e1071 (version 1.7-4)

# 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 kurtosis detailed below.

## Value

The estimated kurtosis of x.

## 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.

## References

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

## Examples

# NOT RUN {
x <- rnorm(100)
kurtosis(x)
# }