Invariant decision procedures

The paper deals with statistical model of decision theory which is invariant relative to some group of transformations. The minimax property of an optimal invariant decision procedure is proved under some conditions. The problem of finding an optimal invariant decision procedure is reduced to that of minimizing a numerical function on the decision space. A connection between this problem and the concept of the conjugate (fiducial) distribution family on the parameter space is discussed. The equality between sample and conjugate densities relative to a properly chosen measure is proved. Sufficient conditions are given for the existence of an invariant minimax decision procedure.