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Multinomial approximations for nonparametric experiments which minimize the maximal loss of Fisher information
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  • Published: June 1989

Multinomial approximations for nonparametric experiments which minimize the maximal loss of Fisher information

  • S. Luckhaus1 &
  • W. Sauermann2 

Probability Theory and Related Fields volume 81, pages 159–184 (1989)Cite this article

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Summary

Certain nonparametric product experimentsP n n can asymptotically be approximated by multinomial experiments obtained by a finite interval partition of the sample space, the real line. For specific familiesP n defined in terms of bounded Fisher information and monotone likelihood ratios with bounded derivatives we study the problem to calculate a partition which is optimal in the sense that it minimizes the maximal loss of Fisher information caused by the discretization. This leads to a minimax problem. By considering partitions of the sample space intok intervals which have equal probability under a densityh and then lettingk→∞ we obtain an expansion for the quantity “loss of Fisher information” which is of orderk -2 under regularity conditions. The corresponding minimax problem for the first order term of this expansion is shown to be the unique solution of a free boundary problem.

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Author information

Authors and Affiliations

  1. Sonderforschungsbereich 123, Universität Heidelberg, Im Neuenheimer Feld 293, D-6900, Heidelberg 1, Germany

    S. Luckhaus

  2. Biometrics Department, Gödecke AG, Mooswaldallee 1-9, D-7800, Freiburg, Germany

    W. Sauermann

Authors
  1. S. Luckhaus
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  2. W. Sauermann
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Additional information

This work has been supported by the Deutsche Forschungsgemeinschaft

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Cite this article

Luckhaus, S., Sauermann, W. Multinomial approximations for nonparametric experiments which minimize the maximal loss of Fisher information. Probab. Th. Rel. Fields 81, 159–184 (1989). https://doi.org/10.1007/BF00319548

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  • Received: 21 December 1987

  • Revised: 22 July 1988

  • Issue Date: June 1989

  • DOI: https://doi.org/10.1007/BF00319548

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Keywords

  • Likelihood Ratio
  • Stochastic Process
  • Unique Solution
  • Order Term
  • Free Boundary
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