Resulting distribution when repeatedly sampling without replacement from weighted elements

Question

I have a model where I need to sample from $m$ elements without replacement, where each element exists exactly once. The element have different weights and the probability of an element being drawn is equal to the fraction of the element's weight relative to the sum of the weights of the remaining elements. In R a single draw would simply be:

m <- 4                  # number fo elements
w <- c(.4,.3,.2,.1)     # weights
n <- 3                  # number of draws
sample(seq_along(w), size = n, prob=w, replace=FALSE)

> [1]  3  1  2 

Simulating many draws and calculating the probability of occurrence of each element we get

v <- replicate(1e5, sample(seq_along(w), size = n, prob=w, replace=FALSE))
p.hat <- table(v) / sum(table(v))
p.hat

>
        1         2         3         4 
0.3075767 0.2902900 0.2526967 0.1494367 

What is the name of the resulting distribution and how could I calculate the results above without simulation?

PS. I stumbled across Wallenius' noncentral hypergeometric distribution but my knowledge in this field is too limited as to decide if this distribution is given or not.

Answer 1

I would not bother trying to come up with a name, or find this in the books. This is an example of a sampling distribution in finite population sampling, and that's about it. You can probably describe all the samples explicitly for a population this small, and derive all the probabilities (again, knowing exactly what your sampling scheme is), so you don't have to rely on sample() simulation.

Unequal probability sampling is very, very complicated. Brewer and Hanif (1983, https://books.google.com/books/about/Sampling_with_unequal_probabilities.html?id=SyPvAAAAMAAJ) listed about 50 methods. Their premise though is to come up with a method that guarantees (or at least closely approximates) the target probabilities of selection (which, as you have seen, aren't even remotely close in your simulation example).