The EFM approach for single-index models

By Xia Cui, Wolfgang Karl Härdle, Lixing Zhu. 2011

177 citations recorded by the source corpus.

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 $\bolds{\beta}$ is one of the most important objectives in the statistical analysis. However, the commonly used assumption on the index coefficients, $\|\bolds{\beta}\|=1$, 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 $\bolds{\beta}$ 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 $\bolds{\beta}$ 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.

Subfields: Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/da99ff5f58aa28fbee9ee5ec3afea6494db926ef

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Citations in the corpus

SIMEX estimation for single-index model with covariate measurement error (2016, background): “Similar to Cui et al. (2011), we discuss the solution of the estimating equation.”
Nonparametric Shape-Restricted Regression (2017, background): “, [111], [87], [37] and the references therein), estimation and inference in shape-restricted single index models are not very well-developed, despite their numerous applications.”
Score estimation in the monotone single‐index model (2017, methodology): “Examples of this type are the semi-parametric least squares estimators of Ichimura (1993) andHärdle et al. (1993) and the pseudo-maximum likelihood estimator of Delecroix et al.”