Density estimation for log-transformed data

For data after the logarithmic transformation, we compared performances of two types of estimators. The transformed data (TD) estimator is simply the kernel density estimator (KDE) computed from the log-transformed data. The plug-in (PI) estimator uses KDE to replace the unknown density f of the original sample in the formula for the density g of the log-transformed sample. We developed the large sample theory and proposed practical implementations for the PI estimator. We found that a single-bandwidth PI estimator tends to produce density estimates that are bumpy in the right tail, whereas the adaptive PI estimators based on the estimated mean squared error (MSE) optimal bandwidth function overcome this disadvantage. Our best performing adaptive PI estimator is based on replacing g by the normal reference density. Such an estimator is referred to as the adaptive normal PI estimator . Even though the TD estimator is difficult to beat in the considered setting, still the adaptive normal PI estimator showed superior performance around the modes of g and, overall, in the multimodal settings.