Electronic Journal of Statistics

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

André Mas

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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.

Article information

Source
Electron. J. Statist., Volume 6 (2012), 1745-1778.

Dates
First available in Project Euclid: 27 September 2012

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1348753351

Digital Object Identifier
doi:10.1214/12-EJS726

Mathematical Reviews number (MathSciNet)
MR2988463

Zentralblatt MATH identifier
1295.62043

Subjects
Primary: 62G08: Nonparametric regression 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 62G20: Asymptotic properties

Keywords
Functional data small ball problems nonparametric regression regular variation Gaussian random elements lower bound

Citation

Mas, André. Lower bound in regression for functional data by representation of small ball probabilities. Electron. J. Statist. 6 (2012), 1745--1778. doi:10.1214/12-EJS726. https://projecteuclid.org/euclid.ejs/1348753351


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