标题： 无偏预测推断在错误指定的高维回归模型中非稀疏载荷的情况 显示英文标题

标题： Debiased Prediction Inference with Non-sparse Loadings in Misspecified High-dimensional Regression Models

摘要： 高维回归模型与正则化稀疏估计被广泛应用。对于统计推断，在可能存在模型误指的情况下，关于单个系数或具有稀疏新协变量向量（也称为载荷）的预测，有去偏方法可用。然而，关于具有非稀疏载荷的预测的统计推断仅在正确指定模型的假设下进行研究。在本工作中，我们为从可能误指的高维回归模型中具有通用载荷的预测开发了去偏估计和相关的Wald置信区间。我们的去偏估计器涉及去偏向量的估计，该向量是在线性模型（LM）设置中由非中心精度矩阵左乘通用载荷，或在广义线性模型（GLM）设置中目标系数向量处的目标函数的逆Hessian矩阵。我们在LM或GLM设置中分别提出了精度矩阵或逆Hessian的合适估计器，并首次在精度矩阵或逆Hessian的稀疏性条件下建立了去偏预测的根n渐近展开式，这些条件与现有工作中关于所有单个系数推断所需的稀疏性条件相当。我们还提供了数值结果，进一步证明了我们提出的置信区间在可能误指的回归模型中对具有通用载荷的预测的有效性。 显示英文摘要

摘要： High-dimensional regression models with regularized sparse estimation are widely applied. For statistical inferences, debiased methods are available about single coefficients or predictions with sparse new covariate vectors (also called loadings), in the presence of possible model misspecification. However, statistical inferences about predictions with non-sparse loadings are studied only under the assumption of correctly specified models. In this work, we develop debiased estimation and associated Wald confidence intervals for predictions with general loadings, allowed to be non-sparse, from possibly misspecified high-dimensional regression models. Our debiased estimator involves estimation of a debiasing vector, which is the general loading left-multiplied by the non-centered precision matrix in the linear model (LM) setting or the inverse Hessian of the objective function at the target coefficient vector in the generalized linear model (GLM) setting. We propose suitable estimators of the precision matrix or the inverse Hessian respectively in the LM or GLM settings and, for the first time, establish a root-n asymptotic expansion for the debiased prediction and justify associated Wald confidence intervals under sparsity conditions on the precision matrix or the inverse Hessian which are comparable to the conjunction of sparsity conditions required for inferences about all single coefficients in existing works. We also provide numerical results which further demonstrate the validity of our proposed confidence intervals for predictions with general loadings from possibly misspecified regression models.