  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.
