Modelling joint distribution of tree diameter and height using Frank and Plackett copulas

  • Friday Nwabueze OganaEmail author
  • Jose Javier Gorgoso-Varela
  • Johnson Sunday Ajose Osho
Original Paper


Bivariate distribution models are veritable tools for improving forest stand volume estimations. Their accuracy depends on the method of construction. To-date, most bivariate distributions in forestry have been constructed either with normal or Plackett copulas. In this study, the accuracy of the Frank copula for constructing bivariate distributions was assessed. The effectiveness of Frank and Plackett copulas were evaluated on seven distribution models using data from temperate and tropical forests. The bivariate distributions include: Burr III, Burr XII, Logit-Logistic, Log-Logistic, generalized Weibull, Weibull and Kumaraswamy. Maximum likelihood was used to fit the models to the joint distribution of diameter and height data of Pinus pinaster (184 plots), Pinus radiata (96 plots), Eucalyptus camaldulensis (85 plots) and Gmelina arborea (60 plots). Models were evaluated based on negative log-likelihood (−ΛΛ). The result show that Frank-based models were more suitable in describing the joint distribution of diameter and height than most of their Plackett-based counterparts. The bivariate Burr III distributions had the overall best performance. The Frank copula is therefore recommended for the construction of more useful bivariate distributions in forestry.


Bivariate distributions Frank copula Plackett copula Diameter Height 



Cumulative distribution function


Probability density function

Burr III-2F

Bivariate Burr III distribution from Frank copula

Burr III-2P

Bivariate Burr III distribution from Plackett copula

Burr XII-2F

Bivariate Burr XII distribution from Frank copula

Burr XII-2P

Bivariate Burr XII distribution from Plackett copula


Bivariate Logit-Logistic distribution from Frank copula


Bivariate Logit-Logistic distribution from Plackett copula


Bivariate Log-Logistic distribution from Frank copula


Bivariate Log-Logistic distribution from Plackett copula


Bivariate generalized Weibull distribution from Frank copula


Bivariate generalized Weibull distribution from Plackett copula


Bivariate Weibull distribution from Frank copula


Bivariate Weibull distribution from Plackett copula


Bivariate Kumaraswamy distribution from Frank copula


Bivariate Kumaraswamy distribution from Plackett copula


Negative loglikelihood value

Supplementary material

11676_2018_869_MOESM1_ESM.docx (81 kb)
Supplementary material 1 (DOCX 81 kb)


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Copyright information

© Northeast Forestry University 2018

Authors and Affiliations

  • Friday Nwabueze Ogana
    • 1
    Email author
  • Jose Javier Gorgoso-Varela
    • 2
  • Johnson Sunday Ajose Osho
    • 1
  1. 1.Department of Social and Environmental ForestryUniversity of IbadanIbadanNigeria
  2. 2.Department of InnovationFöra Forest TechnologiesLugoSpain

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