# 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!

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• 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 $\hat{\mu} = \int x f(x;\hat{\theta})dx$