kurtosis {e1071}

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 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) * (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)