Representation of small ball probabilities in Hilbert space and lower bound in regression for functional data
Résumé
Let $S=\sum_{i=1}^{+\infty}\lambda_{i}Z_{i}$ where the $Z_{i}$'s are i.d.d. positive with $\mathbb{E}\left\vert Z\right\vert ^{3}<+\infty$ and $\left( \lambda_{i}\right) _{i\in\mathbb{N}}$ a positive nonincreasing sequence such that $\sum\lambda_{i}<+\infty$. We study the small ball probability $\mathbb{P}\left( S<\varepsilon\right) $ when $\varepsilon\downarrow0$. We start from a result by Lifshits (1997) who computed this probability by means of the Laplace transform of $S$. We prove that $\mathbb{P}\left( S<\cdot\right) $ belongs to a class of functions introduced by de Haan, well-known in extreme value theory, the class of Gamma-varying functions, for which an exponential-integral representation is available. This approach allows to derive bounds for the rate in nonparametric regression for functional data at a fixed point $x_{0}$ : $\mathbb{E}\left( y|X=x_{0}% \right) $ where $\left( y_{i},X_{i}\right) _{1\leq i\leq n}$ is a sample in $\left( \mathbb{R},\mathcal{F}\right) $ and $\mathcal{F}$ is some space of functions. It turns out that, in a general framework, the minimax lower bound for the risk is of order $\left( \log n\right) ^{-\tau}$ for some $\tau>0$ depending on the regularity of the data and polynomial rates cannot be achieved.
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