2020/02/23 by Alessandro Lanteri, Lanteri, Alessandro, Mauro Maggioni +3 · 1 citation
Mathematics · #62G05 (Primary) 62G08 #62H99 (Secondary) #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Random Matrices and Applications #Statistical Methods and Inference #Statistics Theory (math.ST) #math.ST #msc:62G05 #msc:62G08 #msc:62H99 #stat.TH
paper · pdf · doi:10.48550/arxiv.2002.10008
openalex publication_date 2020/02/23 · arxiv created 2022/05/27 · arxiv updated 2022/05/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
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 𝔼 [ Y | X ] = f ( ⟨ v , X ⟩ ) for some unknown index vector v and 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 √(n) rate under an explicit parameter characterization. Moreover, we prove that polynomial partitioning estimates achieve the 1-dimensional min-max rate for regression of Hölder functions when combined to any √(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.