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On the risk-equivalence of two methods of randomization in statistics

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  • Kirschner, H. P.

Abstract

For statistical decision problems, there are two well-known methods of randomization: on the one hand, randomization by means of mixtures of nonrandomized decision functions (randomized decision rules) in the game "statistician against nature," on the other hand, randomization by means of randomized decision functions. In this paper, we consider the problem of risk-equivalence of these two procedures, i.e., imposing fairly general conditions on a nonsequential decision problem, it is shown that to each randomized decision rule, there is a randomized decision function with uniformly the same risk, and vice versa. The crucial argument is based on rewriting risk-equivalence in terms of Choquet's integral representation theorem. It is shown, in addition, that for certain special cases that do not fulfill the assumptions of the Main Theorem, risk-equivalence holds at least partially.

Suggested Citation

  • Kirschner, H. P., 1976. "On the risk-equivalence of two methods of randomization in statistics," Journal of Multivariate Analysis, Elsevier, vol. 6(1), pages 159-166, March.
    • Handle: RePEc:eee:jmvana:v:6:y:1976:i:1:p:159-166
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