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摘要翻译：
在应用中，条件期望的确切形式是未知的，具有灵活的函数形式可以导致改进。级数法提供了一种基于$k$基函数逼近未知函数的方法，其中$k$允许随样本量$n$增长。我们考虑了条件均值的级数估计：(i)由非交换Khinchin不等式导出的矩阵的尖锐LLNs，(ii)控制逼近误差的$L^\infty$和$L2$-范数之比的Lebesgue因子的界，(iii)熵积分发散过程的极大不等式，(iv)级数型过程的强逼近。这些技术工具使我们能够为系列文献做出贡献，特别是Newey(1997)的开创性工作，如下所示。首先，我们将级数估计中的逼近函数数目的条件从典型的$k2/n\0到$k/n\0到$k/n\0,直至对数因子，而这一条件以前只适用于样条级数。其次，当逼近误差消失时，我们得到了$L_2$速率和逐点中心极限定理的结果。在模型不正确的情况下，即当逼近误差不消失时，也给出了类似的结果。第三，在更强的条件下，我们得到了一致速率和泛函中心极限定理，无论逼近误差是否消失，这些定理都成立。也就是说，我们得到了非参数函数整个估计的强逼近。我们对条件期望函数的一个广泛的线性函数集合导出了一致的速率、高斯近似和一致的置信带。
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英文标题：
《Some New Asymptotic Theory for Least Squares Series: Pointwise and
Uniform Results》
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作者：
Alexandre Belloni, Victor Chernozhukov, Denis Chetverikov and Kengo
Kato
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最新提交年份：
2015
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分类信息：
一级分类：Statistics 统计学
二级分类：Methodology 方法论
分类描述：Design, Surveys, Model Selection, Multiple Testing, Multivariate Methods, Signal and Image Processing, Time Series, Smoothing, Spatial Statistics, Survival Analysis, Nonparametric and Semiparametric Methods
设计，调查，模型选择，多重检验，多元方法，信号和图像处理，时间序列，平滑，空间统计，生存分析，非参数和半参数方法
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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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英文摘要：
In applications it is common that the exact form of a conditional expectation is unknown and having flexible functional forms can lead to improvements. Series method offers that by approximating the unknown function based on $k$ basis functions, where $k$ is allowed to grow with the sample size $n$. We consider series estimators for the conditional mean in light of: (i) sharp LLNs for matrices derived from the noncommutative Khinchin inequalities, (ii) bounds on the Lebesgue factor that controls the ratio between the $L^\infty$ and $L_2$-norms of approximation errors, (iii) maximal inequalities for processes whose entropy integrals diverge, and (iv) strong approximations to series-type processes. These technical tools allow us to contribute to the series literature, specifically the seminal work of Newey (1997), as follows. First, we weaken the condition on the number $k$ of approximating functions used in series estimation from the typical $k^2/n \to 0$ to $k/n \to 0$, up to log factors, which was available only for spline series before. Second, we derive $L_2$ rates and pointwise central limit theorems results when the approximation error vanishes. Under an incorrectly specified model, i.e. when the approximation error does not vanish, analogous results are also shown. Third, under stronger conditions we derive uniform rates and functional central limit theorems that hold if the approximation error vanishes or not. That is, we derive the strong approximation for the entire estimate of the nonparametric function. We derive uniform rates, Gaussian approximations, and uniform confidence bands for a wide collection of linear functionals of the conditional expectation function.
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PDF链接：
https://arxiv.org/pdf/1212.0442
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