Accuracy from biased sampling of boomerang throws Suppose I keep throwing boomerangs every 10 seconds, and I throw 50,000 of them. 20% of the boomerangs are hit by drones so they don’t come back. The rest come back but the time it takes for them to come back varies. Each boomerang is tagged so for each boomerang that returns I can exactly find out how long it took for it to return.
I want to know at what point in time would I be able to accurately estimate the mean duration of all the boomerangs that return.
How would one go about answering that?
I imagine the answer would be in the following format: You’ll be 95% certain with a margin of error 5% after n boomerangs return.
If that is the case then how can we account for the fact that this way of sampling is biased towards having more shorter duration boomerang trips? (Because at the beginning of this experiment, among the boomerangs that have been thrown at more or less the same time, shorter trips will have returned more than their long-trip counterparts)
Any pointers (other posts, resources) here would be appreciated.

This is a missing data problem, and in a literature, this would be called a missing not at random (MNAR) mechanism. So the missingness is related to the value of the missing variable. This is common, e.g., people with too high/low income are more likely not to report it, people that are too depressed are more likely not to fill the depression questionnaire, or people that are too sick won't go for follow-up tests.
In general, this is considered hard or impossible to solve unless you find some other variables that are related to outcome/missingness that are not missing or you make some additional assumptions.
I can imagine that if you assume that e.g., throw times should be normally distributed and missingness depends linearly on throw times, then you should be able to write a likelihood function for such a situation, but I have no idea how to do it, and how to estimate it. Searching for keywords like MNAR estimation might be a good start
