ABSTRACT
We saw in Chapters 1–4 that the GLD and EGLD are very successful in practice in fitting a variety of datasets. Since data is often bivariate (the simplest multivariate setting), it is desirable to have a way of fitting a bivariate GLD, a distribution whose univariate components (not necessarily independent) have univariate GLD (or EGLD) distributions. A bivariate GLD, called GLD–2, was developed by Beckwith and Dudewicz (1996), and its essential features are described in this chapter; for full details, see Beckwith and Dudewicz (1996) and Karian and Dudewicz (1999a). The algorithm developed by Beckwith and Dudewicz (1996) is based on the following procedure.
Fit univariate GLDs separately to the components https://www.w3.org/1998/Math/MathML"> X https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429114502/6d8c8b81-08fc-4b5a-8166-f9463cfdc02d/content/eq4566.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and https://www.w3.org/1998/Math/MathML"> Y https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429114502/6d8c8b81-08fc-4b5a-8166-f9463cfdc02d/content/eq4567.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> of the bivariate random variable https://www.w3.org/1998/Math/MathML"> ( X , Y ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429114502/6d8c8b81-08fc-4b5a-8166-f9463cfdc02d/content/eq4568.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> .
Use the method R. L. Plackett (1965) proposed for generating bivariate distributions with specified marginals to develop a bivariate distribution with these marginals.
Optimize over the infinite set of distributions in the Plackett class to obtain a fit to the actual dataset.
Calculate a bivariate plot and various quantitative measures to visually and quantitatively assess the quality of the fit.
