Lower bound in regression for functional data by representation of small ball probabilities

Abstract

The minimax rate for estimating the regression function $r(\cdot)=\mathbb{E}(y|X=\cdot)$ when $y\in\mathbb{R}$ and $X$ takes values in a function space is the initial motivation of this work. Recent articles underline the major role of the shifted small ball probability $\mathbb{P}(\Vert X-x_{0}\Vert <\cdot)$ in the variance of classical estimates. The main results are twofold. First, starting from a theorem by [41], we study the small ball probability $\mathbb{P}(S<\varepsilon)$ when $\varepsilon\downarrow0$ with $S=\sum_{i=1}^{+\infty}\lambda_{i}Z_{i}$ where the $Z_{i}$’s are i.i.d. positive and $(\lambda_{i})_{i\in\mathbb{N}}$ a positive nonincreasing sequence such that $\sum\lambda_{i}<+\infty$. It is shown that $\mathbb{P}(S<\cdot)$ 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. Second this approach allows to derive minimax lower bounds for the risk at a fixed point $x_{0}$ when $X\in\mathcal{H}$ some Hilbert space of functions. Denoting this minimax risk: \[\mathcal{R}_{n}^{\ast}=\inf_{T_{n}}\sup_{r\in\mathcal{E}}\mathbb{E}\left\vert T_{n}-r(x_{0})\right\vert ^{2}\] where $T_{n}$ is any estimate of $r(x_{0})$ and $\mathcal{E}$ is some class of smooth functions from $\mathcal{H}$ to $\mathbb{R}$ it turns out that, in a general framework, $n^{\tau}\mathcal{R}_{n}^{\ast}\rightarrow+\infty$ for any $\tau>0.$ This negative result may pave the way towards new approaches for modeling regression with functional data.