Computes the kurtosis.

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

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.

The estimated kurtosis of `x`

.

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.

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

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