Computes the kurtosis.
a numeric vector containing the values whose kurtosis is to be computed.
a logical value indicating whether
an integer between 1 and 3 selecting one of the algorithms for computing skewness detailed below.
x contains missings and these are not removed, the skewness
Otherwise, write x_i for the non-missing elements of
n for their number, mu for their mean, s for
their standard deviation, and
m_r = ∑_i (x_i - mu)^r / n
for the sample moments of order r.
Joanes and Gill (1998) discuss three methods for estimating kurtosis:
g_2 = m_4 / m_2^2 - 3. This is the typical definition used in many older textbooks.
G_2 = ((n+1) g_2 + 6) * (n-1) / ((n-2)(n-3)). Used in SAS and SPSS.
b_2 = m_4 / s^4 - 3 = (g_2 + 3) (1 - 1/n)^2 - 3. Used in MINITAB and BMDP.
Only G_2 (corresponding to
type = 2) is unbiased under
The estimated kurtosis of
D. N. Joanes and C. A. Gill (1998), Comparing measures of sample skewness and kurtosis. The Statistician, 47, 183–189.