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Nonparametric likelihood based estimation for a multivariate Lipschitz density

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  • Carando, Daniel
  • Fraiman, Ricardo
  • Groisman, Pablo

Abstract

We consider a problem of nonparametric density estimation under shape restrictions. We deal with the case where the density belongs to a class of Lipschitz functions. Devroye [L. Devroye, A Course in Density Estimation, in: Progress in Probability and Statistics, vol. 14, Birkhuser Boston Inc., Boston, MA, 1987] considered these classes of estimates as tailor-made estimates, in contrast in some way to universally consistent estimates. In our framework we get the existence and uniqueness of the maximum likelihood estimate as well as strong consistency. This NPMLE can be easily characterized but it is not easy to compute. Some simpler approximations are also considered.

Suggested Citation

  • Carando, Daniel & Fraiman, Ricardo & Groisman, Pablo, 2009. "Nonparametric likelihood based estimation for a multivariate Lipschitz density," Journal of Multivariate Analysis, Elsevier, vol. 100(5), pages 981-992, May.
    • Handle: RePEc:eee:jmvana:v:100:y:2009:i:5:p:981-992
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    References listed on IDEAS

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    1. Deheuvels, Paul & Einmahl, John H. J. & Mason, David M. & Ruymgaart, Frits H., 1988. "The almost sure behavior of maximal and minimal multivariate kn-spacings," Journal of Multivariate Analysis, Elsevier, vol. 24(1), pages 155-176, January.
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    Cited by:

    1. Federico Ferraccioli & Eleonora Arnone & Livio Finos & James O. Ramsay & Laura M. Sangalli, 2021. "Nonparametric density estimation over complicated domains," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 83(2), pages 346-368, April.
    2. Madeleine Cule & Richard Samworth & Michael Stewart, 2010. "Maximum likelihood estimation of a multi‐dimensional log‐concave density," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 72(5), pages 545-607, November.

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