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摘要翻译：
在这里，我们给出了一个关于低维目标参数$\alpha$的有效的后选择或后正则化推理的解释性的一般分析，在存在一个非常高维的干扰参数$\eta$的情况下，该参数是用现代选择或正则化方法估计的。我们的分析依赖于高层次的、易于解释的条件，这些条件允许人们清楚地看到实现有效的后正则化推理所需的结构。给出了一类仿射二次模型的简单、易验证的充分条件。我们集中讨论基于使用理论方程$$m(\alpha,\eta)=0$$的经验模拟的估计和推断过程，这些方程识别$\alpha$。在这个结构中，我们证明了这样的方程的建立使得真参数值处的正交性/免疫条件$$\partial\eta M(\alpha,\eta)=0$$被满足，加上$M$的光滑性和估计量$\hat\eta$的质量的合理条件，保证了基于下面讨论的测试或点估计方法对主要参数$\alpha的推断将是规则的，尽管在$\eta$的估计中出现了选择或正则化偏差。特别地，$\alpha$的估计量通常在根-$n$率上一致相合，并且一致渐近正态，即使估计量$\hat\eta$通常不是渐近线性和正则的。这种一致性适用于大类模型，而这些模型并没有施加非常不可信的“贝塔-最小”条件。我们还证明了由Neyman的$C(\alpha)$（正交分数）统计量形成的反向检验可以进行推理。
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英文标题：
《Valid Post-Selection and Post-Regularization Inference: An Elementary,
General Approach》
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作者：
Victor Chernozhukov and Christian Hansen and Martin Spindler
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最新提交年份：
2015
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分类信息：
一级分类：Mathematics 数学
二级分类：Statistics Theory 统计理论
分类描述：Applied, computational and theoretical statistics: e.g. statistical inference, regression, time series, multivariate analysis, data analysis, Markov chain Monte Carlo, design of experiments, case studies
应用统计、计算统计和理论统计：例如统计推断、回归、时间序列、多元分析、数据分析、马尔可夫链蒙特卡罗、实验设计、案例研究
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一级分类：Economics 经济学
二级分类：Econometrics 计量经济学
分类描述：Econometric Theory, Micro-Econometrics, Macro-Econometrics, Empirical Content of Economic Relations discovered via New Methods, Methodological Aspects of the Application of Statistical Inference to Economic Data.
计量经济学理论，微观计量经济学，宏观计量经济学，通过新方法发现的经济关系的实证内容，统计推论应用于经济数据的方法论方面。
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一级分类：Statistics 统计学
二级分类：Statistics Theory 统计理论
分类描述：stat.TH is an alias for math.ST. Asymptotics, Bayesian Inference, Decision Theory, Estimation, Foundations, Inference, Testing.
Stat.Th是Math.St的别名。渐近，贝叶斯推论，决策理论，估计，基础，推论，检验。
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英文摘要：
Here we present an expository, general analysis of valid post-selection or post-regularization inference about a low-dimensional target parameter, $\alpha$, in the presence of a very high-dimensional nuisance parameter, $\eta$, which is estimated using modern selection or regularization methods. Our analysis relies on high-level, easy-to-interpret conditions that allow one to clearly see the structures needed for achieving valid post-regularization inference. Simple, readily verifiable sufficient conditions are provided for a class of affine-quadratic models. We focus our discussion on estimation and inference procedures based on using the empirical analog of theoretical equations $$M(\alpha, \eta)=0$$ which identify $\alpha$. Within this structure, we show that setting up such equations in a manner such that the orthogonality/immunization condition $$\partial_\eta M(\alpha, \eta) = 0$$ at the true parameter values is satisfied, coupled with plausible conditions on the smoothness of $M$ and the quality of the estimator $\hat \eta$, guarantees that inference on for the main parameter $\alpha$ based on testing or point estimation methods discussed below will be regular despite selection or regularization biases occurring in estimation of $\eta$. In particular, the estimator of $\alpha$ will often be uniformly consistent at the root-$n$ rate and uniformly asymptotically normal even though estimators $\hat \eta$ will generally not be asymptotically linear and regular. The uniformity holds over large classes of models that do not impose highly implausible "beta-min" conditions. We also show that inference can be carried out by inverting tests formed from Neyman's $C(\alpha)$ (orthogonal score) statistics.
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PDF链接：
https://arxiv.org/pdf/1501.03430
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