Consistency of the $k$-Nearest Neighbor Regressor under Complex Survey Designs
This addresses a gap in theoretical guarantees for a common algorithm in survey data analysis, but it is incremental as it extends known results to a specific data setting.
The paper tackles the consistency of the k-nearest neighbor regressor under complex survey designs, showing it is consistent with derived lower bounds for convergence rates that exhibit the curse of dimensionality, as illustrated by empirical studies.
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.