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)x |
numeric vector of observations. |
na.rm |
logical scalar indicating whether to remove missing values from |
method |
character string specifying what method to use to compute the sample coefficient
of kurtosis. The possible values are
|
l.moment.method |
character string specifying what method to use to compute the
L-moments when |
plot.pos.cons |
numeric vector of length 2 specifying the constants used in the formula for
the plotting positions when |
excess |
logical scalar indicating whether to compute the kurtosis ( |
Let \underline{x} denote a random sample of n observations from some distribution with mean μ and standard deviation σ.
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:
η_4 = β_2 = \frac{μ_4}{σ^4} \;\;\;\;\;\; (1)
where
η_r = E[(\frac{X-μ}{σ})^r] = \frac{1}{σ^r} E[(X-μ)^r] = \frac{μ_r}{σ^r} \;\;\;\;\;\; (2)
and
μ_r = E[(X-μ)^r] \;\;\;\;\;\; (3)
denotes the r'th moment about the mean (central moment).
The coefficient of excess kurtosis is defined as:
β_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{η}_4 = \frac{\hat{μ}_4}{σ^4} = \frac{\frac{1}{n} ∑_{i=1}^n (x_i - \bar{x})^4}{[\frac{1}{n} ∑_{i=1}^n (x_i - \bar{x})^2]^2} \;\;\;\;\; (5)
where
\hat{σ}^2_m = s^2_m = \frac{1}{n} ∑_{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{σ}^2 = s^2 = \frac{1}{n-1} ∑_{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:
τ_4 = \frac{λ_4}{λ_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{τ}_4 = \frac{\tilde{λ}_4}{\tilde{λ}_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.
A numeric scalar – the sample coefficient of kurtosis or excess kurtosis.
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.
Steven P. Millard (EnvStats@ProbStatInfo.com)
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.
var, sd, cv,
skewness, summaryFull,
Summary Statistics.
# 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 kurtosis(dat, method = "moment", excess = FALSE) #[1] 2.687146 kurtosis(dat, method = "l.moment", excess = FALSE) #[1] 0.1444779 # Now compute excess kurtosis #---------------------------- kurtosis(dat) #[1] -0.0353876 kurtosis(dat, method = "moment") #[1] -0.3128536 kurtosis(dat, method = "l.moment") #[1] -2.855522 #---------- # Clean up rm(dat)
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