I have a simple random sample (SRS) $S$ of size $n$ drawn from a population $D_1$ containing $N_1$ members. A new set $D_2$ with $N_2$ members is added to the population. I would like to create a new set $S'$ that is a SRS from $D_1 \cup D_2$.
One strategy that won't work is to draw a SRS of size $nN_2/N_1$ (or any other size) from $D_2$ and add is to $S$. (The result would be a stratified random sample from $D1 \cup D2$, but not a SRS from it.)
A strategy that will work is to first make one trial on a random variable with a hypergeometric distribution (population size $N_1+N_2$, number of draws $n'$, and number of success states in the population $N_1$). The resulting value $n'_{1}$ is the number of items in $S'$ that should be taken from $D_1$. If $n'_{1} < n$, we discard a SRS of size $n - n'_{1}$ from $S$ to form the initial $S'$. If $n'_{1} > n$, we select an additional SRS of size $n'_{1} - n$ from $D_1 \setminus S$, and put these items, plus all items from $S$ in $S'$. We also draw an SRS of size $n'_{2} = n' - n'_{1}$ from $D_2$ and add it to $S'$.
I'm interested in choosing $n'$ to minimize $E[|S' \setminus S|]$, the expected size of $S' \setminus S$. In other words, the expected value of the number of items from $D_2$ plus the number of new items, if any, from $D_1$.
I have three questions:
Q1. If we have the additional constraint that $n' \ge j$ for some $j$, is there a simple argument that the optimal $n'$ is $j$? Seems like it has to be, but I'd rather not work out the expected value formula and slog through the derivatives.
Q2. Does anyone know of a better algorithm for the above problem?
Q3. The problem I'd really like to solve is more complex. Each item in $D_1$ and $D_2$ is either an A or a B. $S$ has at least $k$ A's, and I want to minimize $E[|S' \setminus S|]$ subject to the constraint that $S'$ has at least $k$ A's. The above algorithm no longer works, because we can't pick $n'$ in advance. (We don't know how rich $D_2$ is in A's, and we might get unlucky anyway.) Is there a simple algorithm for this case? I've perused
Thompson, S. K. and Seber, G. A. G. Adaptive Sampling. Wiley, New York, 1996
but can't find exactly this problem treated. But my gut tells me that both the above problems have been faced and solved many times before.