----- Original Message -----
From: Glen Barnett <[EMAIL PROTECTED]>
To: David A. Heiser <[EMAIL PROTECTED]>; <[EMAIL PROTECTED]>
Sent: Friday, September 01, 2000 5:06 PM
Subject: Re: Skewness and Kurtosis Questions
> > >It is correct if you measure skewness in terms of mean-median. If you
> > >measure it some other way, it is no longer true. Note in particular
> > >that zero third central moment does not imply symmetry (contrary
> > >to what some books assert).
> >
> > If you use form 1) or form 3) then a zero value represents complete
> > symmetry.
>
> (I snipped them, but both forms were moment/cumulant based
> measures)
>
> I'm sorry, but this is wrong.
> Counterexamples are easy to construct and can be found
> in the literature. You can even set *all* odd moments to zero
> and still have non-symmetry. See, for example, Kendall and Stuart.
--------------------------------------------------------
The only solid reference you gave was Kendall and Stuart.
I went into "Kendall's Advanced Theory of Statistics", by Alan Stuart and J.
Keith Ord, fifth edition.
The only references in "Kendall.s" to your statement are:
1) On page 107, they say, "The coefficient b1 itself is also a measure of
skewness. Clearly if the distribution is symmetrical, b1 vanishes with mu3.
In general the ratio of mu3 to (mu2 to tne 3/2 power) will give some
indication of the extent of departure from symmetry...... However Ord
(1968b) gives some asymmetrical distributions with as many zero odd-order
moments as desired, and excercise 3.26 gives one with odd-order moments all
zero, so the value of gamma1 must be interpreted with some caution."
2) The reference given to Ord (1968b) {Biometrika 55,243) contains
information about an approximation to distribution functions which are
hypergeometric, and says nothing about the third moment.
3) Excercise 3.26 gives an unusual function that has more than one peak,
characteristic of damped oscillations on either side of zero. Weird distribu
tions consisting of a dominate single modal function combined with sin/cos
functions as secondary modes are encountered in physical data. They are
known to have problems of model and parameter characterization. Moments of
these functions do not represent uniqueness. The statistical properties of
all these wierd functions remain unknown, and therefore their use should be
avoided, if statistical properties are needed (such as confidence intervals
about the parameters).
Since you have given no other specific references, I have to assume that
your statement was one of generality.
For the large set of distributions having a central position in statistics,
the statement that a b1 or g1 value is a measure of assymmetry (both Fisher
and K. Pearson made this statement), is a valid statement. I certainly would
not want textbooks to be more confusing, by discussing all the exceptions,
that the student in practice would unlikely (very rarely) encounter. The
questions I read in EDSTAT are basically about application problems
involving the binomial or normal distribution in some way or other. Under
these circumstances, moment ratios are applicable.
DAH
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