Density estimation for power transformations

Consider a random sample X 1 , ..., X n from a density f and a positive α. The density g of t ( X 1 )=&7C X 1 &7C-super-αsign( X 1 ) can be estimated in two ways: by a kernel estimator based on the transformed data t ( X 1 ), ..., t ( X n ) or by a plug-in estimator that replaces in the expression for g the unknown density f by a kernel estimator based on the original data. We compare the performance of these two estimators pointwise using the MSE and globally using the mean integrated squared error. From the pointwise comparison, we found that the plug-in estimator is mostly better in the case α>1 when f is symmetric and unimodal, and in the case α≥2.5 when f is right-skewed and/or bimodal. For α>1, the plug-in estimator performs better around the modes of g , while the transformed data estimator is better in the tails of g . Our global comparison shows that the plug-in estimator has a faster rate of convergence for 0.4≤α>1 and 1>α>2. For α>0.4, the plug-in estimator is preferable for a symmetric density f with exponentially decaying tails, while the transformed data estimator is preferable when f is right-skewed or heavy-tailed. Applications to real and simulated data illustrate our theoretical findings.