[R] skewness and kurtosis in e1071 correct?

Dirk Enzmann dirk.enzmann at jura.uni-hamburg.de
Tue May 24 14:27:57 CEST 2005

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

phone: +49-040-42838.7498 (office)
        +49-040-42838.4591 (Billon)
fax:   +49-040-42838.2344
email: dirk.enzmann at jura.uni-hamburg.de

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