Kurtosis

Computes the kurtosis.

kurtosis(x, na.rm = FALSE, type = 3)

Arguments

x	a numeric vector containing the values whose kurtosis is to be computed.
na.rm	a logical value indicating whether NA values should be stripped before the computation proceeds.
type	an integer between 1 and 3 selecting one of the algorithms for computing skewness detailed below.

Details

If x contains missings and these are not removed, the skewness is NA.

Otherwise, write \(x_i\) for the non-missing elements of x, \(n\) for their number, \(\mu\) for their mean, \(s\) for their standard deviation, and \(m_r = \sum_i (x_i - \mu)^r / n\) for the sample moments of order \(r\).

Joanes and Gill (1998) discuss three methods for estimating kurtosis:

Type 1:

\(g_2 = m_4 / m_2^2 - 3\). This is the typical definition used in many older textbooks.

Type 2:

\(G_2 = ((n+1) g_2 + 6) * (n-1) / ((n-2)(n-3))\). Used in SAS and SPSS.

Type 3:

\(b_2 = m_4 / s^4 - 3 = (g_2 + 3) (1 - 1/n)^2 - 3\). Used in MINITAB and BMDP.

Only \(G_2\) (corresponding to type = 2) is unbiased under normality.

Value

The estimated kurtosis of x.

References

D. N. Joanes and C. A. Gill (1998), Comparing measures of sample skewness and kurtosis. The Statistician, 47, 183--189.

Examples

x <- rnorm(100)
kurtosis(x)
#> [1] -0.3256976