Error on the standard deviation, skewness & Kurtosis

poi

New Member
#1
Dear all,

Given a sample of N points extracted from a distribution, I would like to know how to estimate the error on the higher moments of the distribution, such as the standard deviation, the skewness and kurtosis. In other words, I know the error on the mean is simply sigma/sqrt(n), but what about the others?
Thanks
 

BGM

TS Contributor
#2
The variance/standard error of the sample variance can be calculated in closed form and it is distributional free.

However for the sample standard deviation, sample skewness and sample kurtosis, they involve some non-linear transformation of sample moments and thus their exact variance/standard error depends on the distribution of the underlying sample.

Of course when the sample size is large, you can use the asymptotic variance for those estimators (by delta's method), which is usually the leading term of the actual one.
 

poi

New Member
#3
Nice.
Could you please give me a reference where to find the variance/standard error of the sample variance? Sorry about my ignorance, but is there an analytical expression for this asymptotic variance for the other estimators given by the delta's method?

Thanks a lot!

Alex
 

BGM

TS Contributor
#4
For the sample variance, you may take a look at

http://en.wikipedia.org/wiki/Mean_squared_error

Since it is unbiased estimator, its variance is equal to its MSE which you can find in the example section.

For the other two, you may first take a look at

http://en.wikipedia.org/wiki/Skewness#Sample_skewness
http://en.wikipedia.org/wiki/Kurtosis#Sample_kurtosis

However it only gives the case for the underlying sample following normal distribution. I have done some calculations before following the book "Elements of Large Sample Theory" example 5.2.7. I may type the result out later.