Local measures of association: Estimating the derivative of the regression line

A local measure of association that allows both heteroscedasticity and a non‐linear association was developed during the 1990s. The basic goal is to measure the strength of the association between X and Y, given X, when Y=θ(X)+τ(X)ε for some unknown functions θ(X) and τ(X). Application of this method requires the estimation of the derivative of θ(X). The focus in this paper is on four alternatives to a very slight modification of the method used by Doksum et al. when estimating this derivative. The main result is that in simulations, a certain robust analogue of their method dominates in terms of mean squared error, even under normality. The bias of the method is found to be small but a little larger than the bias associated with the method used by Doksum et al. The method is based in part on bootstrap bagging followed by a lowess smooth.