[Math] Standard error of standard deviation, skewness and kurtosis

standard deviationstandard error

I want to use this formula (shown below) for my work (not math based) to calculate the uncertainty in the sample standard deviation (obtained from the link below):
Calculating uncertainty in standard deviation

$SE(\sigma) = \frac{\sigma}{\sqrt(2N-2)}$

where $N$ is the size of the sample record.

However I cannot find any statistics books which cover this in anywhere? They all cover the standard error of the mean. The first sentence of the wikipedia article suggests that it should be possible to obtain a standard error for the standard deviation:

The standard error (SE) of a statistic (usually an estimate of a parameter) 
is the standard deviation of its sampling distribution[1] or an estimate of 
that standard deviation. If the parameter or the statistic is the mean, it 
is called the standard error of the mean (SEM).

Can anyone show me how this formula is derived and any simple(!) books which at least mention it?

Also, can a similar formula be found for standard error in kurtosis and skewness?

PS: I am not yet able to make a comment on the original thread. I made a previous question but it was asking too much so I'm asking again (with less stuff).

Best Answer

Although I don't think it's quite what you were looking for, the following may be helpful. In Mathematica, we can say things like:

In[1]:= Expectation[(Variance[{w, x, y, z}]-1)^2, {w \[Distributed] 
   NormalDistribution[], x \[Distributed] NormalDistribution[], 
  y \[Distributed] NormalDistribution[], 
  z \[Distributed] NormalDistribution[]}]
Out[1]:= 2/3

If you try a few, you find that the expected variance comes out to be 2/(n-1) (when the original normal distribution had a standard deviation of 1). We can confirm this by Monte-Carlo methods.

In[2]:= Table[{N[2/(n - 1)], 
  Mean[Table[(Variance[RandomVariate[NormalDistribution[], n]] - 1)^2,
     100000]]}, {n, 2, 20}]
Out[2]:= {{2., 2.02504}, {1., 0.990561}, {0.666667, 0.671811}, {0.5, 
  0.503348}, {0.4, 0.403083}, {0.333333, 0.335526}, {0.285714, 
  0.286752}, {0.25, 0.250503}, {0.222222, 0.22472}, {0.2, 
  0.200344}, {0.181818, 0.180828}, {0.166667, 0.168419}, {0.153846, 
  0.154251}, {0.142857, 0.141677}, {0.133333, 0.13277}, {0.125, 
  0.125321}, {0.117647, 0.117411}, {0.111111, 0.111469}, {0.105263, 
  0.104944}}

If we continue in this vein and consider the case where the standard deviation of the distribution, $\sigma$, isn't 1, we find the standard deviation of the variance for a sample of size $n$ is $$ \sigma^2\sqrt{\frac{2}{n-1}} $$

This result and much more can be found in this useful summary paper: Standard errors: A review and evaluation of standard error estimators using Monte Carlo simulations (DOI: 10.20982/tqmp.10.2.p107)

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