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What is Learnable in Valiant's Theory of the Learnable? 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Estimation and testing for partially linear single-index models
Hua Liang, Xiang Liu, Runze Li, Chih-Ling Tsai · 2012-11-15 · via math.ST updates on arXiv.org

In partially linear single-index models, we obtain the semiparametrically efficient profile least-squares estimators of regression coefficients. We also employ the smoothly clipped absolute deviation penalty (SCAD) approach to simultaneously select variables and estimate regression coefficients. We show that the resulting SCAD estimators are consistent and possess the oracle property. Subsequently, we demonstrate that a proposed tuning parameter selector, BIC, identifies the true model consistently. Finally, we develop a linear hypothesis test for the parametric coefficients and a goodness-of-fit test for the nonparametric component, respectively. Monte Carlo studies are also presented.

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