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 15 hours ago
Jeff
1788
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 15 hours ago
Jeff
1788
What is the domain of $f$? Positive or also negative input?
– LinAlg
15 hours ago
edited the question accordingly. ${p_i}'s$ are probability simplex.
– Jeff
14 hours ago
then the answer is trivial, right?
– LinAlg
14 hours ago
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
10 hours ago
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 15 hours ago
Jeff
1788
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 15 hours ago
Jeff
1788
asked 15 hours ago
Jeff
1788
edited 10 hours ago
asked 15 hours ago
Jeff
1788
asked 15 hours ago
Jeff
1788
asked 15 hours ago
Jeff
1788
1788
What is the domain of $f$? Positive or also negative input?
– LinAlg
15 hours ago
edited the question accordingly. ${p_i}'s$ are probability simplex.
– Jeff
14 hours ago
then the answer is trivial, right?
– LinAlg
14 hours ago
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
10 hours ago
add a comment |
What is the domain of $f$? Positive or also negative input?
– LinAlg
15 hours ago
edited the question accordingly. ${p_i}'s$ are probability simplex.
– Jeff
14 hours ago
then the answer is trivial, right?
– LinAlg
14 hours ago
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
10 hours ago
What is the domain of $f$? Positive or also negative input?
– LinAlg
15 hours ago
What is the domain of $f$? Positive or also negative input?
– LinAlg
15 hours ago
edited the question accordingly. ${p_i}'s$ are probability simplex.
– Jeff
14 hours ago
edited the question accordingly. ${p_i}'s$ are probability simplex.
– Jeff
14 hours ago
then the answer is trivial, right?
– LinAlg
14 hours ago
then the answer is trivial, right?
– LinAlg
14 hours ago
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
10 hours ago
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
10 hours ago
add a comment |
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What is the domain of $f$? Positive or also negative input?
– LinAlg
15 hours ago
edited the question accordingly. ${p_i}'s$ are probability simplex.
– Jeff
14 hours ago
then the answer is trivial, right?
– LinAlg
14 hours ago
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
10 hours ago