_{BB}
were evaluated. To date, Johnson's S
_{BB}
has been fitted by either indirect (two-stage) method using well-known procedures for the marginal diameter and heights, or direct methods, where all parameters are estimated at once. Application of bivariate Johnson's S
_{BB}
for predicting height and improving volume estimation requires a suitable fitting method.
^{2}
), root mean square Error (RMSE), model efficiency (MEF), Bayesian Information Criterion (BIC) and Hannan-Quinn Criterion (HQC) were used to assess the model predictions. Significant difference between observed and predicted tree height and volumes were tested using paired sample t-test at 5% level for each plot by species.
^{2}
and RMSE for height prediction ranged from 0.994 - 0.820 and 1.454 - 1.676, respectively. The percentage of plot in which the observed and predicted heights were significant was 0.32%. The direct method was the least performed method especially for height prediction in
_{BB}
distribution.

Researchers realize that volume, the primary variable that forest managers are interested in, is heavily dependent on both tree diameter and height. A traditional practice is to fit a marginal distribution to the diameter frequency data and then use an empirical height-diameter relationship to estimate the average height per diameter class and hence the volume (

The traditional method does not quantify the distribution of heights for a given diameter and one approach for modelling the conditional height distribution for the different diameters is to use the height residuals (

One of the most important elements of forest structure is the relationship between tree diameters and heights because information about size-class distributions of the trees within a forest stand is important for estimating product yields (

For many years, the bivariate extension of the S
_{B}
distribution, the S
_{BB}
(
_{BB}
is developed by applying a four-parameter logistic transformation to each of the component variables of a standard bivariate normal distribution (

The parameters of the Johnson's S
_{BB}
distribution have been estimated with both the indirect and direct methods. The indirect method fits the bivariate Johnson's S
_{BB}
distribution by two-stage where the marginals are first fitted separately for the diameter and height; the estimates are then used to compute the parameter of association. Examples of the indirect method that have been reported in forestry literature include Conditional Maximum Likelihood (CML), moments, mode and Knoebel and Burkhart methods (
_{BB}
distribution have never been compared using the same sample. The method used to calibrate the distribution matters in order to evaluate the accuracy of diameter and height predictions (
_{B}
based on two-stage method by the Conditional Maximum Likelihood and moments methods with the direct method where the parameters are estimated simultaneously.

The data used for this study were obtained from three temperate species - Tasmanian blue gum (
^{2}
; to achieve a minimum of 30 trees per plot. Diameter at breast height (Dbh at 1.3 m above the ground) and total height were measured with calliper and hypsometer to a precision of the nearest 0.1 cm and 0.1 m, respectively. A total of 16382, 17845 and 12722 trees were measured from

The univariate Johnson's S
_{B}
distribution (

Where: ξ <

ξ shape parameters (asymmetry and kurtosis parameters, respectively). The Johnson's S
_{BB}
(
_{B}
distribution; given by:

Where:

Where: ρ is the correlation coefficient between Z
_{d}
and Z
_{h}
, the subscripts

Indirect method: this method fits the bivariate Johnson's S
_{BB}
distribution by a two-stage approach where the marginals are first fitted separately for the diameter and height. The estimates from the first stage are then used to compute the correlation parameter (as shown in Eq 3). Generally, fitting Johnson's S
_{BB}
distribution requires the location (

Where:
_{i}
(

The location (
_{min}
) and maximum diameter, respectively for the marginal distribution of diameter. These parameters were set at 1.3 and maximum height for the marginal height distribution. The factor 0.75 was adopted because
_{min}
for this parameter
_{BB}
by different authors, including

_{BB}
distribution.
_{BB}
distribution. It is given by:

Where:
_{x}
= plot diameter and height standard deviations and

Direct method: in this method, all parameters in the bivariate Johnson's S
_{BB}
distribution were estimated simultaneously using maximum likelihood estimation. However, the location (
_{BB}
function derived from the normal copula is given by:

and the likelihood function is expressed as:

where

The indirect method (CML and moments) estimations were obtained using SAS/STAT
^{TM}
software (SAS Institute 2003). The computation of the direct method was carried out using the 'optim' (optimal) function in R (

The suitability of the direct and indirect methods for fitting S
_{BB}
distribution was evaluated by tree height and volume predictions. Prediction of expected tree height given diameter and the estimation of stand volume are the main applications of bivariate distribution. Individual tree heights were predicted with the S
_{BB}
median regression expressed as:

are the estimated parameters from the direct and indirect methods.
_{d}
= δ
_{h}
and ρℽ
_{d}
= ℽ
_{h}
.

Individual tree volume equation developed by
_{BB}
height-diameter median regression using direct and indirect methods:

^{-5}
^{1.973}
^{1.059}

^{-5}
^{1.876}
^{1.079}

^{-5}
^{1.883}
^{1.004}

Where:
^{3}
);

Different fit indices including coefficient of determination (R
^{2}
), root mean square error (RMSE), model efficiency (MEF) (

Where: RSS = residual sum of square, n = sample size, p = number of parameters;
_{i}
= average tree height or volume; Y
_{i}
is the observed value and
_{i}
is the theoretical value predicted by the model.

The descriptive statistics i.e., the mean, maximum, minimum and standard deviation of the estimated parameters of the bivariate Johnson's S
_{BB}
distribution are presented in

The assessment of the suitability of indirect (CML and moments) and direct methods for tree height prediction showed that the indirect method, especially the CML method, had the highest R
^{2}
and the lowest RMSE, MEF, BIC and HQC in
^{2}
, RMSE, MEF, BIC and HQC of CML ranged from 0.994 - 0.820, 1.454 - 1.676, 0.055 - 0.179, 4559 - 13400 and 4492 - 13335, respectively. However, the direct method had relatively lower R
^{2}
and larger RMSE, MEF, BIC and HQC in

In the case of tree volume prediction, indirect (CML and moments) and direct methods performed relatively the same (^{2}
, RMSE and MEF values up to 2 decimal places. The direct method had lowest BIC and HQC in

The mean of the residuals from the height-diameter median regression was computed across DBH classes and plotted accordingly for the three species. The data were group into DBH classes of 5 cm interval and the mean residual prediction for each class by species was assessed. The graphs showed that CML and moments over- and under-estimated the tree height in the larger diameter classes (> 42.5 cm) in

Furthermore, the graph of residual against predicted tree volume showed that CML and moments and the direct methods occupied the same horizontal band (

The direct and indirect methods of fitting Johnson's S
_{BB}
have been evaluated. The methods follow similar trend across

Furthermore, the values of the BIC and HQC difference show that the method of CML fits the diameter-height data better than direct methods for height prediction in

It is obvious from this study that when a complex fitting approach does not outperform another simpler one it is convenient to take the latter. Estimating the parameters of Johnson's S
_{BB}
by CML or moments is much easier and pose little computational difficulty. The CML and moments estimation procedure can even be carried out on Microsoft excel platform. However, the direct method involves complicated algorithm using maximum likelihood technique. This often requires writing the log-likelihood function and the specification of initial values for the parameters which may not achieve convergence (
_{BB}
has nine parameters which makes fitting somewhat complicated. Often constraints are imposed on the two location and scale parameters of the bivariate distribution to make the estimates more plausible (
_{BB}
distribution and compared the result with power-normal and hyperbolic height prediction model. The prediction from power-normal and hyperbolic models were better than bivariate Johnson S
_{BB}
fitted with maximum likelihood. Other studies that have used maximum likelihood technique to fit the bivariate Johnson's S
_{BB}
distributions include
_{BB}
relative to other bivariate functions evaluated in their study.

The bivariate Johnson's S
_{BB}
height-diameter median regression remains the most applied function amidst several bivariate distributions in forestry literature. This is because of its flexibility and, more importantly, because its implied relationship between height and diameter is biologically reasonable (
_{BB}
height-diameter model has also enhanced its continuous application to forestry. This median regression is frequently used for computing percentile lines wherein bounds on height are set. These percentile lines can be used to show how the variation in height decreases with increasing tree size for a specified diameter (
_{BB}
median regression using indirect method fitted by CML from some sample plots in

Modelling the joint distribution of diameter and height by Johnson S
_{BB}
remains an important tool for assessing the variation of tree height for a given diameter, detailed stand structure and volume estimation. Its accuracy is affected by the method of estimating the parameters of the distribution. In this study, we found the indirect (two-stage) method, especially by conditional maximum likelihood, to be the most suitable method for the bivariate Johnson's S
_{BB}
distribution. This method predicted tree height and volume in