Weighted samples, kernel density estimators and convergence

This note extends the standard kernel density estimator to the case of weighted samples in several ways. In the first place I consider the obvious extension by substituting the simple sum in the definition of the estimator by a weighted sum, but I also consider other alternatives of introducing weights, based on adaptive kernel density estimators, and consider the weights as indicators of the informational content of the observations and in this sense as signals of the local density of the data. All these ideas are shown using the Penn World Table in the context of the macroeconomic convergence issue.