An algorithm for calculating \(\Gamma\)-minimax decision rules under generalized moment conditions. (English) Zbl 1041.62005

Noubiap, Roger Fandom; Seidel, Wilfried

Ann. Stat. 29, No. 4, 1094-1116 (2001).

Summary: We present an algorithm for calculating a \(\Gamma\)-minimax decision rule, when \(\Gamma\) is given by a finite number of generalized moment conditions. Such a decision rule minimizes the maximum of the integrals of the risk function with respect to all distributions in \(\Gamma\). The inner maximization problem is approximated by a sequence of linear programs. This approximation is combined with an elimination technique which quickly reduces the domain of the variables of the outer minimization problem. To test for convergence in a final step, the inner maximization problem has to be completely solved once for the candidate of the \(\Gamma\)-minimax rule found by the algorithm. For an infinite, compact parameter space, this is done by semi-infinite programming. The algorithm is applied to calculate robustified Bayesian designs in a logistic regression model and \(\Gamma\)-minimax tests in monotone decision problems.

MSC:

62C20	Minimax procedures in statistical decision theory
90C34	Semi-infinite programming
90C90	Applications of mathematical programming
90C05	Linear programming
65C60	Computational problems in statistics (MSC2010)
62F35	Robustness and adaptive procedures (parametric inference)

Keywords:

Bayesian robustness; experimental design; minimax problems

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