  We introduce a new deal of kernel density estimation using an exponentiated
form of kernel density estimators. The density estimator has two
hyperparameters flexibly controlling the smoothness of the resulting density.
We tune them in a data-driven manner by minimizing an objective function based
on the Hyv\"arinen score to avoid the optimization involving the intractable
normalizing constant due to the exponentiation. We show the asymptotic
properties of the proposed estimator and emphasize the importance of including
the two hyperparameters for flexible density estimation. Our simulation studies
and application to income data show that the proposed density estimator is
appealing when the underlying density is multi-modal or observations contain
outliers.
