Symmetric function: Difference between revisions

Revision as of 17:41, 21 April 2009

Revision as of 19:23, 21 April 2009

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For these specific uses, see the articles [[symmetric polynomial]]s and [[ring of symmetric functions]]; the remainder of this article addresses general properties of symmetric functions in ''n'' variables.

== Symmetrization ==

{{main|Symmetrization}}

Given any function in ''n'' variables, <math>f(x_1,\dots,x_n),</math> it can be made into a symmetric function by averaging over permutations. Similarly, it can be made into an anti-symmetric function by averaging over [[even permutation]]s and subtracting the average over [[odd permutation]]s.

== Applications ==

=== U-statistics ===

{{main|U-statistic}}

In [[statistics]], an ''n''-sample statistic (a function in ''n'' variables) that is obtained by [[bootstrapping (statistics)|bootstrapping]] symmetrization of a ''k''-sample statistic, yielding a symmetric function in ''n'' variables, is called a [[U-statistic]]. Examples include the [[sample mean]] and [[sample variance]].

[[Category:Symmetric functions| ]]