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Hypergeometric distribution with heterogeneous sampling probabilities

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0

$begingroup$

I have been trying to find a proper framework for finding the corresponding generalization of the hypergeometric distribution when the sampling probability from the total items $N$ is not uniform.

That is, let's assume we have a categorical distribution $p_i$ that gives the drawing probability of item $i$. Now, if we take two samples of sizes $K$ and $n$, I am interested in the distributions of the overlap $k$ between these samples. Also, the form of $p_i$ is known, but I expect that the final distribution could be written in a closed form with some well-defined sum of powers of $p_i$ (based on my very preliminary calculations on $n$ and $K$ values less than 3...).

So far what I know is that the generalizations of the hypergeometric distribution commonly known in the literature don't really correspond to this problem (multivariate hypergeometric or Wallenius / Fisher noncentral hypergeometric).

Any suggestion would be highly appreciated.

probability-distributions sampling

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edited Mar 13 at 9:44

Cettt

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asked Mar 12 at 21:26

enysenys

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enys is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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  $begingroup$
  Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer. For equations, please use MathJax.
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  – dantopa
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add a comment | 

0

$begingroup$

I have been trying to find a proper framework for finding the corresponding generalization of the hypergeometric distribution when the sampling probability from the total items $N$ is not uniform.

That is, let's assume we have a categorical distribution $p_i$ that gives the drawing probability of item $i$. Now, if we take two samples of sizes $K$ and $n$, I am interested in the distributions of the overlap $k$ between these samples. Also, the form of $p_i$ is known, but I expect that the final distribution could be written in a closed form with some well-defined sum of powers of $p_i$ (based on my very preliminary calculations on $n$ and $K$ values less than 3...).

So far what I know is that the generalizations of the hypergeometric distribution commonly known in the literature don't really correspond to this problem (multivariate hypergeometric or Wallenius / Fisher noncentral hypergeometric).

Any suggestion would be highly appreciated.

probability-distributions sampling

share|cite|improve this question

edited Mar 13 at 9:44

Cettt

1,888622

asked Mar 12 at 21:26

enysenys

1012

New contributor

enys is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer. For equations, please use MathJax.
  $endgroup$
  – dantopa
  Mar 12 at 21:41
  

  
  
  
  

add a comment | 

$begingroup$

I have been trying to find a proper framework for finding the corresponding generalization of the hypergeometric distribution when the sampling probability from the total items $N$ is not uniform.

That is, let's assume we have a categorical distribution $p_i$ that gives the drawing probability of item $i$. Now, if we take two samples of sizes $K$ and $n$, I am interested in the distributions of the overlap $k$ between these samples. Also, the form of $p_i$ is known, but I expect that the final distribution could be written in a closed form with some well-defined sum of powers of $p_i$ (based on my very preliminary calculations on $n$ and $K$ values less than 3...).

So far what I know is that the generalizations of the hypergeometric distribution commonly known in the literature don't really correspond to this problem (multivariate hypergeometric or Wallenius / Fisher noncentral hypergeometric).

Any suggestion would be highly appreciated.

probability-distributions sampling

share|cite|improve this question

edited Mar 13 at 9:44

Cettt

1,888622

asked Mar 12 at 21:26

enysenys

1012

New contributor

enys is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

share|cite|improve this question

edited Mar 13 at 9:44

Cettt

1,888622

asked Mar 12 at 21:26

enysenys

1012

New contributor

enys is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

share|cite|improve this question

edited Mar 13 at 9:44

Cettt

1,888622

asked Mar 12 at 21:26

enysenys

1012

New contributor

enys is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

edited Mar 13 at 9:44

Cettt

1,888622

edited Mar 13 at 9:44

Cettt

1,888622

asked Mar 12 at 21:26

enysenys

1012

New contributor

enys is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked Mar 12 at 21:26

enysenys

1012