Schur-convex function: Difference between revisions

Revision as of 03:49, 26 November 2016

Revision as of 05:43, 17 January 2018

Line 19:

* <math> \sum_{i=1}^d{x_i^k},k \ge 1 </math> is Schur-convex.

* The function <math> f(x) = \prod_{i=1}^n x_i </math> is Schur-concave, when we assume all <math> x_i > 0 </math>. In the same way, all the [[Elementary symmetric polynomial|Elementary symmetric function]]s are Schur-concave, when <math> x_i > 0 </math>.

* A natural interpretation of [[majorization]] is that if <math> x \succ y </math> then <math> x </math> is moreless spread out than <math> y </math>. So it is natural to ask if statistical measures of variability are Schur-convex. The [[variance]] and [[standard deviation]] are Schur-convex functions, while the [[Median absolute deviation]] is not.

* If <math> g </math> is a convex function defined on a real interval, then <math> \sum_{i=1}^n g(x_i) </math> is Schur-convex.

* A probability example: If <math> X_1, \dots, X_n </math> are [[exchangeable random variables]], then the function <math> \text{E} \prod_{j=1}^n X_j^{a_j} </math> is Schur-convex as a function of <math> a=(a_1, \dots, a_n) </math>, assuming that the expectations exist.