Comparing Johnson’s SBB, Weibull and Logit-Logistic bivariate distributions for modeling tree diameters and heights using copulas

Jose Javier Gorgoso-Varela, Juan Daniel García-Villabrille, Alberto Rojo-Alboreca, Klaus von Gadow, Juan Gabriel Álvarez-González


Aim of study: In this study we compare the accuracy of three bivariate distributions: Johnson’s SBB, Weibull-2P and LL-2P functions for characterizing the joint distribution of tree diameters and heights.

Area of study: North-West of Spain.

Material and methods: Diameter and height measurements of 128 plots of pure and even-aged Tasmanian blue gum (Eucalyptus globulus Labill.) stands located in the North-west of Spain were considered in the present study. The SBB bivariate distribution was obtained from SB marginal distributions using a Normal Copula based on a four-parameter logistic transformation. The Plackett Copula was used to obtain the bivariate models from the Weibull and Logit-logistic univariate marginal distributions. The negative logarithm of the maximum likelihood function was used to compare the results and the Wilcoxon signed-rank test was used to compare the related samples of these logarithms calculated for each sample plot and each distribution.

Main results: The best results were obtained by using the Plackett copula and the best marginal distribution was the Logit-logistic.

Research highlights: The copulas used in this study have shown a good performance for modeling the joint distribution of tree diameters and heights. They could be easily extended for modelling multivariate distributions involving other tree variables, such as tree volume or biomass.


Plackett copula; normal copula; Eucalyptus globulus.

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DOI: 10.5424/fs/2016251-08487