Continuing the discussion from Constraining Output of Multinomial Distribution:
Over time, I have realized that the Wallenius non-central hyoergeometric distribution is not the correct distribution for my needs. The reason is: I do not believe the probability of selection depends upon the number of items available.
An example that I constructed that enumerates the type of problem I am trying to solve:
Assume we have
c different t-shirts, each with a different probability of being purchased,
p_i, i=1...c. For each t-shirt, we have
m_i, i=1...c copies of it.
Now, a customer is shown all of the t-shirts we have in stock (that is, each t-shirt where
m_i > 0). They purchase one according to the probabilities of the shirts that are available. Then this t-shirt is selected,
m_i is decremented by one and the next customer is shown the t-shirts.
This example that I listed seems to be a more “categorical sampling at each realization” where the probabilities only change if we “stick out” of an item.
Does this correspond to a Fisher non-central hypergeometric distribution? If not, does anyone know of a non-central hypergeometric distribution that satisfies this type of sampling?
If so, and
c is large, does anyone have insight as to how to compute the denominator (the
P_0) in the Fisher distribution?