e1071 (version 1.6-7)

kurtosis: Kurtosis

Description

Computes the kurtosis.

Usage

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.

Value

The estimated kurtosis of x.

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.

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)