证明了k近邻回归在复杂抽样设计下的一致性及其收敛速度。
Consistency of the $k$-Nearest Neighbor Regressor under Complex Survey Designs
- 在复杂抽样设计下建立k近邻回归的理论一致性条件。
- 推导出收敛速率下界，证实维度灾难现象存在。
- 适用于带复杂抽样结构的数据分析，如调查数据建模。
我们研究了在复杂调查设计下k-最近邻回归的一致性。尽管该算法在独立同分布数据下的一致性结果已很成熟，但针对复杂调查数据的结果仍不充分。本文证明，在抽样设计和数据分布满足一定正则性条件下，k-最近邻回归具有一致性。我们推导了收敛速率的下界，发现其表现出与独立同分布情形相同的维度灾难特性。基于模拟数据和真实数据的实证研究验证了理论结论。
原文摘要 · Abstract (English)
We study the consistency of the $k$-nearest neighbor regressor under complex survey designs. While consistency results for this algorithm are well established for independent and identically distributed data, corresponding results for complex survey data are lacking. We show that the $k$-nearest neighbor regressor is consistent under regularity conditions on the sampling design and the distribution of the data. We derive lower bounds for the rate of convergence and show that these bounds exhibit the curse of dimensionality, as in the independent and identically distributed setting. Empirical studies based on simulated and real data illustrate our theoretical findings.
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