基于深度需求神经网络的单指标模型高效估计

IF 0.8 3区 数学 Q2 MATHEMATICS
Zhihuang Yang, Siming Zheng, Niansheng Tang
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引用次数: 0

摘要

单指标模型提供了比广义线性模型更大的建模灵活性,并在一定程度上保留了模型的可解释性。虽然提出了许多标准方法,如核函数或惩罚/光滑样条来估计光滑连杆函数,但由于它们在有限样本量下的近似能力较差,无法有效地逼近复杂的未知连杆函数及其导数。为了解决这一问题,本文提出了一种基于整流二次单元(ReQU)激活深度神经网络的单指标模型半参数最小二乘估计方法,称为深度半参数最小二乘估计方法。在一些正则性条件下,我们证明了所提出的DSLS估计量的非渐近性质,并证明了指标系数估计量可以达到半参数效率。特别地,我们得到了当响应变量为条件次指数时所提出的DSLS估计量的一致性和收敛速度。这是将深度学习技术结合到单指标模型的半参数有效估计中的一次尝试。仿真研究和实例数据分析验证了所提出的DSLS估计器。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Efficient Estimation of Single-index Models with Deep ReQU Neural Networks

Single-index model offers the greater flexibility of modelling than generalized linear models and also retains the interpretability of the model to some extent. Although many standard approaches such as kernels or penalized/smooothing splines were proposed to estimate smooth link function, they cannot approximate complicated unknown link functions together with the corresponding derivatives effectively due to their poor approximation ability for a finite sample size. To alleviate this problem, this paper proposes a semiparametric least squares estimation approach for a single-index model using the rectifier quadratic unit (ReQU) activated deep neural networks, called deep semiparametric least squares (DSLS) estimation method. Under some regularity conditions, we show non-asymptotic properties of the proposed DSLS estimator, and evidence that the index coefficient estimator can achieve the semiparametric efficiency. In particular, we obtain the consistency and the convergence rate of the proposed DSLS estimator when response variable is conditionally sub-exponential. This is an attempt to incorporate deep learning technique into semiparametrically efficient estimation in a single index model. Several simulation studies and a real example data analysis are conducted to illustrate the proposed DSLS estimator.

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来源期刊
CiteScore
1.00
自引率
0.00%
发文量
138
审稿时长
14.5 months
期刊介绍: Acta Mathematica Sinica, established by the Chinese Mathematical Society in 1936, is the first and the best mathematical journal in China. In 1985, Acta Mathematica Sinica is divided into English Series and Chinese Series. The English Series is a monthly journal, publishing significant research papers from all branches of pure and applied mathematics. It provides authoritative reviews of current developments in mathematical research. Contributions are invited from researchers from all over the world.
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