Kurtosis

kurt() returns the excess kurtosis, therefore the kurtosis calculates as kurt(x) + 3 if required.

Usage

kurt(
  x,
  conf.level = NA,
  sides = c("two.sided", "left", "right"),
  method = c("boot", "classic"),
  estimator = 3,
  weights = NULL,
  na.rm = FALSE,
  ...
)

Arguments

x: a numeric vector. An object that is not a vector is coerced by as.vector if possible.
conf.level: confidence level of the interval. If set to NA (the default), only the point estimate is returned.
sides: character string specifying the sidedness of the confidence interval (one of "two.sided" (default), "left" or "right"). See ConfidenceIntervals().
method: character string specifying the confidence interval method. "boot" (default) uses a nonparametric bootstrap, with BCa intervals unless another bootstrap type is supplied through \dots; "classic" uses a Wald interval based on the asymptotic standard error. See Details and ConfidenceIntervals().
estimator: integer, either 1, 2 or 3 (default) defining the algorithm used for calculation. See Details.
weights: a numerical vector of weights the same length as x giving the weights to use for elements of x
na.rm: logical, indicating whether NA values should be stripped before the computation proceeds. Defaults to FALSE.
...: further arguments passed to boot::boot() when confidence intervals are calculated

Value

if conf.level = NA, a numeric scalar. Otherwise a named numeric vector with elements:

est: kurtosis estimate
lci: lower confidence interval bound
uci: upper confidence interval bound

Details

If na.rm is TRUE then missing values are removed before computation proceeds.

The estimator for calculating kurtosis can either be:
1: g_2 = m_4 / m_2^2 - 3
2: G_2 = ((n+1) g_2 + 6) * (n-1) / ((n-2)(n-3))
3: b_2 = m_4 / s^4 - 3 = (g_2 + 3) (1 - 1/n)^2 - 3

1 is the typical definition used in Stata and in many older textbooks.
2 is used in SAS and SPSS.
3 is used in MINITAB and BMDP.

Cramer (1997) mentions the asymptotic standard error of the kurtosis:

ASE.kurt = sqrt((24*n*(n - 1)^2) / ((n - 3)*(n - 2)*(n + 3)*(n + 5)))

to be used for calculating the confidence intervals. This is implemented here with method="classic".
However, Joanes and Gill (1998) advise against this approach, pointing out that the normal assumptions would virtually always be violated. They suggest using the bootstrap method. That's why the default method for the confidence interval type is set to "boot". If not further specified the boot ci type will be chosen as "bca".

This implementation is comparably fast, as the expensive sums are coded in C.

Random number generation

method = "boot" - the default - resamples and therefore advances R's global random number generator. Call base::set.seed() beforehand for reproducible intervals.

References

Cramer, D. (1997): Basic Statistics for Social Research Routledge.

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

Examples


kurt(Pizza$price, na.rm=TRUE)
#> [1] 0.1076097

# use sapply to calculate skewness for a data.frame
sapply(Pizza[,c("temperature","price","delivery_min")], kurt, na.rm=TRUE)
#>  temperature        price delivery_min 
#>   0.05058327   0.10760970   0.09541410 

# or apply to do that columnwise with a matrix
apply(as.matrix(Pizza[,c("temperature","price","delivery_min")]), 2, 
      kurt, na.rm=TRUE)
#>  temperature        price delivery_min 
#>   0.05058327   0.10760970   0.09541410