skew(x, na.rm = TRUE,type=3)
kurtosi(x, na.rm = TRUE,type=3)
mardia(x,na.rm = TRUE,plot=TRUE)
As of version 1.2.3,when finding the skew and the kurtosis, there are three different options available. These match the choices available in skewness and kurtosis found in the e1071 package (see Joanes and Gill (1998) for the advantages of each one).
If we define $m_r = [\sum(X- mx)^r]/n$ then
Type 1 finds skewness and kurtosis by $g_1 = m_3/(m_2)^{3/2}$ and $g_2 = m_4/(m_2)^2 -3$.
Type 2 is $G1 = g1 * \sqrt{n *(n-1)}/(n-2)$ and $G2 = (n-1)*[(n+1)g2 +6]/((n-2)(n-3))$.
Type 3 is $b1 = [(n-1)/n]^{3/2} m_3/m_2^{3/2}$ and $b2 = [(n-1)/n]^{3/2} m_4/m_2^2)$. For consistency with e1071 and with the Joanes and Gill, the types are now defined as above. However, from revision 1.0.93 to 1.2.3, kurtosi by default gives an unbiased estimate of the kurtosis (DeCarlo, 1997). Prior versions used a different equation which produced a biased estimate. (See the kurtosis function in the e1071 package for the distinction between these two formulae. The default, type 1 gave what is called type 2 in e1071. The other is their type 3.) For comparison with previous releases, specifying type = 2 will give the old estimate. These type numbers are now changed.
L.DeCarlo. 1997) On the meaning and use of kurtosis, Psychological Methods, 2(3):292-307,
K.V. Mardia (1970). Measures of multivariate skewness and kurtosis with applications. Biometrika, 57(3):pp. 519-30, 1970.
describe
, describe.by
, mult.norm in QuantPsyc, Kurt in QuantPsycround(skew(attitude),2) #type 3 (default)
round(kurtosi(attitude),2) #type 3 (default)
#for the differences between the three types of skew and kurtosis:
round(skew(attitude,type=1),2) #type 1
round(skew(attitude,type=2),2) #type 2
mardia(attitude)
x <- matrix(rnorm(1000),ncol=10)
describe(x)
mardia(x)
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