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A reviewer asked for a citation that the sample mean is a consistent and unbiased estimator of the expected value and therefore converges towards the expected value. I know I can easily do the calculations myself based on the variance of the sample mean etc. However, since the reviewer specifically asked for a citation, I am not sure a proof instead would be acceptable. Also, I don't like adding proofs that have almost nothing to do with the actual subject of the article.

In most books I looked at, this fact is either only provided for the normal distribution (i.e. sample mean is an estimator of the parameter $\mu$), or included as an exercise.

Is anybody aware of a citation that I could use to support the claim that the sample mean is an unbiased estimator for any distribution (assuming of course expected value exists and variance is finite).

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Unbiasedness: Casella and Berger, Statistical Inference, 2nd Ed., Theorem 5.2.6 p213-214

Consistency: Casella and Berger, Statistical Inference, 2nd Ed., Example 10.1.2 et seq., p468-469

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  • $\begingroup$ Casella was my first thought, too. I wonder if it would be necessary, however, to go to peer-reviewed, primary literature. $\endgroup$
    – Dave
    Jul 31 '20 at 10:36
  • $\begingroup$ @Dave I don't really know why the reviewer wants a citation for this, as I considered it general knowledge. So I just hope they are satisfied with this citation. The paper isn't about statistics and it is just a small part of the error estimation of the analysis, so I really believe this is sufficient. $\endgroup$
    – LiKao
    Jul 31 '20 at 10:58
  • $\begingroup$ @Dave Good luck with that. $\endgroup$
    – Glen_b
    Jul 31 '20 at 17:12
  • $\begingroup$ @Glen_b that’s what I was thinking, that it would be quite difficult to find it in the primary literature. $\endgroup$
    – Dave
    Jul 31 '20 at 17:24
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    $\begingroup$ I think a textbook will be adequate for what's essentially undergraduate work; it's not like physicists expect you to cite Philosophiæ Naturalis Principia Mathematica and/or give the quote "Lex II: Mutationem motus proportionalem esse vi motrici impressae, et fieri secundum lineam rectam qua vis illa imprimitur." when you just want to write $F=ma$. $\endgroup$
    – Glen_b
    Jul 31 '20 at 17:34

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