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Estimating a density and its derivatives via the minimum distance method
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  • Published: February 1989

Estimating a density and its derivatives via the minimum distance method

  • Lesław Gajek1 

Probability Theory and Related Fields volume 80, pages 601–617 (1989)Cite this article

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  • 3 Citations

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Summary

This paper deals with minimum distance (MD) estimators and minimum penalized distance (MPD) estimators which are based on the L p distance. Rates of strong consistency of MPD density estimators are established within the family of density functions which have a bounded m-th derivative. For the case p=2, it is also proved that the MPD density estimator achieves the optimum rate of decrease of the mean integrated square error and the L 1 error. Estimation of derivatives of the density is considered as well.

In a class parametrized by entire functions, it is proved that the rate of convergence of the MD density estimator (and its derivatives) to the unknown density (its derivatives) is of order \(1{\text{/}}\sqrt n\)in expected L 1 and L 2 distances. In the same class of distributions, MD estimators of unknown density and its derivatives are proved to achieve an extraordinary rate (log log n/n)1/2 of strong consistency.

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Authors and Affiliations

  1. Institute of Mathematics, Technical University of Łódź, Al. Politechniki 11, PL-90-924, łódź, Poland

    Lesław Gajek

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  1. Lesław Gajek
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Gajek, L. Estimating a density and its derivatives via the minimum distance method. Probab. Th. Rel. Fields 80, 601–617 (1989). https://doi.org/10.1007/BF00318908

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  • Received: 30 June 1987

  • Revised: 20 July 1988

  • Issue Date: February 1989

  • DOI: https://doi.org/10.1007/BF00318908

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Keywords

  • Density Function
  • Stochastic Process
  • Probability Theory
  • Minimum Distance
  • Statistical Theory
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