Global correction of projection estimators under local constraint

Abstract

This paper presents a general methodology for nonparametric estimation of a function s related to a nonnegative real random variable X, under a constraint of type $s(0)=c$. When a projection estimator of the target function is available, we explain how to modify it in order to obtain an estimator which satisfies the constraint. We extend risk bounds from the initial to the new estimator, and propose and study adaptive procedures for both estimators. The example of cumulative distribution function estimation illustrates the method for two different models: the multiplicative noise model ($Y=XU$ is observed, with U following a uniform distribution) and the additive noise model $(Y=X+V$ is observed where V is a nonnegative nuisance variable with known density).

Appendix: Integrated Talagrand’s inequality

The following result follows from the Talagrand concentration inequality (see Klein and Rio 2005).

Theorem A.1

Consider $n \in{\mathbb{N}}^*$, ${\mathcal{F}}$ a class at most countable of measurable functions, and $(X_i)_{i\in \{1,\ldots ,n\}}$ a family of real independent random variables. Define, for $f\in{\mathcal{F}}$, $\nu _n(f) =(1/n) \sum _{i=1}^{n} (f(X_i)-{\mathbb{E}}[f(X_i)])$, and assume that there are three positive constants M, H and v such that ${\sup }_{f\in{\mathcal{F}}} \Vert f\Vert _{\infty } \le M$, ${\mathbb{E}}[{\sup }_{f\in{\mathcal{F}}} |\nu _n(f)| ] \le H$, and ${\sup }_{f\in{\mathcal{F}}} ({1}/{n})\sum _{i=1}^{n} \text{Var}(f(X_i)) \le v$. Then for all $\alpha >0$, setting $C(\alpha )=(\sqrt{1+\alpha }-1) \wedge 1$, and $b=1/6$,

$$\begin{aligned}&{\mathbb{E}}\left[ \left( \underset{f\in{\mathcal{F}}}{\sup } |\nu _n(f)|^2-2(1+2\alpha )H^2 \right) _+ \right] \\&\quad \le \frac{4}{b} \left( \frac{v}{n} \exp \left( -b \alpha \frac{n H^2}{v} \right) + \frac{49M^2}{b C^2(\alpha )n^2} \exp \left( -\frac{\sqrt{2}b C(\alpha )\sqrt{\alpha }}{7}\frac{nH}{M} \right) \right) . \end{aligned}$$