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.