估計單指標模型中單一解釋變數的主效應

Efromovich (2005)提出利用輔助變數估計一元無母數迴歸的方法，並委展出在輔助變數存在時無母數迴歸的漸進最佳預利(asymptotically optimal nonparametric univariate regression estimation in the presence of auxiliary covariate)的定理。Efromovich(2005)先估計h(X,Z)=E(Y|X,Z)-e(Y|X)然後用Y扣掉它，產生峰噪散點圖(denoised scattergram)，以此來估計E(Y|X=x)，這裡Z是一個輔助變數。本篇論文討論在單指標模型中，如何估計單一解釋變數的主效應，我們以Efromovich (2005)文章中的方法爲基礎，對h(x, z)重新估計，產生降噪散點圖(denoised scattergram)來估計f(x)=E(Y|X=x),並證明此估計也是一個漸進尖銳極小化最大值估計(asymptotic sharp minimax estimate)。

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) = E(Y|X = x,Z = z) − E(Y|X = x) 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 models. The problem is to estimate the main effect of a covariate in single-index models. We employ the techniques of Efromovich (2005) to estimate the main effect and prove the obtained denoise scattergram yields an asymptotic sharp minimax estimate.