Schur-convexity of multinomial distribution
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Consider ${p_i in [0,c]}_i$, $c<1$, such that $sum_{i=1}^{k} p_i = P$ for some positive $k>2$ and constant $P < 1$. Let $(x_i in mathbb{N})_i$. Is
$$f(p_1,ldots,p_k)
= sum_{x_1,x_2,ldots,x_k=0}^{c} frac{p_1^{x_1}}{x_1} frac{p_2^{x_2}}{x_2!}
cdots
frac{p_k^{x_k}}{x_k!}
$$
Schur-concave in ${p_i}$?
The function is symmetric, but it is a bit hard to check the cond $partial f/partial p_i - partial f/partial p_j$. Any suggestion?
PS: The function is similar to the marginal of a Multinomial distribution.
statistics convex-analysis convex-optimization
asked Nov 20 at 14:20
Jeff
1838
add a comment |
up vote
0
down vote
favorite
Consider ${p_i in [0,c]}_i$, $c<1$, such that $sum_{i=1}^{k} p_i = P$ for some positive $k>2$ and constant $P < 1$. Let $(x_i in mathbb{N})_i$. Is
$$f(p_1,ldots,p_k)
= sum_{x_1,x_2,ldots,x_k=0}^{c} frac{p_1^{x_1}}{x_1} frac{p_2^{x_2}}{x_2!}
cdots
frac{p_k^{x_k}}{x_k!}
$$
Schur-concave in ${p_i}$?
The function is symmetric, but it is a bit hard to check the cond $partial f/partial p_i - partial f/partial p_j$. Any suggestion?
PS: The function is similar to the marginal of a Multinomial distribution.
statistics convex-analysis convex-optimization
asked Nov 20 at 14:20
Jeff
1838
What is the domain of $f$? Positive or also negative input?
– LinAlg
Nov 20 at 14:45
edited the question accordingly. ${p_i}'s$ are probability simplex.
– Jeff
Nov 20 at 14:57
then the answer is trivial, right?
– LinAlg
Nov 20 at 15:15
you were right. Actually, it should be Schur-concave, and also I had a summation, which was missing in the original question. Now, it is not trivial to me whether it is Schur-concave or not...
– Jeff
Nov 20 at 18:56
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Consider ${p_i in [0,c]}_i$, $c<1$, such that $sum_{i=1}^{k} p_i = P$ for some positive $k>2$ and constant $P < 1$. Let $(x_i in mathbb{N})_i$. Is
$$f(p_1,ldots,p_k)
= sum_{x_1,x_2,ldots,x_k=0}^{c} frac{p_1^{x_1}}{x_1} frac{p_2^{x_2}}{x_2!}
cdots
frac{p_k^{x_k}}{x_k!}
$$
Schur-concave in ${p_i}$?
The function is symmetric, but it is a bit hard to check the cond $partial f/partial p_i - partial f/partial p_j$. Any suggestion?
PS: The function is similar to the marginal of a Multinomial distribution.
statistics convex-analysis convex-optimization
asked Nov 20 at 14:20
Jeff
1838
Consider ${p_i in [0,c]}_i$, $c<1$, such that $sum_{i=1}^{k} p_i = P$ for some positive $k>2$ and constant $P < 1$. Let $(x_i in mathbb{N})_i$. Is
$$f(p_1,ldots,p_k)
= sum_{x_1,x_2,ldots,x_k=0}^{c} frac{p_1^{x_1}}{x_1} frac{p_2^{x_2}}{x_2!}
cdots
frac{p_k^{x_k}}{x_k!}
$$
Schur-concave in ${p_i}$?
The function is symmetric, but it is a bit hard to check the cond $partial f/partial p_i - partial f/partial p_j$. Any suggestion?
PS: The function is similar to the marginal of a Multinomial distribution.
statistics convex-analysis convex-optimization
statistics convex-analysis convex-optimization
asked Nov 20 at 14:20
Jeff
1838
asked Nov 20 at 14:20
Jeff
1838
edited Nov 20 at 18:54
asked Nov 20 at 14:20
Jeff
1838
asked Nov 20 at 14:20
Jeff
1838
asked Nov 20 at 14:20
Jeff
1838
1838
What is the domain of $f$? Positive or also negative input?
– LinAlg
Nov 20 at 14:45
edited the question accordingly. ${p_i}'s$ are probability simplex.
– Jeff
Nov 20 at 14:57
then the answer is trivial, right?
– LinAlg
Nov 20 at 15:15
you were right. Actually, it should be Schur-concave, and also I had a summation, which was missing in the original question. Now, it is not trivial to me whether it is Schur-concave or not...
– Jeff
Nov 20 at 18:56
add a comment |
What is the domain of $f$? Positive or also negative input?
– LinAlg
Nov 20 at 14:45
edited the question accordingly. ${p_i}'s$ are probability simplex.
– Jeff
Nov 20 at 14:57
then the answer is trivial, right?
– LinAlg
Nov 20 at 15:15
you were right. Actually, it should be Schur-concave, and also I had a summation, which was missing in the original question. Now, it is not trivial to me whether it is Schur-concave or not...
– Jeff
Nov 20 at 18:56
What is the domain of $f$? Positive or also negative input?
– LinAlg
Nov 20 at 14:45
What is the domain of $f$? Positive or also negative input?
– LinAlg
Nov 20 at 14:45
edited the question accordingly. ${p_i}'s$ are probability simplex.
– Jeff
Nov 20 at 14:57
edited the question accordingly. ${p_i}'s$ are probability simplex.
– Jeff
Nov 20 at 14:57
then the answer is trivial, right?
– LinAlg
Nov 20 at 15:15
then the answer is trivial, right?
– LinAlg
Nov 20 at 15:15
you were right. Actually, it should be Schur-concave, and also I had a summation, which was missing in the original question. Now, it is not trivial to me whether it is Schur-concave or not...
– Jeff
Nov 20 at 18:56
you were right. Actually, it should be Schur-concave, and also I had a summation, which was missing in the original question. Now, it is not trivial to me whether it is Schur-concave or not...
– Jeff
Nov 20 at 18:56
add a comment |
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What is the domain of $f$? Positive or also negative input?
– LinAlg
Nov 20 at 14:45
edited the question accordingly. ${p_i}'s$ are probability simplex.
– Jeff
Nov 20 at 14:57
then the answer is trivial, right?
– LinAlg
Nov 20 at 15:15
you were right. Actually, it should be Schur-concave, and also I had a summation, which was missing in the original question. Now, it is not trivial to me whether it is Schur-concave or not...
– Jeff
Nov 20 at 18:56