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

Abstract

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.