On the characterization of fisher information and stability of the least favorable lattice distributions |
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Authors: | N O Vil'chevskii G L Shevlyakov |
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Institution: | (1) Department of Mathematics, St. Petersburg State Technical University, Polytekhnicheskaya. 29, 195251 St Petersburg, Russia |
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Abstract: | The paper is concerned with the stability properties of the least favorable distributions minimizing the Fisher information
in a given class of distributions. The derivation of a least favorable distribution (the solution of a variational problem)
is a necessary stage of the Huber minimax approach in robust estimation of a location parameter. Generally, the solutions
of variational problems essentially depend on the regularity restrictions of a functional class. The stability of these optimal
solutions to violations of the smoothness restrictions is studied under the lattice distribution classes. The discrete analogues
of Fisher information are obtained in these cases. They have the form of the Hellinger metrics with the estimation of a real
continuous location parameter and the form of the X2 metrics with the estimation of an integer discrete location parameter. The analytical expressions for the corresponding least
favorable discrete distributions are derived in some classes of lattice distributions by means of generating functions and
Bellman's recursive functional equations of dynamic programming. These classes include the class of nondegenerate distributions
with a restriction on the value of the density in the center of symmetry, the class of finite distributions, and the class
of contaminated distributions. The obtained least favorable lattice distributions preserve the structure of their prototypes
in the continuous case. These results show the stability of robust minimax solutions under different types of transitions
from the continuous distribution to the discrete one.
Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajduszoboszló, Hungary, 1997, Part II. |
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