The linearity condition and adaptive estimation in single-index regressions

TL;DR

This paper demonstrates that under a linearity condition, the coefficient in single-index regression can be estimated efficiently without knowing the link function, effectively replacing the need for the exact conditional distribution.

Contribution

It introduces a linearity condition that allows for efficient estimation of the coefficient in single-index models without requiring the link function to be known.

Findings

Coefficient estimation efficiency matches the known link function case

Linearity condition substitutes for knowledge of the response distribution

Estimation remains effective under the linearity assumption

Abstract

We show that under a linearity condition on the distribution of the predictors, the coefficient in single-index regression can be estimated with the same efficiency as in the case when the link function is known. Thus, the linearity condition seems to substitute for knowing the exact conditional distribution of the response given the linear combinations of the predictors.