Randomization

A statistical procedure in which a decision is randomly taken. Suppose that, given a realization $ x $ of a random variable $ X $ with values in a sample space $ ( \overline{X}\; , {\mathcal B} , {\mathsf P} _ \theta ) $, $ \theta \in \Theta $, one has to choose a solution $ \xi $ from a measurable space $ ( \Xi , {\mathcal A} ) $, and suppose that a family of so-called transition probability distributions $ \{ Q _ {x} ( \cdot ) \} $, $ x \in \overline{X}\; $, has been defined on $ ( \Xi , {\mathcal A} ) $ such that the function $ Q _ {\mathbf . } ( A) $ is $ {\mathcal B} $- measurable in $ x $ for every fixed event $ A \in {\mathcal A} $. Then randomization is the statistical procedure of decision taking in which, given a realization $ x $ of $ X $, the decision is made by drawing lots subject to the probability law $ Q _ {x} ( \cdot ) $.

References

[1]	N.N. [N.N. Chentsov] Čentsov, "Statistical decision rules and optimal inference" , Amer. Math. Soc. (1982) (Translated from Russian)

Comments

The statistical procedure of randomization is also called a randomized decision rule.

References

[a1]	J.O. Berger, "Statistical decision theory and Bayesian analysis" , Springer (1985)
[a2]	E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1986)