英文标题：
《Minimax Risk and Uniform Convergence Rates for Nonparametric Dyadic
Regression》
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作者：
Bryan S. Graham, Fengshi Niu, James L. Powell
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最新提交年份：
2021
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英文摘要：
Let $i=1,\\ldots,N$ index a simple random sample of units drawn from some large population. For each unit we observe the vector of regressors $X_{i}$ and, for each of the $N\\left(N-1\\right)$ ordered pairs of units, an outcome $Y_{ij}$. The outcomes $Y_{ij}$ and $Y_{kl}$ are independent if their indices are disjoint, but dependent otherwise (i.e., \"dyadically dependent\"). Let $W_{ij}=\\left(X_{i}\',X_{j}\'\\right)\'$; using the sampled data we seek to construct a nonparametric estimate of the mean regression function $g\\left(W_{ij}\\right)\\overset{def}{\\equiv}\\mathbb{E}\\left[\\left.Y_{ij}\\right|X_{i},X_{j}\\right].$ We present two sets of results. First, we calculate lower bounds on the minimax risk for estimating the regression function at (i) a point and (ii) under the infinity norm. Second, we calculate (i) pointwise and (ii) uniform convergence rates for the dyadic analog of the familiar Nadaraya-Watson (NW) kernel regression estimator. We show that the NW kernel regression estimator achieves the optimal rates suggested by our risk bounds when an appropriate bandwidth sequence is chosen. This optimal rate differs from the one available under iid data: the effective sample size is smaller and $d_W=\\mathrm{dim}(W_{ij})$ influences the rate differently.
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中文摘要：
让$i=1、\\ldots，N$索引从一些大群体中抽取的简单随机样本单位。对于每个单元，我们观察回归向量$X{i}$，对于每个$N\\left（N-1\\right）$有序单元对，结果$Y{ij}$。如果结果$Y_{ij}$和$Y_{kl}$的指数是不相交的，则它们是独立的，但在其他情况下是相依的（即“二元相依”）。设$W_{ij}=\\left（X_{i}\'，X_{j}\'\\right）$；利用采样数据，我们试图构造均值回归函数$g\\left（W_{ij}\\right）\\overset{def}{\\equiv}\\mathbb{E}\\left[\\left.Y_{ij}\\right | X_{i}，X_{j}\\right]的非参数估计我们给出了两组结果。首先，我们计算在（i）点和（ii）无穷范数下估计回归函数的极小极大风险的下界。第二，我们计算了（i）点态和（ii）一致收敛速度为二元模拟熟悉的纳达拉亚-沃森（NW）核回归估计。我们证明，当选择适当的带宽序列时，NW核回归估计达到了风险界建议的最佳速率。该最佳速率与iid数据下的可用速率不同：有效样本量较小，$d_W=\\mathrm{dim}（W_{ij}）$对速率的影响不同。
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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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