gld

R package for the Generalised Lambda Distribution. This is available on CRAN as https://cran.r-project.org/package=gld

The generalised $\lambda$ distribution, also known as the Asymmetric Lambda or Asymmetric Tukey Lambda Distribution provides a wide variety of shapes with one functional form.

This package provides a number of different types of the distribution, each defined by their Quantile function:

  1. FMKL type (default) -

$$Q(u)= \lambda_1 + { { \frac{u^{\lambda_3}-1}{\lambda_3} - \frac{(1-u)^{\lambda_4}-1}{\lambda_4} } \over \lambda_2 } \qquad\text{ for } \quad\lambda_2 > 0$$

Freimer Mudholkar, Kollia and Lin (1988) A study of the generalized tukey lambda family, Communications in Statistics - Theory and Methods 17 3547--3567.

  1. RS type -

$$ Q(u)= \lambda_1 + \frac{u^{\lambda_3} - (1-u)^{\lambda_4}}{\lambda_2} $$

Ramberg, J. S. & Schmeiser, B. W. (1974) An approximate method for generating asymmetric random variables, Communications of the ACM 17 78--82

This type has a complex complex series of rules determining which values of the parameters produce valid statistical distributions. See the gl.check.lambda function in this package for details.

  1. GPD type -

$$ Q(u) = \alpha + \beta \left((1-\delta)\frac{(u^\lambda -1)}{\lambda} - \delta\frac{((1-u)^\lambda -1)}{\lambda} \right) \quad\text{ where }\beta > 0 \text{ and }-1 \leq \delta \leq 1$$

Van Staden, Paul J., & M.T. Loots. (2009), Method of L-moment Estimation for the Generalized Lambda Distribution. In Proceedings of the Third Annual ASEARC Conference

  1. FM5 type -

$$ Q(u) = \lambda_1 + \frac{ \frac{(1-\lambda_5)(u^{\lambda_3}-1)}{\lambda_3} - \frac{(1+\lambda_5)((1-u)^{\lambda_4}-1)}{\lambda_4} }{ \lambda_2 }$$

The package provides the density, probability, quantile, and density quantile functions, Quantile-Quantile plots and random number generation for all 4 types.

Since four types of the distribution are defined by their quantile function and its distribution and density functions do not exist in closed form (except for some special cases). Accordingly, the results from pgl and dgl are the result of numerical solutions to the quantile function, using the Newton-Raphson method. Since the density quantile function, f(Q(u)), does exist, an additional function, dqgl, computes this.

The package provides a variety of estimation methods for the distribution, with the table below showing which estimation methods are available for which type.

Estimation Method FMKL RS GPD FM5
Starship ✓ ✓ ✓ ✓
L Moments ✓ ✓
Maximum likelihood ✓
Maximum product of spacings ✓
Titterington's ✓
Moments ✓
Trimmed L-Moments ✓
Distributional Least Absolutes ✓