Suppose we perform n different multinomial experiments, each time with k possible outcomes: category 1 to category k. The probability of drawing category j in the i-th experiment is pij. That is, the probabilities of drawing a certain category differ for different experiments. We do know all the pij (i=1,…,n, j=1,…,k) (and for each i these pij sum up to 1, as they should).
Let xi be the outcome of the i-th experiment.
Now we are told that we have to draw category j (j=1,...,k) exactly nj times (where the sum of the nj is equal to n, of course).
My problem is to sample the xi (i=1,…,n) given that category 1 is drawn n1 times, category 2 n2 times, …, and category k nk times using the probabilities pij.
Does anyone know how to do this in an efficient way?
For small n one could generate all possible samples, select the ones where category 1 is drawn n1 times, category 2 n2 times, …, and category k nk times, calculate the probabilities of these samples conditional on the constraint that one of them has to be chosen, and then draw one of these samples. However, in my case n is large and this approach does not work.
(What complicates the problem further is that some of the pij may be equal to zero. This means that drawing the categories sequentially does not always work as for some categories there may not be enough experiments left where this category has non-zero probability).
What do you mean with sample the xi ?
That`s because the outcomes may not be random variables.
A random sample is a succesion of random variables independients and identically distributed. Of course you could make empirical observations and use parameters for determining their common distribution function or their joint distribution function.
Dear Alejandra,
Thank you for your reply.
Maybe I should have written that I want to draw a vector x=(x1,...,xn). Each such a vector has a certain probabilty of being selected, which can be calculated easily using the pij. I want to draw such a vector x given that each category i occurs exactly ni times (i=1,...k) in x.
What we have here is a distribution "hipergeometrica multivariada" (sorry I didn't know how to write this in english here you have the answer:
Dear Alejandro,
Thank you for your reply. Both the sample size n and population size N are very large in my case (a few million). That means that I can't calculate your summations exactly. Perhaps I'll be able to use some approximation.
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