To me your answer is opaque (but that seems to be rather a problem of
language ;-) ). Perhaps my question has not been expressed clearly
enough. Let me state it differently:
In the R package e1071 the formulas (implicit) used are (3) and (4) (see
below), the standard deviation used in these formulas, however is based
on (2) (see below). This seems to be inconsistent and my question is,
whether there is a commonly used third definition of skewness and
kurtosis in which the formulas for the "biased" skewness and kurtosis
_but_ with the "unbiased" standard deviation are employed.
The standard deviation can be defined as the _sample_ statistic:
sd = 1/n * sum( (x - mean(x))^2 ) # (1)
and as the estimated population parameter:
sd = 1/(n-1) * sum( (x-mean(x))^2 ) # (2).
In R the function sd() calculates the latter.
In the same way, expressed via z-values skewness and kurtosis can be
defined as the _sample_ statistic (also called "biased estimator" , see:
http://www.mathdaily.com/lessons/Skewness ):
skewness = mean(z^3) # (3)
kurtosis = mean(z^4)-3 # (4)
with z = (x - mean(x))/sd(x)
with sd = 1/n * sum( (x - mean(x)^2 )
(thus: here sd is the _sample_ statistic, see (1) above!)
but they can also be defined as the estimated population parameters
(also called "unbiased", see:
http://www.mathdaily.com/lessons/Kurtosis#Sample_kurtosis ):
skewness = n/((n-1)*(n-2)) * sum(z^3) # (5)
kurtosis = n*(n+1)/((n-1)*(n-2)*(n-3)) * sum(z^4) -
3*(n-1)^2/((n-2)*(n-3)) # (6)
with z = (x - mean(x))/sd(x)
with sd = 1/(n-1) * sum( (x - mean(x)^2 )
(thus: here sd is the estimated population parameter, see (2)
above!. BTW: The R function scale() calculates the z-values based on
this definition, as well.)
Campbell wrote:
This is probably an issue over definitions rather than the correct
answer. To me skewness and kurtosis are functions of the distribution
rather than the population, they are equivalent to expectation rather
than mean. For the normal distribution it makes no sense to estimate
them as the distribution is uniquely defined by its first two moments.
However there are two defnitions of kurotsis as it is often
standardized such that the expectation is 0.
*************************************************
Dr. Dirk Enzmann
Institute of Criminal Sciences
Dept. of Criminology
Edmund-Siemers-Allee 1
D-20146 Hamburg
Germany
phone: +49-040-42838.7498 (office)
+49-040-42838.4591 (Billon)
fax: +49-040-42838.2344
email: [EMAIL PROTECTED]
www:
http://www2.jura.uni-hamburg.de/instkrim/kriminologie/Mitarbeiter/Enzmann/Enzmann.html
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