The Annals of Statistics

The EFM approach for single-index models

Xia Cui, Wolfgang Karl Härdle, and Lixing Zhu

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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.

Article information

Ann. Statist., Volume 39, Number 3 (2011), 1658-1688.

First available in Project Euclid: 25 July 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G08: Nonparametric regression 62G08: Nonparametric regression 62G20: Asymptotic properties

Single-index models index coefficients estimating equations asymptotic properties iteration


Cui, Xia; Härdle, Wolfgang Karl; Zhu, Lixing. The EFM approach for single-index models. Ann. Statist. 39 (2011), no. 3, 1658--1688. doi:10.1214/10-AOS871.

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