标题： 高维单指数模型的潜在因子推断 显示英文标题

标题： High-dimensional inference for single-index model with latent factors

摘要： 带有潜在因子的模型最近引起了广泛关注。 然而，大多数研究集中在线性回归模型上，因此无法捕捉非线性。 为了解决这个问题，我们提出了一种新颖的因子增强单指数模型。 我们首先通过引入一个得分类型检验统计量来解决是否需要考虑增强部分的问题。 与之前的检验统计量相比，我们提出的检验统计量不需要估计高维回归系数，也不需要估计高维精度矩阵，从而在实现上更为简单。 我们还提出了一种高斯乘子引导法来确定临界值。 在适当条件下，我们程序的有效性得到了理论上的证明。 我们进一步研究了回归模型的惩罚估计。 在估计出潜在因子后，我们建立了估计量的误差界限。 最后，我们介绍了去偏估计量，并基于渐近正态性构建了个体系数的置信区间。 我们的方法没有对误差项施加任何矩条件。 因此，当随机误差服从重尾分布或存在异常值时，我们的方法表现良好。 我们通过全面的数值研究和其在FRED-MD宏观经济数据集上的应用来展示所提方法的有限样本性能。 显示英文摘要

摘要： Models with latent factors recently attract a lot of attention. However, most investigations focus on linear regression models and thus cannot capture nonlinearity. To address this issue, we propose a novel Factor Augmented Single-Index Model. We first address the concern whether it is necessary to consider the augmented part by introducing a score-type test statistic. Compared with previous test statistics, our proposed test statistic does not need to estimate the high-dimensional regression coefficients, nor high-dimensional precision matrix, making it simpler in implementation. We also propose a Gaussian multiplier bootstrap to determine the critical value. The validity of our procedure is theoretically established under suitable conditions. We further investigate the penalized estimation of the regression model. With estimated latent factors, we establish the error bounds of the estimators. Lastly, we introduce debiased estimator and construct confidence interval for individual coefficient based on the asymptotic normality. No moment condition for the error term is imposed for our proposal. Thus our procedures work well when random error follows heavy-tailed distributions or when outliers are present. We demonstrate the finite sample performance of the proposed method through comprehensive numerical studies and its application to an FRED-MD macroeconomics dataset.