_{BB}, Weibull-2^{P} and LL-2^{P} functions for characterizing the joint distribution of tree diameters and heights.

_{BB} bivariate distribution was obtained from S_{B} 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.

Stand volume, one of the most important variables in forest management, is usually estimated based on sampled tree diameters and heights (

An alternative approach for improving stand volume estimation, which takes into account those variations, involves the use of a bivariate distribution (

Hence there has been considerable interest in identifying suitable bivariate distributions to describe diameter-height frequency data. For many years, the bivariate extension of the _{B}_{BB}_{BB} is developed by applying a four-parameter logistic transformation to each of the component variables of a standard bivariate normal distribution (

The objective of the present study is to fit and compare the accuracy of three bivariate distributions: Johnson’s S_{BB}, Weibull and Logit-Logistic fitted to diameter-height data from pure and even-aged stands of ^{P} and LL-2^{P}, were obtained from Weibull and LL marginal distributions by using the Plackett copula whereas the S_{BB} bivariate distribution was obtained from S_{B} marginal distributions using the Normal copula, i.e., a four-parameter logistic transformation to each of the component variables.

All tree diameters and heights were measured in 128 field plots in Tasmanian blue gum^{2}, depending on stand density. The objective was to assess a minimum of 30 trees per plot.

All trees in each plot were numbered; diameters at breast height were measured with a caliper, to the nearest 0.1 cm, and heights were measured with hypsometer to the nearest 0.1 m. The stand variables calculated in each inventory included the quadratic mean diameter, the number of trees per hectare, dominant height, basal area and mean height. A total of 17,588 trees were measured. The summary statistics of the main stand variables are presented in

where

Frequently used copulas are the Normal (

Plackett copula

where z_{x} and z_{y} are specific transformations of

The S_{BB} distribution was obtained from S_{B} marginal distributions using the normal copula. In this case, the variables _{1})/_{1} and _{2})/_{2} where _{1} and _{2} are the location parameters and _{1} and _{2} are the observed ranges of diameter (_{x} and z_{y} were obtained from a four-parameter logistic transformation of _{x}_{1} + _{1} log(_{y}_{2} + _{2} log(

The parameters _{min}-0.5 and h_{min}-0.5 for diameter and height, respectively, whereas the parameter

The Weibull-2^{P} and the LL-2^{P} bivariate distributions

Logit-logistic density function

where

The parameters were estimated by minimizing the negative log-likelihood function of ^{P} and LL-2^{P} and (4) for S_{BB} using the R function optim (^{P} and LL-2^{P}) have a closed form of their cumulative distributions. If this is not the case, numerical methods should be used for evaluating the cumulative distribution in the model-fitting process.

Each bivariate model considered in this study has the same number of parameters, namely five: two specific parameters for each marginal distribution and one common parameter. Thus, the parameter values were used as goodness-of-fit criteria for comparison. The Wilcoxon signed-rank test was used to compare the related samples of the negative log-likelihood function calculated for each sample plot and each of the three distributions. This is a non-parametric paired difference test to assess whether the population mean ranks differ when the population cannot be assumed to be normally distributed.

The means, maxima, minima and standard deviations of the estimated parameters for the three bivariate distributions (bivariate Johnson’s S_{BB}, Weibull-2^{P} and LL-2^{P}) are presented in ^{P} bivariate and bivariate beta distributions, probably due to these plots having J-shaped marginal distributions (

The S_{BB} distribution showed better results in terms of goodness-of-fit statistics than the Weibull distribution (^{P} over S_{BB} and Weibull was expected since the logit-logistic univariate distribution is more flexible than the other two, covering a wide range of skewness-kurtosis combinations. However, the good results of the Weibull distribution were unexpected, because the Weibull univariate turned out to be the least flexible of the three univariate distributions used. The reason for this may be the very regular shape of the marginal diameter and height distributions of our even-aged stands. Moreover, it also should be taken into account that the locations (ε) and the ranges (λ) of diameters (^{P} and S_{BB} bivariate distributions.

Both the normal and the Plackett copulas 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. However, it should be noted that the normal copula, in general, does not have a closed form for its joint density, except for the Normal or Johnson’s marginal distributions. Another point to consider is the fact that the Plackett copula requires that the marginal has a closed form for its cumulative distribution (

_{BB}

_{BB}distribution