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 vertical bar 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 root n rate under an explicit parameter characterization. Moreover, we prove that polynomial partitioning estimates achieve the 1-dimensional min-max rate for regression of Holder functions when combined to any root 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.

Conditional regression for single-index models / A. Lanteri, M. Maggioni, S. Vigogna. - In: BERNOULLI. - ISSN 1350-7265. - 28:4(2022 Nov), pp. 3051-3078. [10.3150/22-BEJ1482]

Conditional regression for single-index models

A. Lanteri
Primo
;
2022-11

Abstract

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 vertical bar 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 root n rate under an explicit parameter characterization. Moreover, we prove that polynomial partitioning estimates achieve the 1-dimensional min-max rate for regression of Holder functions when combined to any root 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.
Single-index model; dimension reduction; nonparametric regression; finite-sample bounds;
Settore SECS-S/01 - Statistica
Settore MAT/06 - Probabilita' e Statistica Matematica
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2434/938368
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