Schur-convex function: Difference between revisions

Revision as of 22:23, 26 January 2024

Revision as of 04:51, 21 April 2024

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* <math> \sum_{i=1}^d{x_i^k},0 < k < 1 </math> is Schur-concave.

* The function <math> f(x) = \prod_{i=1}^d 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.

* 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.

* The [[Gini coefficient]] is strictly Schur convex.