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A statistical procedure in which a decision is randomly taken. Suppose that, given a realization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077390/r0773901.png" /> of a random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077390/r0773902.png" /> with values in a sample space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077390/r0773903.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077390/r0773904.png" />, one has to choose a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077390/r0773905.png" /> from a measurable space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077390/r0773906.png" />, and suppose that a family of so-called transition probability distributions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077390/r0773907.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077390/r0773908.png" />, has been defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077390/r0773909.png" /> such that the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077390/r07739010.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077390/r07739011.png" />-measurable in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077390/r07739012.png" /> for every fixed event <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077390/r07739013.png" />. Then randomization is the statistical procedure of decision taking in which, given a realization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077390/r07739014.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077390/r07739015.png" />, the decision is made by drawing lots subject to the probability law <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077390/r07739016.png" />.
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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====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.N. [N.N. Chentsov] Čentsov,  "Statistical decision rules and optimal inference" , Amer. Math. Soc.  (1982)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.N. [N.N. Chentsov] Čentsov,  "Statistical decision rules and optimal inference" , Amer. Math. Soc.  (1982)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 08:09, 6 June 2020


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
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article