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# A personal list of favorite unbiased estimators in statistics

Mathias Fuchs, March 2019
This is my personal list of favorite unbiased estimators.
I'm using the following abbreviations:
• $n=$ number of observations
• $m = \bar{x}=$ sample mean of observations
• $s^2=$ (unbiased) sample variance of observations
• $y_2 = \frac{1}{n}\sum (x_i - m)^2=$ biased second central sample moment. Same as $\frac{n-1}{n}s^2$.
• $y_3 = \frac{1}{n}\sum (x_i - m)^3=$ biased third central sample moment, aka sample skewness
• $y_4 = \frac{1}{n}\sum (x_i - m)^4=$ biased fourth central sample moment, aka sample kurtosis
So, here is the list of unbiased estimators:
 mean $m = \bar{x} = \frac{1}{n}\sum x_i$ variance $s^2 = \frac{1}{n-1}\sum\bigl(x_i - \bar{x}\bigr)^2$ square of mean $\frac{1}{n}\sum x_i^2 - s^2 = m^2 - \frac{s^2}{n}$ fourth power of mean $m^4 - \frac{6 }{n}m^2s^2 + \bigl(\frac{-1}{2n} - \frac{3}{n-2} + \frac{2}{n-3}\bigr)y_2^2 + \bigl(\frac{-8}{n-1} + \frac{8}{n-2}\bigr) m y_3 + \bigl(\frac{-3}{n-1} + \frac{6}{n-2} -\frac{3}{n-3}\bigr) (y_4 - 3) - \frac{9}{n-1} + \frac{18}{n-2} - \frac{9}{n-3}$ square of population variance $\bigl(1 + \frac{1}{2(n-1)} + \frac{5}{2(n-2)} + \frac{9}{2(n-2)(n-3)}\bigr) y_2^2 - \big(\frac{1}{n-2} + \frac{3}{(n-2)(n-3)}\bigr) y_4$ variance of $s^2$ $(s^2)^2 - \text{preceding line}$ population kurtosis $\mathbb{E}[(X-\mathbb{E}(X))^4]$ $\bigl(\frac{3}{2(n-1)} + \frac{6}{n-2} - \frac{27}{2(n-3)}\bigr) y_2^2 + \bigl(1+\frac{1}{n-1}-\frac{6}{n-2} + \frac{9}{n-3}\bigr) y_4$ variance of $s^2$ - alternative $\frac{\text{preceding line}}{n} - \frac{\text{(second-to-last preceding line)} (n-3)}{n(n-1)}$ variance of $s^2$ - another alternative, shortest version $\bigl(\frac{1}{2(n - 2)} + \frac{1}{2n} - \frac{2}{n - 3} \bigr) (s^2)^2 + (\frac{3}{n - 3} - \frac{2}{n - 2}) y_4$ error of a supervised learning algorithm U-statistic associated with the kernel of degree $g+1$ where $g$ is the learning set size