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Adaptive density estimation based on real and artificial data

Author

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  • Tina Felber
  • Michael Kohler
  • Adam Krzyżak

Abstract

Let X, X 1 , X 2 , ... be independent and identically distributed ℝ-super- d -valued random variables and let m :ℝ-super- d →ℝ be a measurable function such that a density f of Y = m ( X ) exists. The problem of estimating f based on a sample of the distribution of ( X,Y ) and on additional independent observations of X is considered. Two kernel density estimates are compared: the standard kernel density estimate based on the y -values of the sample of ( X,Y ), and a kernel density estimate based on artificially generated y -values corresponding to the additional observations of X . It is shown that under suitable smoothness assumptions on f and m the rate of convergence of the L 1 error of the latter estimate is better than that of the standard kernel density estimate. Furthermore, a density estimate defined as convex combination of these two estimates is considered and a data-driven choice of its parameters (bandwidths and weight of the convex combination) is proposed and analysed.

Suggested Citation

  • Tina Felber & Michael Kohler & Adam Krzyżak, 2015. "Adaptive density estimation based on real and artificial data," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 27(1), pages 1-18, March.
    • Handle: RePEc:taf:gnstxx:v:27:y:2015:i:1:p:1-18
    DOI: 10.1080/10485252.2014.969729
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    Cited by:

    1. Kohler, Michael, 2014. "Optimal global rates of convergence for noiseless regression estimation problems with adaptively chosen design," Journal of Multivariate Analysis, Elsevier, vol. 132(C), pages 197-208.
    2. Benedict Götz & Sebastian Kersting & Michael Kohler, 2021. "Estimation of an improved surrogate model in uncertainty quantification by neural networks," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 73(2), pages 249-281, April.
    3. Michael Kohler & Adam Krzyżak, 2020. "Estimating quantiles in imperfect simulation models using conditional density estimation," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 72(1), pages 123-155, February.

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