See http://en.wikipedia.org/wiki/Pearson_distribution for an introduction to the Pearson distribution, and http://en.wikipedia.org/wiki/Gamma_distribution for an introduction to the Gamma distribution (the Pearson type III distribution is, essentially, a Gamma distribution with 3 parameters).Definition
Parameters (3): $\mu$ (location), $\sigma$ (scale), $\gamma$ (shape).
If $\gamma \ne 0$, let $\alpha=4/\gamma^2$, $\beta=\frac{1}{2}\sigma |\gamma|$, and $\xi= \mu - 2 \sigma/\gamma$.
If $\gamma > 0$, then the range of $x$ is $\xi \le x < \infty$ and
$$f(x) = \frac{(x - \xi)^{\alpha - 1} e^{-(x-\xi)/\beta}}{\beta^{\alpha} \Gamma(\alpha)}$$
$$F(x) = G \left(\alpha, \frac{x-\xi}{\beta}\right)/ \Gamma(\alpha)$$
If $\gamma=0$, then the distribution is Normal, the range of $x$ is $-\infty < x < \infty$ and
$$f(x) = \phi \left(\frac{x-\mu}{\sigma}\right)$$
$$F(x) = \Phi \left(\frac{x-\mu}{\sigma}\right)$$
where
$\phi(x)=(2\pi)^{-1/2}\exp(-x^2/2)$ and
$\Phi(x)=\int_{-\infty}^x \phi(t)dt$.
If $\gamma < 0$, then the range of $x$ is $-\infty < x \le \xi$ and
$$f(x) = \frac{(\xi - x)^{\alpha - 1} e^{-(\xi-x)/\beta}}{\beta^{\alpha} \Gamma(\alpha)}$$
$$F(x) = G \left(\alpha, \frac{\xi-x}{\beta}\right)/ \Gamma(\alpha)$$
In each case, $x(F)$ has no explicit analytical form.
Here $\Gamma$ is the gamma function, defined as $$\Gamma (x) = \int_0^{\infty} t^{x-1} e^{-t} dt$$
and
$$G(\alpha, x) = \int_0^x t^{\alpha-1} e^{-t} dt$$
is the incomplete gamma function.
$\gamma=2$ is the exponential distribution; $\gamma=0$ is the Normal distribution; $\gamma=-2$ is the reverse exponential distribution.
The parameters $\mu$, $\sigma$ and $\gamma$ are the conventional moments of the distribution.
L-moments
Assuming $\gamma>0$, L-moments are defined for $0<\alpha<\infty$.< p="">
$$\lambda_1 = \xi + \alpha \beta$$
$$\lambda_2 = \pi^{-1/2} \beta \Gamma(\alpha + 1/2)/\Gamma(\alpha)$$
$$\tau_3 = 6 I_{1/3} (\alpha, 2 \alpha)-3$$
where $I_x(p,q)$ is the incomplete beta function ratio
$$I_x(p,q) = \frac{\Gamma(p+q)}{\Gamma(p)\Gamma(q)} \int_0^x t^{p-1} (1-t)^{q-1} dt$$
There is no simple expression for $\tau_4$.
Here we use the rational-funcion approximation given by Hosking and Wallis (1997, pp. 201-202).
The corresponding results for $\gamma <0$ are="" obtained="" by="" changing="" the="" signs="" of="" $\lambda_1$,="" $\tau_3$="" and="" $\xi$="" wherever="" they="" occur="" above.<="" p="">
Parameters
$alpha$ is obtained with an approximation.
If $0<|\tau_3|<1 3$,="" let="" $z="3" \pi="" \tau_3^2$="" and="" use="" $$\alpha="" \approx="" \frac{1+0.2906="" z}{z="" +="" 0.1882="" z^2="" 0.0442="" z^3}$$="" if="" $1="" 3<|\tau_3|<1$,="" \frac{0.36067="" z="" -="" 0.59567="" 0.25361="" z^3}{1-2.78861="" 2.56096="" -0.77045="" z^3}$$<="" p="">
Given $\alpha$, then
$\gamma=2 \alpha^{-1/2} sign(\tau_3)$,
$\sigma=\lambda_2 \pi^{1/2} \alpha^{1/2} \Gamma(\alpha)/\Gamma(\alpha+1/2)$,
$\mu=\lambda_1$.
Lmom.gamma and par.gamma accept input as vectors of equal length.
In f.gamma, F.gamma, invF.gamma and rand.gamma parameters (mu, sigma, gamm) must be atomic.