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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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