I have a large population and I want select a sample based on some characteristics hence the selected sample is not random since not all members of the population have an equal chance of being selected. From that non-random sample, I will pick a random subset. My question is that random subset unbiased and can get a consistent estimate from it? For example, can I prove that the mean of that subset converges to the mean of the non-random sample and the mean of population? Assuming discrete and iid. I liked at some posts like This one, this one and this but did not answer my questions.
Another way: What if I start with the random sample first and then non-random sample second? Will I get an consistent estimator and unbiased mean?
Example :
Suppose I am trying to find the mean of the even numbers of the populations. Then I have to pick a non-random sample of even numbers right? And then I take a random subset of that non-random sample to show that the mean of that subset is consistent estimator.
