The EFM approach for single-index models

Abstract

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

Published in at http://dx.doi.org/10.1214/10-AOS871 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (1)

• Dimension reduction based on constrained canonical correlation and variable filtering

Cited by in corpus (14)

• Nonparametric Shape-restricted Regression
• Profile least squares estimators in the monotone single index model
• The recovery of ridge functions on the hypercube suffers from the curse of dimensionality
• Semiparametric Efficiency in Convexity Constrained Single Index Model
• Fused kernel-spline smoothing for repeatedly measured outcomes in a generalized partially linear model with functional single index
• High-dimensional varying index coefficient quantile regression model
• Convergence guarantee for the sparse monotone single index model
• A semi-parametric model for target localization in distributed systems
• On an extension of the promotion time cure model
• SIMEX estimation for single-index model with covariate measurement error
• Variable Selection and Estimation for Partially Linear Single-index Models with Longitudinal Data
• Bayesian Variable Selection for Single Index Logistic Model
• A semiparametric single-index estimator for a class of estimating equation models
• Bias-corrected GEE estimation and smooth-threshold GEE variable selection for single-index models with clustered data