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摘要翻译：
研究了内生回归时的非参数回归问题，这是计量经济学中一个重要的非参数工具变量回归问题，也是统计学中一个难以解决的带未知算子的不适定反问题。首先建立了一个筛子估计的超范数（一致）收敛速度的一般上界，允许内生回归和弱相关数据。这一结果给出了样条和小波最小二乘回归估计在弱相关数据和重尾误差项下的最优超范数收敛速度。这个上界也给出了I.I.D.下筛NPIV估计的超范数收敛速度。数据：对于严重不适定问题，这些速率与已知的最优$L^2$-范数速率一致，对于轻度不适定问题，这些速率比最优$L^2$-范数速率慢$\log(n)$。然后，我们建立了超模损失的极大极小风险下界，该下界与样条和小波筛NPIV估计的超模率上界一致。这种超范数最优性为筛分NPIV估计器的广泛应用提供了另一个理由。对弱相关随机矩阵也给出了有用的结果。
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英文标题：
《Optimal Uniform Convergence Rates for Sieve Nonparametric Instrumental
Variables Regression》
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作者：
Xiaohong Chen and Timothy Christensen
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最新提交年份：
2013
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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：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：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：
We study the problem of nonparametric regression when the regressor is endogenous, which is an important nonparametric instrumental variables (NPIV) regression in econometrics and a difficult ill-posed inverse problem with unknown operator in statistics. We first establish a general upper bound on the sup-norm (uniform) convergence rate of a sieve estimator, allowing for endogenous regressors and weakly dependent data. This result leads to the optimal sup-norm convergence rates for spline and wavelet least squares regression estimators under weakly dependent data and heavy-tailed error terms. This upper bound also yields the sup-norm convergence rates for sieve NPIV estimators under i.i.d. data: the rates coincide with the known optimal $L^2$-norm rates for severely ill-posed problems, and are power of $\log(n)$ slower than the optimal $L^2$-norm rates for mildly ill-posed problems. We then establish the minimax risk lower bound in sup-norm loss, which coincides with our upper bounds on sup-norm rates for the spline and wavelet sieve NPIV estimators. This sup-norm rate optimality provides another justification for the wide application of sieve NPIV estimators. Useful results on weakly-dependent random matrices are also provided.
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PDF链接：
https://arxiv.org/pdf/1311.0412
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