Global Optimization Decomposition Methods (Gourdin et al.)

======================================================================
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

----

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.


======================================================================

SIAM
3600 University City Science Center
Philadelphia, PA  19104-2688, USA

Phone:  215-382-9800, 800-447-7426 (USA only)
Fax:  215-386-7999
E-mail:  journals@siam.org