Conditional regression for single-index models
| Autor: | Lanteri, A., Maggioni, M., Vigogna, S. |
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| Rok vydání: | 2020 |
| Předmět: |
Statistics and Probability
Single-index model finite-sample bounds nonparametric regression dimension reduction FOS: Mathematics 62G05 (Primary) 62G08 62H99 (Secondary) Mathematics - Statistics Theory Statistics Theory (math.ST) Settore SECS-S/01 - Statistica Settore MAT/06 - Probabilita' e Statistica Matematica |
| DOI: | 10.48550/arxiv.2002.10008 |
| Popis: | The single-index model is a statistical model for intrinsic regression where responses are assumed to depend on a single yet unknown linear combination of the predictors, allowing to express the regression function as $ \mathbb{E} [ Y | X ] = f ( \langle v , X \rangle ) $ for some unknown \emph{index} vector $v$ and \emph{link} function $f$. Conditional methods provide a simple and effective approach to estimate $v$ by averaging moments of $X$ conditioned on $Y$, but depend on parameters whose optimal choice is unknown and do not provide generalization bounds on $f$. In this paper we propose a new conditional method converging at $\sqrt{n}$ rate under an explicit parameter characterization. Moreover, we prove that polynomial partitioning estimates achieve the $1$-dimensional min-max rate for regression of H\"older functions when combined to any $\sqrt{n}$-convergent index estimator. Overall this yields an estimator for dimension reduction and regression of single-index models that attains statistical optimality in quasilinear time. |
| Databáze: | OpenAIRE |
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