An algorithm for calculating Γ-minimax decision rules under generalized moment conditions

Abstract

We present an algorithm for calculating a $\Gamma$-minimax decision rule, when 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.