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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This work has been supported by the Deutsche Forschungsgemeinschaft
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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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DOI: https://doi.org/10.1007/BF00319548