Estimating the main effect of a covariate in single-index models

The article Efromovich (2005) addresses the problem of finding a relationship between the univariate predictor and the response when regression errors, created in part by known auxiliary covariates, are too large for a reliable regression estimation. The proposed solution of Efromovich (2005) is to estimate the noise component
h(x,z) and substract it from the response and the obtained denoise scattergram yields the optimal estimation of the regression function. Besides, Efromovich (2005) develops a theory of asymptotically optimal nonparametric univariate regression estimation in the presence of auxiliary covariates. This article discusses the problem under single index model. The problem is to estimate
the main effect of a covariate in single-index models. I estimate h(x,z) and substract it from the response and prove the obtained denoise scattergram yields
an asymptotic sharp minimax estimate。