e1071 (version 1.7-16)

kurtosis: Kurtosis

Description

Computes the kurtosis.

Usage

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

Value

The estimated kurtosis of x.

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 kurtosis detailed below.

Details

If x contains missings and these are not removed, the kurtosis 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:

$G_2=((n+1)g_2+6)\ast (n-1)/((n-2)(n-3))$. 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)