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I face a random variable whose distribution I don't know.

Someone draws a sample of k observations from a population and tells me their average. He repeats the process m times.

I assume m is in order of magnitude of hundreds.

If 1 < k < 20, What can I tell about the population variance?

What about other lower moments?

If k=1, I can trivially draw the emplirical distribution. What is the closest analoug for 1 < k < 20?

Amitai
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    What do you mean by "approximating the original population"? – Tim Aug 07 '16 at 15:55
  • @Tim, I mean to have the best fit I can of the original distribution. Having a good approximation of its lower moments would be a nice start. Another direction would be an analoug of the empirical distribution (with k=1 it is trivial, what if k=2 or k=3 ?) – Amitai Aug 07 '16 at 16:08
  • Could you edit to make it more precise (e.g. by defining what "small" k means, by defining what do you want to learn about the distribution etc.)? – Tim Aug 07 '16 at 16:27
  • Even for the case $k=1$ this is a broad question: it covers the entire theory of estimation. Could you focus it by giving more information about any assumptions that can be made about the unknown distribution? – whuber Aug 07 '16 at 20:30
  • @whuber and Tim, thank you for your focusing comments. I rephrased my question. – Amitai Aug 08 '16 at 08:01

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