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→Examples: reverting March 19, 2023 edit by 149.125.125.47 (an outcome that majorizes another outcome is *less* spread out than that other outcome) |
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* The [[Shannon entropy]] function <math>\sum_{i=1}^d{P_i \cdot \log_2{\frac{1}{P_i}}}</math> is Schur-concave.
* The [[Rényi entropy]] function is also Schur-concave.
* <math>x \mapsto \sum_{i=1}^d{x_i^k},k \ge 1 </math> is Schur-convex if <math>k \geq 1</math>, and Schur-concave if <math>k \in (0, 1)</math>.
* 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 less 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.
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