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June 2011 The EFM approach for single-index models
Xia Cui, Wolfgang Karl Härdle, Lixing Zhu
Ann. Statist. 39(3): 1658-1688 (June 2011). DOI: 10.1214/10-AOS871


Single-index models are natural extensions of linear models and circumvent the so-called curse of dimensionality. They are becoming increasingly popular in many scientific fields including biostatistics, medicine, economics and financial econometrics. Estimating and testing the model index coefficients β is one of the most important objectives in the statistical analysis. However, the commonly used assumption on the index coefficients, ‖β‖ = 1, represents a nonregular problem: the true index is on the boundary of the unit ball. In this paper we introduce the EFM approach, a method of estimating functions, to study the single-index model. The procedure is to first relax the equality constraint to one with (d − 1) components of β lying in an open unit ball, and then to construct the associated (d − 1) estimating functions by projecting the score function to the linear space spanned by the residuals with the unknown link being estimated by kernel estimating functions. The root-n consistency and asymptotic normality for the estimator obtained from solving the resulting estimating equations are achieved, and a Wilks type theorem for testing the index is demonstrated. A noticeable result we obtain is that our estimator for β has smaller or equal limiting variance than the estimator of Carroll et al. [J. Amer. Statist. Assoc. 92 (1997) 447–489]. A fixed-point iterative scheme for computing this estimator is proposed. This algorithm only involves one-dimensional nonparametric smoothers, thereby avoiding the data sparsity problem caused by high model dimensionality. Numerical studies based on simulation and on applications suggest that this new estimating system is quite powerful and easy to implement.


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Xia Cui. Wolfgang Karl Härdle. Lixing Zhu. "The EFM approach for single-index models." Ann. Statist. 39 (3) 1658 - 1688, June 2011.


Published: June 2011
First available in Project Euclid: 25 July 2011

zbMATH: 1221.62062
MathSciNet: MR2850216
Digital Object Identifier: 10.1214/10-AOS871

Primary: 62G08 , 62G08 , 62G20

Keywords: asymptotic properties , estimating equations , index coefficients , iteration , Single-index models

Rights: Copyright © 2011 Institute of Mathematical Statistics

Vol.39 • No. 3 • June 2011
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