously the best rates of convergence across the whole scale. For example, the $L_q-$ Sobolev scale is a set of Sobolev function classes $W_q^s\left(C\right)$ indexed by the parameters $s$ and $C$ , see [link] for the definition of a Sobolev class. We now formalize the notion of an adaptive estimator.

Let $A$ be a given set and let $\{{\mathcal{F}}_{\alpha },\alpha \in A\}$ be the scale of functional classes ${\mathcal{F}}_{\alpha }$ indexed by $\alpha \in A.$ Denote $R_n(\alpha ,p)$ the minimax risk over ${\mathcal{F}}_{\alpha }$ for the $L_p-$ loss:

The estimator $m_n^{*}$ is called adaptive up to a logarithmic factor for the $L_p-$ loss and the scale of classes ${\mathcal{F}}_{\alpha },\alpha \in A$ if for any $\alpha \in A$ there exist $c_{\alpha }>0$ and $\gamma ={\gamma }_{\alpha }>0$ such that

Thus, adaptive estimators have an optimal rate of convergence and behave as if they know in advance in which class the function to be estimated lies.

The VisuShrink procedure is adaptive up to a logarithmic factor for the $L_2-$ loss over every Besov, Hölder and Sobolev class that is contained in $C[0,1]$ , see Theorem 1.2 in [link] . The SureShrink estimator does better: it is adaptive for the $L_2-$ loss, for a large scale of Besov, Hölder and Sobolev classes, see Theorem 1 in [link] .

Conclusion

In this chapter, we saw the basic properties of standard wavelet theory and explained how these are related to the construction of wavelet regression estimators.