How can I find the L-skewness and L-kurtosis values for different distributions?
For example, I'd like to be able to plot the curves or points for the Generalized Extreme Value, Gamma, Generalized Pareto, Pearson type III, Lognormal, and log-logistic distributions on the plot.
Hosking (1990)$^{(1)}$ describes L-moments in detail and draws connection to a variety of related measures. That paper lists explicit expressions for L-moments and L-moment-ratios $\lambda_1$, $\lambda_2$, $\tau_3$ and $\tau_4$ for the following distributions: Uniform, Exponential, Gumbel, Logistic, Normal, Generalized Pareto, Generalized extreme value, Generalized logistic, Log-normal, Gamma.
(... apart from expressions for $\tau_4$ for the last two, which are in Hosking (1986)$^{(2)}$; however, note equation 2.4 in Hosking 1990, which allows computation of these quantities as the shape-parameter changes, so in practice this could be done numerically; e.g. as implemented in Hosking (1991)$^{(3)}$)
If we just look at the L-skewness and L-kurtosis we can form a table akin to that in Hosking (1990) but we can add a few details:
$$\begin{array}{lcc} \text{Distribution} & \tau_3 & \tau_4 \\\hline \text{Uniform} & 0&0 \\ \text{Exponential} &\frac13 &\frac16 \\ \text{Gumbel} & 0.1699&0.1504 \\ \text{Logistic} & 0& \frac16\\ \text{Normal} &0 & 0.1226\\ \text{Generalized Pareto} &\frac{1-k}{3+k} & \tau_3\frac{2-k}{4+k} \\ \text{Generalized extreme value} & 2\frac{1-3^{-k}}{1-2^{-k}}-3&\frac{1-6(2^{-k})+10(3^{-k})-5(4^{-k})}{1-2^{-k}} \\ \text{Generalized logistic} & -k&\frac{1}{6}(1+5k^2) \\ \text{Log normal} & \frac{6}{\sqrt{\pi}\operatorname{erf}(\frac{\sigma}{2})}\int_0^{\sigma/2}\operatorname{erf}(x/\sqrt{3})\exp(-x^2)dx& - \\ \text{Gamma} & 6I_{\frac13}(\alpha,2\alpha)-3& - \\ \text{Bounds} & -1<\tau_3< 1& (5\tau_3^2-1)/4\leq\tau_4 <1 \end{array}$$
$I$ in the second-last row is the incomplete beta function (that $\tau_3$ for the lognormal may actually be computable from the integral but should be fairly easy to do via numerical integration in any case).
Note that Table 2 of Hosking (1992)$^{(4)}$ gives specific points for a number of distributions across a number of families, so if you implement the curves for the lognormal or gamma there are some points (1 and 3 respectively) for which you can check you have the curve right.
Note that a Pearson type III is simply a scaled- and shifted gamma (where the rescaling may include multiplying by a negative number). It will therefore share the curve with the Gamma (except that if you're showing the two-sided version of the plot with both signs of skewness distinguished it also appears symmetrically with $(-\tau_3,\tau_4)$ from the gamma).
Similarly the row for a Weibull may be obtained from the generalized extreme value simply by flipping the sign of $\tau_3$.
These papers give all of the details you should need about the L-moment-ratios to produce a plot. As far as constructing a plot, for the families that constitute curves you can plot them parametrically (i.e. as implicit functions); note that both $\tau_3$ and $\tau_4$ are given as functions of a single variable.
However, note that there's an issue with plotting data on the same plot. See Hosking & Wallis (1995)$^{(5)}$
There are some papers around that give these quantities for some other distributions, but it's also fairly easy to get either exact (numerically) or approximate curves (using a quick approximation to the expected quantiles) for distributions for which you have a suitable cdf or quantile function.
Note that the R package lmom (also by Hosking) may do a lot of what you're trying to achieve; it interfaces to the code from Hosking (1991). See for example the function lmrd (and the functions immediately following it, from page 24-on) in the help here
(1): Hosking, J.R.M. (1990). L-Moments: Analysis and Estimation of Distributions Using Linear Combinations of Order Statistics. Journal of the Royal Statistical Society. Series B (Methodological), 52, 105-124.
(2): Hosking, J. R. M. (1986). The theory of probability weighted moments. Research Report RC12210, IBM Research Division, Yorktown Heights, N.Y.
(3): Hosking, J. R. M. (1991). Fortran Routines for Use With the Method of L-Moments, Version 2, Research Report RC17097, IBM Research Division, Yorktown Heights, NY.
(4): Hosking, J. R. M. (1992) Moments or L Moments? An Example Comparing Two Measures of Distributional Shape, The American Statistician, 46:3, 186-189
(5): Hosking, J. R. M., and J. R. Wallis (1995), A Comparison of Unbiased and Plotting-Position Estimators of L Moments, Water Resour. Res., 31(8), 2019–2025
