# unbiased estimates and MLE of central moments and of standardized moments?

I have heard of unbiased estimate and MLE of variance, and some about those of kurtosis. Are there general results about

• unbiased estimates of k-th order central moments?

• MLE of k-th order central moments?

• unbiased estimates of k-th order standardized moments?

• MLE of k-th order standardized moments?

Thanks!

-

## 1 Answer

• Yes, they can be constructed, although they are not simple. See http://www.jstor.org/stable/2985201 .

• Using the invariance of the MLE, it follows that the MLE of the k-th order central moments and the k-th order central moments is simply obtaining by plugging the MLE into the expression of the corresponding quantities of interest. For instance $\hat{\mu} = \int x f(x;\hat{\theta})dx$

-
Thanks! For standardized moments, is its MLE also obtained by the invariance of MLE, and how about its unbiased estimates? –  Tim Feb 2 '14 at 1:31