Maximum likelihood of Compound Poisson Distributions
I'm trying to compute a maximum likelihood of the compound Poisson-Gamma distribution in R. The distribution is defined by $ sum_j=1^N Y_j $ where $Y_n$ is i.i.d sequence independent $gamma(k,theta)$ values and $N$ is a Poisson distribution with parameter $beta$. I'm trying to estimate the parameters $theta$ and $beta$ without luck.
r
edited Nov 14 '18 at 1:38
Ben Bolker
133k11223311
asked Nov 13 '18 at 10:16
lina binalina bina
11
add a comment |
I'm trying to compute a maximum likelihood of the compound Poisson-Gamma distribution in R. The distribution is defined by $ sum_j=1^N Y_j $ where $Y_n$ is i.i.d sequence independent $gamma(k,theta)$ values and $N$ is a Poisson distribution with parameter $beta$. I'm trying to estimate the parameters $theta$ and $beta$ without luck.
r
edited Nov 14 '18 at 1:38
Ben Bolker
133k11223311
asked Nov 13 '18 at 10:16
lina binalina bina
11
This will probably get more useful results on CrossValidated; I'm voting to migrate it there. If I needed to do this, I would do it by brute force/approximation. We know that the distribution of the sum of j Gamma deviates with parameters (shape=k, scale=theta) is Gamma(j * k, theta), so we can define the distribution as a mixture of Gammas: Prob(x|beta,k,theta) = sum_j=0^infty dPois(j|beta)*Gamma(j * k,theta). The problem is that it's an infinite sum - unless there's a closed-form solution you might have to truncate the summation
– Ben Bolker
Nov 14 '18 at 1:35
I would probably look in Bailey's stochastic processes book or Pielou's statistical ecology book (or Johnson/Kotz) for more information about the distribution/what's known about the probability distribution
– Ben Bolker
Nov 14 '18 at 1:36
Wikipedia suggests that this distribution is equivalent to a Tweedie distribution, for which there are resources available in R (library("sos"); findFn("tweedie")en.wikipedia.org/wiki/…
– Ben Bolker
Nov 14 '18 at 1:37
unfortunately this package does not contain a function to estimate this parameters .
– lina bina
Nov 20 '18 at 21:40
add a comment |
I'm trying to compute a maximum likelihood of the compound Poisson-Gamma distribution in R. The distribution is defined by $ sum_j=1^N Y_j $ where $Y_n$ is i.i.d sequence independent $gamma(k,theta)$ values and $N$ is a Poisson distribution with parameter $beta$. I'm trying to estimate the parameters $theta$ and $beta$ without luck.
r
edited Nov 14 '18 at 1:38
Ben Bolker
133k11223311
asked Nov 13 '18 at 10:16
lina binalina bina
11
I'm trying to compute a maximum likelihood of the compound Poisson-Gamma distribution in R. The distribution is defined by $ sum_j=1^N Y_j $ where $Y_n$ is i.i.d sequence independent $gamma(k,theta)$ values and $N$ is a Poisson distribution with parameter $beta$. I'm trying to estimate the parameters $theta$ and $beta$ without luck.
r
r
edited Nov 14 '18 at 1:38
Ben Bolker
133k11223311
asked Nov 13 '18 at 10:16
lina binalina bina
11
edited Nov 14 '18 at 1:38
Ben Bolker
133k11223311
asked Nov 13 '18 at 10:16
lina binalina bina
11
edited Nov 14 '18 at 1:38
Ben Bolker
133k11223311
edited Nov 14 '18 at 1:38
Ben Bolker
133k11223311
edited Nov 14 '18 at 1:38
Ben Bolker
133k11223311
133k11223311
asked Nov 13 '18 at 10:16
lina binalina bina
11
asked Nov 13 '18 at 10:16
lina binalina bina
11
asked Nov 13 '18 at 10:16
lina binalina bina
11
11
This will probably get more useful results on CrossValidated; I'm voting to migrate it there. If I needed to do this, I would do it by brute force/approximation. We know that the distribution of the sum of j Gamma deviates with parameters (shape=k, scale=theta) is Gamma(j * k, theta), so we can define the distribution as a mixture of Gammas: Prob(x|beta,k,theta) = sum_j=0^infty dPois(j|beta)*Gamma(j * k,theta). The problem is that it's an infinite sum - unless there's a closed-form solution you might have to truncate the summation
– Ben Bolker
Nov 14 '18 at 1:35
I would probably look in Bailey's stochastic processes book or Pielou's statistical ecology book (or Johnson/Kotz) for more information about the distribution/what's known about the probability distribution
– Ben Bolker
Nov 14 '18 at 1:36
Wikipedia suggests that this distribution is equivalent to a Tweedie distribution, for which there are resources available in R (library("sos"); findFn("tweedie")en.wikipedia.org/wiki/…
– Ben Bolker
Nov 14 '18 at 1:37
unfortunately this package does not contain a function to estimate this parameters .
– lina bina
Nov 20 '18 at 21:40
add a comment |
This will probably get more useful results on CrossValidated; I'm voting to migrate it there. If I needed to do this, I would do it by brute force/approximation. We know that the distribution of the sum of j Gamma deviates with parameters (shape=k, scale=theta) is Gamma(j * k, theta), so we can define the distribution as a mixture of Gammas: Prob(x|beta,k,theta) = sum_j=0^infty dPois(j|beta)*Gamma(j * k,theta). The problem is that it's an infinite sum - unless there's a closed-form solution you might have to truncate the summation
– Ben Bolker
Nov 14 '18 at 1:35
I would probably look in Bailey's stochastic processes book or Pielou's statistical ecology book (or Johnson/Kotz) for more information about the distribution/what's known about the probability distribution
– Ben Bolker
Nov 14 '18 at 1:36
Wikipedia suggests that this distribution is equivalent to a Tweedie distribution, for which there are resources available in R (library("sos"); findFn("tweedie")en.wikipedia.org/wiki/…
– Ben Bolker
Nov 14 '18 at 1:37
unfortunately this package does not contain a function to estimate this parameters .
– lina bina
Nov 20 '18 at 21:40
This will probably get more useful results on CrossValidated; I'm voting to migrate it there. If I needed to do this, I would do it by brute force/approximation. We know that the distribution of the sum of j Gamma deviates with parameters (shape=k, scale=theta) is Gamma(j * k, theta), so we can define the distribution as a mixture of Gammas: Prob(x|beta,k,theta) = sum_j=0^infty dPois(j|beta)*Gamma(j * k,theta). The problem is that it's an infinite sum - unless there's a closed-form solution you might have to truncate the summation
– Ben Bolker
Nov 14 '18 at 1:35
This will probably get more useful results on CrossValidated; I'm voting to migrate it there. If I needed to do this, I would do it by brute force/approximation. We know that the distribution of the sum of j Gamma deviates with parameters (shape=k, scale=theta) is Gamma(j * k, theta), so we can define the distribution as a mixture of Gammas: Prob(x|beta,k,theta) = sum_j=0^infty dPois(j|beta)*Gamma(j * k,theta). The problem is that it's an infinite sum - unless there's a closed-form solution you might have to truncate the summation
– Ben Bolker
Nov 14 '18 at 1:35
I would probably look in Bailey's stochastic processes book or Pielou's statistical ecology book (or Johnson/Kotz) for more information about the distribution/what's known about the probability distribution
– Ben Bolker
Nov 14 '18 at 1:36
I would probably look in Bailey's stochastic processes book or Pielou's statistical ecology book (or Johnson/Kotz) for more information about the distribution/what's known about the probability distribution
– Ben Bolker
Nov 14 '18 at 1:36
Wikipedia suggests that this distribution is equivalent to a Tweedie distribution, for which there are resources available in R (
library("sos"); findFn("tweedie") en.wikipedia.org/wiki/…– Ben Bolker
Nov 14 '18 at 1:37
Wikipedia suggests that this distribution is equivalent to a Tweedie distribution, for which there are resources available in R (
library("sos"); findFn("tweedie") en.wikipedia.org/wiki/…– Ben Bolker
Nov 14 '18 at 1:37
unfortunately this package does not contain a function to estimate this parameters .
– lina bina
Nov 20 '18 at 21:40
unfortunately this package does not contain a function to estimate this parameters .
– lina bina
Nov 20 '18 at 21:40
add a comment |
1 Answer
1
active
oldest
votes
If you wanted to do something similar, but for a negative binomial distribution, then you can use the the function negbin.mle from the package Rfast
y <- rpois(100, 2)
Rfast::negbin.mle(y)
Output
$iters
[1] 5
$loglik
[1] -162.855
$param
success probability number of failures mean
0.9963271 480.1317031 1.7700000
Also if you run the command:
Rfast::negbin.mle
You can see what the function is computing.
You can also check the functions manual with:
?Rfast::negbin.mle
Edit:
Unfortunately I haven't found something that perfectly fits your answer.
As Ben states, this answer is for a Poisson with Gamma-distributed mean.
answered Nov 13 '18 at 11:39
StefanosStefanos
335313
1
I don't think this works. NB is a Poisson with Gamma-distributed mean; OP wants a sum of Gammas
– Ben Bolker
Nov 13 '18 at 12:37
It does not work!
– lina bina
Nov 13 '18 at 21:24
@BenBolker You are right, I had misunderstood the question
– Stefanos
Nov 14 '18 at 0:34
@linabina It is working, I just tested it. The problem is that as it appears it doesn't perfectly fit your question. So I will edit my answer.
– Stefanos
Nov 14 '18 at 0:37
add a comment |
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1 Answer
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oldest
votes
1 Answer
1
active
oldest
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oldest
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votes
If you wanted to do something similar, but for a negative binomial distribution, then you can use the the function negbin.mle from the package Rfast
y <- rpois(100, 2)
Rfast::negbin.mle(y)
Output
$iters
[1] 5
$loglik
[1] -162.855
$param
success probability number of failures mean
0.9963271 480.1317031 1.7700000
Also if you run the command:
Rfast::negbin.mle
You can see what the function is computing.
You can also check the functions manual with:
?Rfast::negbin.mle
Edit:
Unfortunately I haven't found something that perfectly fits your answer.
As Ben states, this answer is for a Poisson with Gamma-distributed mean.
answered Nov 13 '18 at 11:39
StefanosStefanos
335313
1
I don't think this works. NB is a Poisson with Gamma-distributed mean; OP wants a sum of Gammas
– Ben Bolker
Nov 13 '18 at 12:37
It does not work!
– lina bina
Nov 13 '18 at 21:24
@BenBolker You are right, I had misunderstood the question
– Stefanos
Nov 14 '18 at 0:34
@linabina It is working, I just tested it. The problem is that as it appears it doesn't perfectly fit your question. So I will edit my answer.
– Stefanos
Nov 14 '18 at 0:37
add a comment |
If you wanted to do something similar, but for a negative binomial distribution, then you can use the the function negbin.mle from the package Rfast
y <- rpois(100, 2)
Rfast::negbin.mle(y)
Output
$iters
[1] 5
$loglik
[1] -162.855
$param
success probability number of failures mean
0.9963271 480.1317031 1.7700000
Also if you run the command:
Rfast::negbin.mle
You can see what the function is computing.
You can also check the functions manual with:
?Rfast::negbin.mle
Edit:
Unfortunately I haven't found something that perfectly fits your answer.
As Ben states, this answer is for a Poisson with Gamma-distributed mean.
answered Nov 13 '18 at 11:39
StefanosStefanos
335313
1
I don't think this works. NB is a Poisson with Gamma-distributed mean; OP wants a sum of Gammas
– Ben Bolker
Nov 13 '18 at 12:37
It does not work!
– lina bina
Nov 13 '18 at 21:24
@BenBolker You are right, I had misunderstood the question
– Stefanos
Nov 14 '18 at 0:34
@linabina It is working, I just tested it. The problem is that as it appears it doesn't perfectly fit your question. So I will edit my answer.
– Stefanos
Nov 14 '18 at 0:37
add a comment |
If you wanted to do something similar, but for a negative binomial distribution, then you can use the the function negbin.mle from the package Rfast
y <- rpois(100, 2)
Rfast::negbin.mle(y)
Output
$iters
[1] 5
$loglik
[1] -162.855
$param
success probability number of failures mean
0.9963271 480.1317031 1.7700000
Also if you run the command:
Rfast::negbin.mle
You can see what the function is computing.
You can also check the functions manual with:
?Rfast::negbin.mle
Edit:
Unfortunately I haven't found something that perfectly fits your answer.
As Ben states, this answer is for a Poisson with Gamma-distributed mean.
answered Nov 13 '18 at 11:39
StefanosStefanos
335313
If you wanted to do something similar, but for a negative binomial distribution, then you can use the the function negbin.mle from the package Rfast
y <- rpois(100, 2)
Rfast::negbin.mle(y)
Output
$iters
[1] 5
$loglik
[1] -162.855
$param
success probability number of failures mean
0.9963271 480.1317031 1.7700000
Also if you run the command:
Rfast::negbin.mle
You can see what the function is computing.
You can also check the functions manual with:
?Rfast::negbin.mle
Edit:
Unfortunately I haven't found something that perfectly fits your answer.
As Ben states, this answer is for a Poisson with Gamma-distributed mean.
answered Nov 13 '18 at 11:39
StefanosStefanos
335313
edited Nov 14 '18 at 0:44
answered Nov 13 '18 at 11:39
StefanosStefanos
335313
answered Nov 13 '18 at 11:39
StefanosStefanos
335313
answered Nov 13 '18 at 11:39
StefanosStefanos
335313
335313
1
I don't think this works. NB is a Poisson with Gamma-distributed mean; OP wants a sum of Gammas
– Ben Bolker
Nov 13 '18 at 12:37
It does not work!
– lina bina
Nov 13 '18 at 21:24
@BenBolker You are right, I had misunderstood the question
– Stefanos
Nov 14 '18 at 0:34
@linabina It is working, I just tested it. The problem is that as it appears it doesn't perfectly fit your question. So I will edit my answer.
– Stefanos
Nov 14 '18 at 0:37
add a comment |
1
I don't think this works. NB is a Poisson with Gamma-distributed mean; OP wants a sum of Gammas
– Ben Bolker
Nov 13 '18 at 12:37
It does not work!
– lina bina
Nov 13 '18 at 21:24
@BenBolker You are right, I had misunderstood the question
– Stefanos
Nov 14 '18 at 0:34
@linabina It is working, I just tested it. The problem is that as it appears it doesn't perfectly fit your question. So I will edit my answer.
– Stefanos
Nov 14 '18 at 0:37
1
1
I don't think this works. NB is a Poisson with Gamma-distributed mean; OP wants a sum of Gammas
– Ben Bolker
Nov 13 '18 at 12:37
I don't think this works. NB is a Poisson with Gamma-distributed mean; OP wants a sum of Gammas
– Ben Bolker
Nov 13 '18 at 12:37
It does not work!
– lina bina
Nov 13 '18 at 21:24
It does not work!
– lina bina
Nov 13 '18 at 21:24
@BenBolker You are right, I had misunderstood the question
– Stefanos
Nov 14 '18 at 0:34
@BenBolker You are right, I had misunderstood the question
– Stefanos
Nov 14 '18 at 0:34
@linabina It is working, I just tested it. The problem is that as it appears it doesn't perfectly fit your question. So I will edit my answer.
– Stefanos
Nov 14 '18 at 0:37
@linabina It is working, I just tested it. The problem is that as it appears it doesn't perfectly fit your question. So I will edit my answer.
– Stefanos
Nov 14 '18 at 0:37
add a comment |
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This will probably get more useful results on CrossValidated; I'm voting to migrate it there. If I needed to do this, I would do it by brute force/approximation. We know that the distribution of the sum of j Gamma deviates with parameters (shape=k, scale=theta) is Gamma(j * k, theta), so we can define the distribution as a mixture of Gammas: Prob(x|beta,k,theta) = sum_j=0^infty dPois(j|beta)*Gamma(j * k,theta). The problem is that it's an infinite sum - unless there's a closed-form solution you might have to truncate the summation
– Ben Bolker
Nov 14 '18 at 1:35
I would probably look in Bailey's stochastic processes book or Pielou's statistical ecology book (or Johnson/Kotz) for more information about the distribution/what's known about the probability distribution
– Ben Bolker
Nov 14 '18 at 1:36
Wikipedia suggests that this distribution is equivalent to a Tweedie distribution, for which there are resources available in R (
library("sos"); findFn("tweedie")en.wikipedia.org/wiki/…– Ben Bolker
Nov 14 '18 at 1:37
unfortunately this package does not contain a function to estimate this parameters .
– lina bina
Nov 20 '18 at 21:40