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We introduce a new deal of kernel density estimation using an exponentiatedform of kernel density estimators. The density estimator has twohyperparameters flexibly controlling the smoothness of the resulting density.We tune them in a data-driven manner by minimizing an objective function basedon the Hyv\"arinen score to avoid the optimization involving the intractablenormalizing constant due to the exponentiation. We show the asymptoticproperties of the proposed estimator and emphasize the importance of includingthe two hyperparameters for flexible density estimation. Our simulation studiesand application to income data show that the proposed density estimator isappealing when the underlying density is multi-modal or observations containoutliers.

Fully Data-driven Normalized and Exponentiated Kernel Density Estimator with Hyv\"arinen Score