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In my Master's thesis project I could not show my (normally-distributed) samples to have a common population variance (through Levene's test or otherwise), so I could not use the n/(n-1) Bessel's correction or similar for averaged sample variances; I used the n/(n+1) sample variance correction for my population variance estimate for each sample (seeking to minimise the mean square error, to get as close as possible to each sample's true population variance in my estimates), but have run into trouble in the requirement that the only citations I am allowed are peer-reviewed research or review articles, or perhaps published textbooks.

If I understand correctly, these below are a number of uncitable sources claiming the validity of the n/(n+1) correction. If the approach is valid, is there any citable source I could use? Searching by myself thus far has not gone well.

https://en.wikipedia.org/wiki/Mean_squared_error#Variance

http://people.missouristate.edu/songfengzheng/Teaching/MTH541/Lecture%20notes/evaluation.pdf

https://math.stackexchange.com/questions/1250333/variance-with-minimal-mse-in-normal-distribution

https://davegiles.blogspot.com/2013/05/variance-estimators-that-minimize-mse.html

https://alemorales.info/post/variance-estimators/

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    $\begingroup$ You can’t just give the derivation yourself? You’re not claiming something with a proof by citation; you’re proving a mathematical fact. $\endgroup$
    – Dave
    Commented Aug 24, 2020 at 9:49
  • $\begingroup$ @Dave : An excellent point! Thank you very much! $\endgroup$
    – MCC
    Commented Aug 24, 2020 at 10:00

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Here are two references from textbooks I frequently turn to for rather "basic" stuff:

  • Kevin P. Murphy - "Machine Learning: A Probabilistic Perspective", Ch. 6.4.2 Unbiased estimators, p. 200
  • Christopher M. Bishop - "Pattern Recognition and Machine Learning", Ch. 1.2.4 The Gaussian distribution, p. 27

I'm pretty sure there are more.

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  • $\begingroup$ I am now looking at those two (Murphy 2012, Bishop 2006): I am only seeing references to N/(N-1) there (equations 6.35 and 1.59), both referring to eliminating the bias of the population-variance estimate rather than minimising its mean square error. In a bullet-cluster analogy, I do not want to shoot at the target so that the shots miss it on all sides while being perfectly centred on the heart, nor do I want the bullets all hitting the same spot on a tree off to the side from the target--I want most bullets hitting the target even if off-centre. Thank you nevertheless. $\endgroup$
    – MCC
    Commented Aug 24, 2020 at 11:47

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