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Controlling risks under different loss functions: The compromise decision problem. (English) Zbl 0666.62008
Two statisticians are presented with the problem of agreeing to use a single decision procedure from D, the collection of all such decision procedures. When each statistician is a Bayesian a “Bayes compromise problem” is to find a decision procedure which minimizes \(r_ i(\delta,\Pi)\) subject to \(r_ i(\delta,\Pi)\leq K_ j\) where (i,j), the roles of the statisticians, is fixed at (1,2) or (2,1) and \(K_ j\) is a fixed constant satisfying \[ r_ j(\delta_ j^ B,\Pi_ j)\leq K_ j\leq r_ j(\delta_ i^ B,\Pi_ j). \] Here \(r_ i(\delta,\Pi_ i)\) denotes the expected risk of \(\delta\) with respect to the prior distribution \(\Pi_ i\) on the parameter space for loss \(L_ i\), and \(\delta_ 1^ B\) and \(\delta_ 2^ B\) denote the Bayes procedures.
The “\(\lambda\)-Bayes compromise” problem is to find a decision procedure which minimizes \[ \lambda r_ 1(\delta,\Pi_ 1)+(1- \lambda)r_ 2(\delta,\Pi_ 2) \] where \(\lambda\) is a fixed constant (0\(\leq \lambda \leq 1)\) “Minimax-Bayes compromise” problem and “\(\lambda\)-minimax-Bayes compromise” problem are defined similarly. The author gives a theory characterizing solutions to Bayes compromise and minimax compromise problems.
Reviewer: K.Alam

MSC:
62C10 Bayesian problems; characterization of Bayes procedures
62C05 General considerations in statistical decision theory
62C25 Compound decision problems in statistical decision theory
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