Skewness

skew computes the skewness, kurt the excess kurtosis of the values in x.

Usage

skew(
  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
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. The weights are read as frequencies, so that their sum takes the place of the sample size in the estimator's bias corrections and in the standard error.
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: skewness 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 the skewness can either be:
1: g_1 = m_3 / m_2^(3/2)
2: G_1 = g_1 * sqrt(n(n-1)) / (n-2)
3: b_1 = m_3 / s^3 = g_1 ((n-1)/n)^(3/2)

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 G_1, that is of estimator = 2:

ASE.skew = sqrt( 6*n*(n-1)/((n-2)*(n+1)*(n+3)) )

to be used for calculating the confidence intervals. The standard errors of the other two estimators follow from it by the same factors that relate the estimators themselves, so that est/se does not depend on the choice of estimator. 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".

The standard error is only defined for n >= 3; for shorter input the variance, and with it any method="classic" interval, is NA.

This implementation of the two functions is comparably fast, as the expensive sums are coded in C++.

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


skew(bedrock::Pizza$price, na.rm=TRUE)
#>       est 
#> 0.4970801 

# use sapply to calculate skewness for a data.frame
sapply(bedrock::Pizza[,c("temperature","price","delivery_min")], 
       skew, na.rm=TRUE)
#>  temperature.est        price.est delivery_min.est 
#>       -0.8418683        0.4970801        0.6106322 

# the estimate lies inside its own confidence interval
set.seed(1)
skew(rlnorm(50), conf.level=0.95, method="classic")
#>       est       lci       uci 
#> 1.2578214 0.6371518 1.8784909