Global Optimization Decomposition Methods (Gourdin et al.)
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SIAM Journal on Scientific Computing
Volume 15-1, January 1994, pp. 16-35
(C) 1994 by Society for Industrial and Applied Mathematics
All rights reserved
Title: Global Optimization Decomposition Methods for
Bounded Parameter Minimax Risk Evaluation
Author: Eric Gourdin, Brigitte Jaumard, and
Brenda MacGibbon
AMS Subject
Classifications: 90C26, 62-04, 62F10
Key words: decomposition, \ih\ constant, bounded parameter,
minimax, discrete least favorable prior
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ABSTRACT
There has been much recent statistical research in the area
of inference under constraints. The problem considered here
is that of bounded parameter estimation, in particular that
of normal and Poisson means, using minimaxity as the
criterion of evaluation. Because of the ease of calculation
of linear minimax rules, the ratio of these risks to the
nonlinear minimax risks for these problems is also studied.
To find the minimax solution, the dual problem of finding
the least favorable prior distribution is often considered.
On bounded parameter spaces the least favorable prior is
often discrete, so the finding of the minimax estimator and
its risk is equivalent to a global optimization problem with
constraints. Previously published numerical specifications
of the priors have used iterative (often heuristic)
procedures. Two global optimization procedures are proposed.
The first is based on multivariate Lipschitz optimization
and makes use of bounds on the first-order derivatives. The
second is a decomposition procedure that utilizes the
partial concavity of the Bayes risk function. Both
procedures are compared, and the decomposition method
appears to be much more efficient. It is shown that the
Ibragimov--Hasminskii constant, the maximum of the ratio of
linear to nonlinear minimax risks, is different for the
Poisson and normal problems.
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