Fully Data-Driven Normalized and Exponentiated Kernel Density Estimator with Hyvärinen Score

Shunsuke Imai, Takuya Koriyama, Shouto Yonekura, Shonosuke Sugasawa, Yoshihiko Nishiyama

Research output: Contribution to journalArticlepeer-review


Recently, Jewson and Rossell proposed a new approach for kernel density estimation using an exponentiated form of kernel density estimators. The density estimator contained two hyperparameters that flexibly controls the smoothness of the resulting density. We tune them in a data-driven manner by minimizing an objective function based on the Hyvärinen score to avoid the optimization involving the intractable normalizing constant caused by 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 promising when the underlying density is multi-modal or when observations contain outliers.

Original languageEnglish
JournalJournal of Business and Economic Statistics
Publication statusAccepted/In press - 2024


  • Bandwidth selection
  • Density estimation
  • Fisher divergence
  • Kernel smoothing
  • Unnormalized model

ASJC Scopus subject areas

  • Statistics and Probability
  • Social Sciences (miscellaneous)
  • Economics and Econometrics
  • Statistics, Probability and Uncertainty


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