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In mathematics, a Schur-convex function, also known as S-convex, isotonic function and order-preserving function is a function $f : ℝ^{d} \to ℝ$ , for which if $\forall x, y \in ℝ^{d}$ where $x$ is majorized by $y$ , then $f (x) \leq f (y)$ . Named after Issai Schur, Schur-convex functions are used in the study of majorization. Every function that is convex and symmetric is also Schur-convex. The opposite implication is not true, but all Schur-convex functions are symmetric (under permutations of the arguments).

Schur-concave function

A function $f$ is 'Schur-concave' if its negative, $- f$ , is Schur-convex.

A simple criterion

If $f$ is Schur-convex and all first partial derivatives exist, then the following holds, where $f_{(i)} (x)$ denotes the partial derivative with respect to $x_{i}$ :

(x_{1} - x_{2}) (f_{(1)} (x) - f_{(2)} (x)) \geq 0

for all

x

. Since

f

is a symmetric function, the above condition implies all the similar conditions for the remaining indexes!

Examples

$f (x) = \min (x)$ is Schur-concave while $f (x) = \max (x)$ is Schur-convex. This can be seen directly from the definition.

The Shannon entropy function $\sum_{i = 1}^{d} P_{i} \cdot \log_{2} \frac{1}{P_{i}}$ is Schur-concave.

The Rényi entropy function is also Schur-concave.

$\sum_{i = 1}^{d} x_{i}^{k}, k \geq 1$ is Schur-convex.

The function $f (x) = \prod_{i = 1}^{n} x_{i}$ is Schur-concave, when we assume all $x_{i} > 0$ . In the same way, all the Elementary symmetric functions are Schur-concave, when $x_{i} > 0$ .

A natural interpretation of majorization is that if $x ≻ y$ then $x$ is more spread out than $y$ . 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 $g$ is a convex function defined on a real interval, then $\sum_{i = 1}^{n} g (x_{i})$ is Schur-convex.

Some probability examples: If $X_{1}, \dots, X_{n}$ are exchangeable random variables, then the function

   : $E \prod_{j = 1}^{n} X_{j}^{a_{j}}$  
   is Schur-convex as a function of  $a = (a_{1}, \dots, a_{n})$ , assuming that the expectations exist.

The Gini coefficient is strictly Schur concave.

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+In mathematics, a '''Schur-convex function''', also known as '''S-convex''', '''isotonic function''' and '''order-preserving function''' is a [[function (mathematics)|function]] <math>f: \mathbb{R}^d\rightarrow \mathbb{R}</math>, for which if <math>\forall x,y\in \mathbb{R}^d </math> where <math>x</math> is [[majorization|majorized]] by <math>y</math>, then <math>f(x)\le f(y)</math>. Named after [[Issai Schur]], Schur-convex functions are used in the study of [[majorization]]. Every function that is [[Convex function|convex]] and [[Symmetric function|symmetric]] is also Schur-convex.  The opposite implication is not  true, but all Schur-convex functions are symmetric (under permutations of the arguments).
+== Schur-concave function ==
+A function <math>f</math> is 'Schur-concave' if its negative,<math>-f</math>, is Schur-convex.
+==A simple criterion==
+If <math>f</math> is Schur-convex and all first partial derivatives exist, then the following holds, where <math> f_{(i)}(x) </math> denotes the partial derivative with respect to <math> x_i </math>:
+:<math>    (x_1 - x_2)(f_{(1)}(x) - f_{(2)}(x)) \ge 0
+ </math>   for all <math> x </math>. Since <math> f </math> is a symmetric function, the above condition implies all the similar conditions for the remaining indexes!
+== Examples ==
+* <math> f(x)=\min(x) </math> is Schur-concave while <math> f(x)=\max(x) </math> is Schur-convex. This can be seen directly from the definition.
+* 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> \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 more 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.
+* Some probability examples: 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 concave.
+==See also==
+* [[majorization]]
+* [[Quasiconvex function]]
+* [[Convex function]]
+[[Category:Convex analysis]]
+[[Category:Inequalities]]
+{{mathanalysis-stub}}

ESP game: Difference between revisions

Revision as of 19:16, 31 October 2013

Contents

Schur-concave function

A simple criterion

Examples

See also

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ESP game: Difference between revisions

Revision as of 19:16, 31 October 2013

Schur-concave function

A simple criterion

Examples

See also

Navigation menu

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