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

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  • $\begingroup$ Thanks! For standardized moments, is its MLE also obtained by the invariance of MLE, and how about its unbiased estimates? $\endgroup$
    – Tim
    Commented Feb 2, 2014 at 1:31

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