Smoothing Bias in Density Derivative Estimation

This article discusses a generic feature of density estimation by local smoothing, namely that estimated derivatives and location score vectors will display a systematic downward (attenuation) bias. We study the behavior of kernel estimators, indicating how the derivative bias arises and showing a simple result. We then consider the estimation of score vectors (negative log-density derivatives), which are motivated by the problem of estimating average derivatives and the adaptive estimation of regression models. Using “fixed bandwidth” limits, we show how scores are proportionally downward biased for normal densities and argue from normal mixture densities that proportional bias can be a reasonable approximation. We propose a simple diagnostic statistic for score bias.