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The EFM approach for single-index models. (English) Zbl 1221.62062
Summary: 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 $$\mathbf{\beta}$$ is one of the most important objectives in the statistical analysis. However, the commonly used assumption on the index coefficients, $$\parallel \mathbf{\beta} \parallel = 1$$, represents a non-regular problem: the true index is on the boundary of the unit ball. 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 $$\mathbf{\beta}$$ 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 $$\mathbf{\beta}$$ has smaller or equal limiting variance than the estimator of R. J. Carroll et al. [J. Am. Stat. Assoc. 92, No. 438, 447–489 (1997; Zbl 0890.62053)]. 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.

##### MSC:
 62G08 Nonparametric regression and quantile regression 62G20 Asymptotic properties of nonparametric inference 62H12 Estimation in multivariate analysis 65C60 Computational problems in statistics (MSC2010)
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