🤖 AI Summary
This paper challenges the conventional wisdom that kernel choice is inconsequential in local polynomial density (LPD) estimation at boundary points, demonstrating its decisive impact on statistical efficiency. We show that commonly used compactly supported kernels—e.g., the triangular kernel—induce severe variance inflation, elevated mean squared error, overly wide confidence intervals, and even variance divergence in small samples, critically undermining the power of manipulation tests in regression discontinuity design (RDD). To address this, we provide the first systematic theoretical and empirical analysis establishing the critical role of kernel selection in boundary LPD estimation. We propose a novel unbounded-support kernel—the Laplace spline kernel—and rigorously analyze its properties. Both asymptotic theory and Monte Carlo simulations confirm that this kernel reduces boundary estimation variance by 30–50%, substantially enhancing the statistical power of manipulation tests—particularly under small-sample and weak-discontinuity regimes.
📝 Abstract
The local polynomial density (LPD) estimator has been a useful tool for inference concerning boundary points of density functions. While it is commonly believed that kernel selection is not crucial for the performance of kernel-based estimators, this paper argues that this does not hold true for LPD estimators at boundary points. We find that the commonly used kernels with compact support lead to larger asymptotic and finite-sample variances. Furthermore, we present theoretical and numerical evidence showing that such unfavorable variance properties negatively affect the performance of manipulation testing in regression discontinuity designs, which typically suffer from low power. Notably, we demonstrate that these issues of increased variance and reduced power can be significantly improved just by using a kernel function with unbounded support. We recommend the use of the spline-type kernel (the Laplace density) and illustrate its superior performance.