Re: most probable value of estimator of k-th central moment


"eric" <questermocolle@xxxxxxxxxxx> wrote in message
news:1172609555.495380.269440@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Thanks. I replied but I don't see my post, so here is it again.
Yes, the problem is: how do I compute the distribution of this
statistics? If the X_i's are normal distributed, it's trivial to
determine the distribution of (X_i)^k, but the distribution of
1/N \Sum_i (X_i)^k seems hard to determine. Someone knows where this
problem is discussed? My guess is that it's not known in closed form.

On Feb 27, 8:42 pm, <kenneth_m_...@xxxxxxxxxxxxx> wrote:
You need to first determine the distribution function for the said
statistics. Then you could determine the "peak" point if it exists.

"eric" <questermoco...@xxxxxxxxxxx> wrote in message

news:1172599814.912070.211310@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

Hi all, I have a (possibly) naive question, hope someone can give me
some
references. Consider the estimator of the k-th central moment from N
i.i.d. data:
mu_k(N) = 1/N \Sum_{i=1,N} (X_i)^k
(let's assume X_i has zero mean). Is there any general result for the
N-dependence
of the pdf of mu_k, for example if the X_i are normal distributed?
Specifically
I'm interested in the most probable value of mu_k(N).
Not asking for exact results, bounds would be good...
thanks.
+++++++++++++++++++++++++++++++++++++++++++++++++
I don't know where the other responders are comming from.

In general, this problem even for a normal distribution has been of
historical interest. I know only of work that has been done on the
normalized third and fourth moments/ as skewness and kurtosis. Fisher
developed his k statistics to develop a theoretical approach.. An inherent
problem is whether the momenst are about the mean or about some some other
central measure. There also has been some work on the higher momensts (see
Kendal, et. al). The historical objective was to use skewness and kurtosis
measures as tests for normality. A source here for some estimating equations
for the distributions is "Goodness-Of-Fit Techniques" by D' Agostino and
Stephens (Marcek Dekker (1986). My URL contains part of an effert to put all
this information on one site, but there was absolutely no outside interest
on this, so I stopped work on it several years ago.

David Heiser


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