A Bivariate Generalized Lambda Distribution (GLD-2) using Plackett's Method of Construction: Distribution, Examples and Applications

SYNOPTIC ABSTRACT

The Generalized Lambda Distribution (GLD), an extension of Tukey's Lambda Distribution, has proved to be a useful family for fitting probability distributions. However, the specification of the distribution via its inverse function would seem to preclude direct bivariate or multivariate extensions (at least in a reasonably straightforward form), since any bivariate F(x,y) is a function of the marginals F₁(x) and F₂(y) called a copula, F(x,y) = C(F₁(x),F₂(y)). Johnson and Kotz (1973) proposed a non-closed-form multivariate extension of Tukey's original (ungeneralized) distribution, but given the lack of a closed form for the univariate GLD it seems likely that a multivariate (even bivariate) extension of the four-parameter GLD would be even more difficult to give a closed-form distribution function for. However, R.L. Plackett (1965) proposed a method for generating bivariate distributions from specified marginals, and his method is employed in this paper to furnish a bivariate GLD-2. Although Plackett's method rarely generates a closed-form expression, this is not a problem as graphs and properties of the resulting distribution are readily calculated on a computer. Several bivariate cases are presented, where the marginals have been fitted using the GLD, and the bivariate distribution, GLD-2, is calculated numerically. Also, to show the variety of possible contour shapes achievable, some mixtures of known univariate distributions are considered. Finally, we investigate the usefulness of the method in fitting real data sets.

Key Words and Phrases:

• Generalized Lambda Distribution
• GLD
• Extended Generalized Lambda Distribution
• EGLD
• GLD-2
• bivariate distributions
• bivariate data
• distribution fitting