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I have some set of items. Each item has a weight and I can sample the items from the population with probabilities proportional to their weights. I know the size of the population. I want to estimate the total/average weight of the population. The population is assumed to be finite.

What is known about this problem? Could you please give me some pointers to the literature?

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  • $\begingroup$ Stephen Thompson, Sampling; en.wikipedia.org/wiki/Horvitz%E2%80%93Thompson_estimator. (You already know about the latter: you asked about it in your previous, now deleted, question.) $\endgroup$ – whuber Feb 9 at 16:42
  • $\begingroup$ Thank you for the reference. Could you elaborate on how it helps? I think I cannot use that when I don't know the probabilities which I don't as otherwise, I would also know the sum. $\endgroup$ – user2316602 Feb 9 at 17:32
  • $\begingroup$ Your question indicates you do know the weights, so please clarify that. $\endgroup$ – whuber Feb 9 at 17:38
  • $\begingroup$ Yes, I do know the weights, I don't know the probabilities. These differ by a factor given by the sum I am trying to estimate. $\endgroup$ – user2316602 Feb 9 at 17:41
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    $\begingroup$ Yes, independent samples $\endgroup$ – user2316602 Feb 11 at 18:31
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So you have a distribution of numbers (interpreted as weights) and you want to estimate the mean of the population? A very well studied problem!

The mean of the sample is an unbiased estimate of the mean of the distribution. However, it is not robust - one outlier can throw out the estimate. How much of a problem this is in practice depends on the variance. If the distribution is normal, with a standard deviation that is small compared to the mean, the estimator will be good. If the distribution has "Black Swans" (rare extreme values) then you are in trouble.

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  • $\begingroup$ The mean of a probability weighted sample is a biased estimate, I'm afraid. You might have overlooked some crucial details in the question. $\endgroup$ – whuber Feb 9 at 16:43

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