# 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) |

**How to Cite This Entry:**

Randomization.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Randomization&oldid=48430