估計單指標模型中單一解釋變數的主效應

Translated title of the contribution: Estimating the main effect of a covariate in single index models

Research output: Contribution to journalJournal articlepeer-review

Abstract

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.
Translated title of the contribution Estimating the main effect of a covariate in single index models Chinese (Traditional) 272-313 41 中國統計學報 47 4 https://doi.org/10.29973/JCSA.200912.0004 Published - 1 Dec 2009

User-Defined Keywords

• 漸進尖銳極小化最大值估計
• 輔助變數
• 降噪散點圖
• 無母數迴歸
• 單指標模型
• Asymptotic sharp minimax estimate
• auxiliary covariates
• denoised scattergram
• nonparametric regression
• single index model

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