MSE of sample standard deviation (cont')
Continue our previous post of computing the MSE of sample standard deviation. We want to have higher order expansion of $E(S_n)$. The algebric computation is a bit tedious, we will use sympy to help us.
The result is quite interesting. For normal distribution, the MSE of $S_n$ and $\hat \sigma$ are the same up to the order $1/n^2$. If the distribution has a thicker tail than normal, i.e. kurtosis > 3, then $\hat \sigma_n$ will have a smaller MSE than $S_n$. On the other hand, if the distribution has a lighter tail than normal, than $S_n$ will have a smaller MSE.
