# Estimate a proportion using priority sampling (I just made that up)

I have this idea in my head that is either bunk or has a name I don't know. (I'm not naive enough to think I'm breaking new ground here!)

Here's my scenario: I would like to know the proportion of a population that has disease x. Instead of taking a random sample of the population, I'm taking what I'm calling a priority sample, which is I start with the people who are most likely to have the disease, then after finding them I go to the next priority or the next most likely. After I test so many, I have a positivity proportion. The more I test, the further down the priority list I get, and the lower my positivity proportion. At some point I stop and then taking my positivity proportion and the number I've tested, I apply it to some distribution (skewed I would guess) and get an estimate of what the actual proportion of the population has the disease with some kind of confidence interval.

Already with this I could imagine I could guess some kind of upper bound after some 10-15 positive tests since the true proportion in the population is unlikely to be higher than my positivity proportion since I started with the highest priority.

Does this method exist? If so, can you steer me in the right direction to learn how to do it properly?

EDIT

After mulling it over a bit, I realize there's one more detail I haven't included here, which is that I know the size of the total population and therefore can tell you the testing coverage.

In lieu of some fantastic mathematical proofs, I made a few simulations.

First I take a population of 1000 and divide them up into 100 groups of 10. Each of these have some proportion of the disease normally distributed around the population mean which I want to estimate. (I'm setting it at 0.2 here.)

I then order these groups into the highest proportion (priority) to lowest.

So if I were keeping track of my 10-person groups, when I reached the median group (50) I would have the population proportion. But the more common way is a cumulative collection. So if I graph the proportion cumulatively I get this.

This is surprisingly linear after about 20% coverage. And the true proportion seems to lie about 10% below the coverage at 20% for all but the extreme population proportions. So if this slope is consistent, then I could take the population coverage and extend it linearly to 100%.