The EFM approach for single-index models
单指标模型的估计函数方法（EFM）AI 解读 ↓
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单指标模型的估计函数方法（EFM）
- 研究问题
- 单指标系数的单位范数约束带来非正则问题时，如何构造便于计算的估计与检验，并改善相对既有估计的渐近方差？
- 方法与思路
- 作者对指标系数作降维重参数化，结合得分投影和核估计函数构造估计方程，并提出仅用一维平滑器的固定点迭代。
- 材料陈述的贡献
- 摘要称获得根号样本量一致性、渐近正态性和Wilks型检验，且指标系数估计的极限方差不大于所比较的既有方法。
- 阅读关注点
- 应核对重参数化的识别约束、投影空间与核平滑条件、方差比较适用范围，以及固定点迭代的收敛和数值实现。
- 阅读边界
- 仅据版本未注明的OpenAlex摘要；定理条件及模拟细节未说明，不能将特定渐近方差比较解释为普遍有限样本优势。
原文摘要
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.