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Using sample moments, how can the mean and variance estimators be improved if e.g. skewness and kurtosis are known exactly? And what about using estimates for these instead, which should imho be of no help?

In general then I would like to know if there are analogous approaches for other kinds of location and scatter estimators, such as L-moments.

kjetil b halvorsen
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Quartz
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    What do you mean by "skewness and kurtosis are known exactly"? Are you referring to the *sample* moments or to properties of the parent distribution? If it's the latter, what assumptions are you making about that distribution? – whuber Jul 09 '13 at 18:03
  • Parent properties in the first question, only sample moments in the second. I am making no specific assumption on the distribution, that's what makes the issue interesting; otherwise if a parametric (in such moments) family was given then it should be pretty straightforward to approach, e.g. via ML, right? – Quartz Jul 10 '13 at 08:43
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    Because (talking about properties of a distribution) skewness and kurtosis tell us nothing about the mean or variance (except that they exist), I take the opposite view: the first question is uninteresting because after your clarification there is nothing left to answer! I do not know what the second question asks because I find it impossible to determine what you mean by "analogous approaches." – whuber Jul 10 '13 at 13:50
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    True, skewness and kurtosis tell us nothing about mean or variance, they're orthogonal by definition, but they do have an influence on their *sample* estimates in terms of efficiency. E.g. large kurtosis will mean that large "outliers" in sample variance will be more probable than otherwise, and thus an appropriate correction could be taken to improve efficiency. Maybe even just a simple weighting would do. That's my main point, sorry if the question is not clear enough as stated. Actually I know how to address the problem, just wanted to spare the effort, it's out there already for sure. – Quartz Jul 10 '13 at 16:46

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