Coefficient of (Excess) Kurtosis

Compute the sample coefficient of kurtosis or excess kurtosis.

kurtosis(x, na.rm = FALSE, method = "fisher", l.moment.method = "unbiased", 
    plot.pos.cons = c(a = 0.35, b = 0), excess = TRUE)

Arguments

x: numeric vector of observations.
na.rm: logical scalar indicating whether to remove missing values from x. If na.rm=FALSE (the default) and x contains missing values, then a missing value (NA) is returned. If na.rm=TRUE, missing values are removed from x prior to computing the coefficient of variation.
method: character string specifying what method to use to compute the sample coefficient of kurtosis. The possible values are "fisher" (ratio of unbiased moment estimators; the default), "moments" (ratio of product moment estimators), or "l.moments" (ratio of $L$-moment estimators).
l.moment.method: character string specifying what method to use to compute the $L$-moments when method="l.moments". The possible values are "ubiased" (method based on the $U$-statistic; the default), or "plotting.position" (method based on the plotting position formula).
plot.pos.cons: numeric vector of length 2 specifying the constants used in the formula for the plotting positions when method="l.moments" and
l.moment.method="plotting.position". The default value is
plot.pos.cons=c(a=0.35, b=0). If this vector has a names attribute with the value c("a","b") or c("b","a"), then the elements will be matched by name in the formula for computing the plotting positions. Otherwise, the first element is mapped to the name "a" and the second element to the name "b".
excess: logical scalar indicating whether to compute the kurtosis (excess=FALSE) or excess kurtosis (excess=TRUE; the default).

Details

Let $\underline{x}$ denote a random sample of $n$ observations from some distribution with mean $\mu$ and standard deviation $\sigma$.

Product Moment Coefficient of Kurtosis
(method="moment" or method="fisher")
The coefficient of kurtosis of a distribution is the fourth standardized moment about the mean: $$\eta_4 = \beta_2 = \frac{\mu_4}{\sigma^4} \;\;\;\;\;\; (1)$$ where $$\eta_r = E[(\frac{X-\mu}{\sigma})^r] = \frac{1}{\sigma^r} E[(X-\mu)^r] = \frac{\mu_r}{\sigma^r} \;\;\;\;\;\; (2)$$ and $$\mu_r = E[(X-\mu)^r] \;\;\;\;\;\; (3)$$ denotes the $r$'th moment about the mean (central moment).

The coefficient of excess kurtosis is defined as: $$\beta_2 - 3 \;\;\;\;\;\; (4)$$ For a normal distribution, the coefficient of kurtosis is 3 and the coefficient of excess kurtosis is 0. Distributions with kurtosis less than 3 (excess kurtosis less than 0) are called platykurtic: they have shorter tails than a normal distribution. Distributions with kurtosis greater than 3 (excess kurtosis greater than 0) are called leptokurtic: they have heavier tails than a normal distribution.

When method="moment", the coefficient of kurtosis is estimated using the method of moments estimator for the fourth central moment and and the method of moments estimator for the variance: $$\hat{\eta}_4 = \frac{\hat{\mu}_4}{\sigma^4} = \frac{\frac{1}{n} \sum_{i=1}^n (x_i - \bar{x})^4}{[\frac{1}{n} \sum_{i=1}^n (x_i - \bar{x})^2]^2} \;\;\;\;\; (5)$$ where $$\hat{\sigma}^2_m = s^2_m = \frac{1}{n} \sum_{i=1}^n (x_i - \bar{x})^2 \;\;\;\;\;\; (6)$$

This form of estimation should be used when resampling (bootstrap or jackknife).

When method="fisher", the coefficient of kurtosis is estimated using the unbiased estimator for the fourth central moment (Serfling, 1980, p.73) and the unbiased estimator for the variance. $$\hat{\sigma}^2 = s^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2 \;\;\;\;\;\; (7)$$

L-Moment Coefficient of Kurtosis (method="l.moments")
Hosking (1990) defines the $L$-moment analog of the coefficient of kurtosis as: $$\tau_4 = \frac{\lambda_4}{\lambda_2} \;\;\;\;\;\; (8)$$ that is, the fourth $L$-moment divided by the second $L$-moment. He shows that this quantity lies in the interval (-1, 1).

When l.moment.method="unbiased", the $L$-kurtosis is estimated by: $$t_4 = \frac{l_4}{l_2} \;\;\;\;\;\; (9)$$ that is, the unbiased estimator of the fourth $L$-moment divided by the unbiased estimator of the second $L$-moment.

When l.moment.method="plotting.position", the $L$-kurtosis is estimated by: $$\tilde{\tau}_4 = \frac{\tilde{\lambda}_4}{\tilde{\lambda}_2} \;\;\;\;\;\; (10)$$ that is, the plotting-position estimator of the fourth $L$-moment divided by the plotting-position estimator of the second $L$-moment.

See the help file for lMoment for more information on estimating $L$-moments.

Value

A numeric scalar – the sample coefficient of kurtosis or excess kurtosis.

References

Berthouex, P.M., and L.C. Brown. (2002). Statistics for Environmental Engineers, Second Edition. Lewis Publishers, Boca Raton, FL.

Ott, W.R. (1995). Environmental Statistics and Data Analysis. Lewis Publishers, Boca Raton, FL.

Taylor, J.K. (1990). Statistical Techniques for Data Analysis. Lewis Publishers, Boca Raton, FL.

Vogel, R.M., and N.M. Fennessey. (1993). $L$ Moment Diagrams Should Replace Product Moment Diagrams. Water Resources Research 29(6), 1745–1752.

Zar, J.H. (2010). Biostatistical Analysis. Fifth Edition. Prentice-Hall, Upper Saddle River, NJ.

Author

Steven P. Millard (EnvStats@ProbStatInfo.com)

Note

Traditionally, the coefficient of kurtosis has been estimated using product moment estimators. Sometimes an estimate of kurtosis is used in a goodness-of-fit test for normality (D'Agostino and Stephens, 1986). Hosking (1990) introduced the idea of $L$-moments and $L$-kurtosis.

Vogel and Fennessey (1993) argue that $L$-moment ratios should replace product moment ratios because of their superior performance (they are nearly unbiased and better for discriminating between distributions). They compare product moment diagrams with $L$-moment diagrams.

Hosking and Wallis (1995) recommend using unbiased estimators of $L$-moments (vs. plotting-position estimators) for almost all applications.

Examples

  # Generate 20 observations from a lognormal distribution with parameters 
  # mean=10 and cv=1, and estimate the coefficient of kurtosis and 
  # coefficient of excess kurtosis. 
  # (Note: the call to set.seed simply allows you to reproduce this example.)

  set.seed(250) 

  dat <- rlnormAlt(20, mean = 10, cv = 1) 

  # Compute standard kurtosis first 
  #--------------------------------
  kurtosis(dat, excess = FALSE) 
#> [1] 2.964612
  #[1] 2.964612
  
  kurtosis(dat, method = "moment", excess = FALSE) 
#> [1] 2.687146
  #[1] 2.687146
  
  kurtosis(dat, method = "l.moment", excess = FALSE) 
#> [1] 0.1444779
  #[1] 0.1444779


  # Now compute excess kurtosis 
  #----------------------------
  kurtosis(dat) 
#> [1] -0.0353876
  #[1] -0.0353876
 
  kurtosis(dat, method = "moment") 
#> [1] -0.3128536
  #[1] -0.3128536
  
  kurtosis(dat, method = "l.moment") 
#> [1] -2.855522
  #[1] -2.855522

  #----------
  # Clean up
  rm(dat)