negative hyper-geometric distribution

[Oscar Ozfidan]

Unlike some other distributions (negative binomial), Stata does not seem to have a function for negative hyper-geometric distribution. This is the the built-in function for hyper-geometric distribution: p = hypergeometric(N, K, n, k).
Does anyone know if there is a user written program or any other way of getting the cumulative probability values for a negative hyper-geometric distribution? Thanks for any help or guidance you can provide.

[William Lisowski]

At

https://encyclopediaofmath.org/wiki/...c_distribution

there is an equation relating the negative hypergeometric distribution function to that for the hypergeometric distribution function. Perhaps that will help you.

[Oscar Ozfidan]

Thank you William. I have seen that but I lack background to implement it on Stata.

[William Lisowski]

Stata tells us

   hypergeometric(N,K,n,k)
       Description:  the cumulative probability of the hypergeometric distribution
        
                     N is the population size, K is the number of elements in the population
                     that have the attribute of interest, and n is the sample size.  
                     Returned is the probability of observing k or fewer elements
                     from a sample of size n that have the attribute of interest.

In the terms of the linked article, the cumulative hypergeometric distribution G(m) with parameters N, M, and n - the probability of obtaining m or fewer "marked" elements when sampling n elements from a population of N elements of which M are marked - would be G(m) which in Stata is given by hypergeometric(N,M,n,m).

The linked article tells us the cumulative negative hypergeometric distribution F(n) with parameters N, M, and m - the probability that it will require a sample of n or fewer elements to yield m "marked" elements when sampling from of population of N elements of which M are marked - would be F(n) = 1 - G(m-1). From the above, G(m-1) = hypergeometric(N,M,n,m-1). Substituting this in tells us that in Stata F(n) would be given by 1 - hypergeometric(N,M,n,m-1).

[Oscar Ozfidan]

Thank you William.