INTRODUCTION
Nowadays, in commercial boreal forest, forest management needs to utilize fundamental principles of forest dynamics. Forest growth, like many other natural processes, is subjected to various disturbances. Thus, the predictions computed from deterministic models limit our understanding of other possible outcomes of stand diameter and height. One way of quantifying stand growth under random perturbations is with diffusion process models.
The processes of growth play an important role in various applied areas, such
as biology, medicine, biochemical industry. The environment of any real system
is in general not constant but shows random fluctuations. Despite this, the
growth model historically has crystallized as the deterministic logistic type
process (Kar and Matsuda, 2007; Sakanoue,
2007). The forest growth is usually modeled with a logistic model (Garcia,
2005). The parameters of logistic diameter models are not directly measurable
but they are estimated from the observed data set.
Stand as a community of trees is the main component of the forest. Stand consists
of trees with different diameters and heights. Those differences depend on a
lot of unsearchable genetic and environmental factors, therefore it leads to
consideration that diameter of a tree is a random variable which depends on
the age and height. Diameter dynamic is affected by many processes and varies
among stands (Temesgen and Gadow, 2004). Stochastic
diameter growth models allow us to reduce the unexplained variability of a diameter
and to implement the randomness phenomenon, which makes a stochastic influence
on diameter growth process useful in practical applications. Over the years
an extensive amount of research has been devoted to the randomness of stand
growth since the pioneer work of Suzuki (1971) and the
successive works of Tanaka (1986) and Rupsys
(2007). There are two types of approaches for this purpose. The first approach
is based on ‘environment’ stochasticity, introducing a diffusion term
in the ordinary differential equation of diameter dynamic (Suzuki,
1971; Tanaka, 1986; Rupsys, 2007;
Rupsys et al., 2007; Rupsys
and Petrauskas, 2009, 2010). The second approach
is based on demographic stochasticity in which the tree size X is a random variable
(Boungiorno, 2001; Lohmander and
Mohammadi, 2007). In this study we follow the first approach.
The main purpose of this study is to develop the ageheightdependent probability density function on diameter size using measurements of tree variables such as age, height and diameter. This study not only provides useful stochastic models for the diameter growth modeling, but shows that it is possible to relate the diameter growth model and the diameter distribution model. The distributions of tree diameter size in stands describe forest structure and can be used for the assessment of stand volume and biomass, forest biodiversity and density management. Knowledge of the predicted ageheightdependent distribution function enables a more differentiated prediction of the assortment for a stand. This is not possible with commonly used distribution functions or yieldtables.
In evenaged stands various distribution functions, such as negative exponential,
Pearson, gamma, lognormal, beta, Weibull, Johnson, GramCharlier, have been
used in describing the diameter distributions (Mehtatalo,
2005). In unevenaged stands have been used bivariate distributions and
density mixtures (Wang and Rennolls, 2007; Wang
et al., 2008). In this work we motivate the use of stochastic differential
equations in forestry. The methodology is to consider a univariate distribution
as arising from univariate diameter growth stochastic dynamical system. The
system fluctuations, generally infiltrated from outside, are defined by a onedimensional
standard Wiener process.
In this study, a univariate agedependent stochastic differential equation
methodology of tree diameter distribution is expanded into ageheightdependent
distribution function methodology. The Gompertz homogeneous and nonhomogeneous
growth models are applied to analyze the trend of tree diameter, taking the
height as an exogeneous variable that affects the diameter of a tree. The choice
of the height among other possible exogenous factors is justified by the significant
correlation with the height and age (Skovsgaard and Vanclay,
2008; Garcia, 2009). This approach is rather different
from the univariate diffusion models (Suzuki, 1971;
Tanaka, 1986; Rupsys and Petrauskas,
2009, 2010), since these distributions are not related
with the dynamic of height.
MATERIALS AND METHODS
Growth model: Let study the dynamic behavior of tree diameter (diameter
at breast height) and its relationship with diameter distribution law. For determination
of diameter growth we suppose that dynamic of tree diameter is expressed in
terms of the Gompertz shape stochastic differential equation with multiplicative
noise. The Gompertz deterministic model is a classical continuous model useful
in describing population dynamic. It was introduced by Gompertz
(1825) to analyze population dynamic and to determinate life contingencies.
We consider a univariate Gompertz diameter growth process facing stochastic
fluctuations in the following Ito (1942) stochastic
differential equation:
where α_{0}, α_{1}, β, σ>0 are unknown
real parameters to be estimated, D(t) is a breast height diameter (in the sequeldiameter)
at the age t, d_{0}≥0, g(t) is an exogenous factor which is expressed
by a time continuous known function, {W(t); tε[t_{0}; T]} is a
onedimensional Wiener process and the differential dD(t) is to be understood
in the sense of Ito (1942). In the sequel, the density
p(d, t) of D(t) at t D(t_{0}) = d_{0} at t = t_{0} is
denominated as transition probability density function or conditional probability
density function.
The heightage models can be used to detect trends in the exogenous factor
g(t) because they indirectly show whether the growing conditions are changing
over time (Skovsgaard and Vanclay, 2008; Garcia,
2009). In the sequel, we relate the exogenous factor g(t) as the heightage
trajectory h(t) of a tree. Chronologies of height increments are a good tool
to quantify the exogenous factor, thus is g(t) ≡ h(t). It is wellknown
that tree growth is sigmoidal (Garcia, 2005) and several
sigmoidal growth models with biologically interpretable parameters have been
proposed, such as Verhulst, Gomperz, Mitcherlich and Bertalanffy models. Recently,
the majority of the newly developed ageheight models are derived by the procedure
named the algebraic difference approach (Cieszewski and
Bella, 1989). In this study, we focus on two types of heightage models.
The first heightage model is defined by the Gomperz shape growth model:
The formula describing the Gomperz heightage trajectory is defined in the following form:
where, γ, K are unknown real parameters to be estimated from the observations of a realization of a tree height for tε[t_{0}; T], γ is the intrinsic growth rate of height, K is called the carrying capacity of the environment and commonly represents the maximum height that can be supported by the resources of the environment and h_{1}(t_{0}) = h_{0}≥0. The second heightage model is defined by the Mitcherlich shape growth model:
The formula describing the Mitcherlich heightage trajectory is defined in the following form:
where, γ_{1}, K_{1} are unknown real parameters to be estimated from the observations of a realization of a tree height for tε[t_{0}, T], γ_{1} is the intrinsic growth rate of height, K_{1}/γ_{1} is called the carrying capacity of tree height and h_{2} (t_{0}), = h_{0}≥0.
According to Eq. 1 the parameter β affects the term D(t) In (D(t)) and acts to slow down the drift term (α_{0}+α_{1}g(t))D(t). When β = 0, the diameter stochastic growth process D(t), tε[t_{0}; T] described by Eq. 1 contains the univariate nonhomogeneous lognormal diffusion process with the exogenous factor g(t).
Using Itô’s formula for the age dependent transformation X(t) = e^{βt} we obtain the explicit solution of original stochastic differential Eq. 1 in the following form:
which is a continuous nonhomogeneous Markov (Gaussian) process with transition
probability density function (Gutierrez et al., 2008):
where,
Noticing that the parameter α_{1} of the homogeneous case is equal to 0, the functions μ(t) and λ^{2}(t) of the homogeneous case take the following forms:
Therefore, the random variable D(t)/D(t_{0}) = d_{0} has onedimensional
lognormal distribution Λ(μ(t), λ^{2} (t)).
The mean and variance of the stochastic process D(t) defined by Eq. 1 take the following forms:
Next we address the approach of estimating the unknown parameters of stochastic
differential Eq. 1 from the following discrete sampling D(t_{i})
= d_{i}, h(t_{i}) = h_{i}, t_{i}), i = 0,1,...,
n, assuming that t_{i}t_{il} = 1. A natural estimation procedure
is maximum likelihood because it is possible to write the likelihood function
explicitly. Explicit knowledge of the transition probability density function
of diameter dynamic allows us to construct the likelihood function L(α_{0},
α_{1}, β, σ). The transition probability density function
denotes
the probability density that tree diameter, D(t), at time t is equal to y given
tree diameter, D(s), at time s is equal to x. The conditional likelihood function
related with the discrete sample
takes the following form:
Derivation of the maximum likelihood function from Eq. 10 and the maximum likelihood estimators are given in the Appendix.
Goodnessoffit tests allow us to verify the correspondence between the estimated
theoretical model and real data set. The quantitative analysis of tree diameter
distribution is usually based on the tests, such as, the Chisquared, the KolmogorovSmirnov,
the AndersonDarling, the Cramervon Mises (Thode, 2002).
Most of these tests are very sensitive to the presence of outliers in the observed
data. In forestry various measures for the deviation of an actual (empirical)
distribution from its estimated theoretical distribution are commonly used,
such as, the Reynolds error index, the absolute discrepancy, the stand stability
index, the bias and standard error of estimate and many more (Reynolds
et al., 1988; Cao, 2004). These measures
of the goodnessoffit can be used for comparisons between observed data sets
and distribution models.
Statistical testing is often based on distributional assumption of normality.
A useful technique for evaluating the normality of small and moderate size samples
is the ShapiroWilk test statistic W (Shapiro and Wilk,
1965). In this study, we test the normality of the pseudoresiduals defined
by Zucchini and MacDonald (1999). The pseudoresiduals,
r_{i}, corresponding to the observation (d_{i}, h_{i},
t_{i}) are defined in the following form:
where, Φ denotes the distribution function of the standard normal distribution,
(d_{i}, h_{i}, t_{i}) is the ith observation of diameter,
height and age. Let denote
an n dimensional vector of ordered pseudoresiduals. Thus, given an assumption
that the transition probability density function p(d, t) of tree diameter is
indeed correct function for the observed data set the
pseudo residuals follow
the standard normal distribution. So, if r is drown from a standard normal distribution
then it is possible to write where
q_{i} is expected values of standard normal order statistics, defined
by
and Φ denotes the distribution function of a standard normal distribution. In case of residuals owning a standard normal distribution the value of statistics W tends to be close 1 and on the contrary tends to be small if residuals are from nonnormal distribution. A normal probability plot of pseudoresiduals (11) is constructed by plotting r_{(i)} against q_{i}. The normal probability plot of pseudoresiduals enables us to evaluate visually the fit of the estimated theoretical diameter distribution to the observations.
In order to rank the performance of each transition probability density function
we utilize Reynolds’ error index measure. The error index is calculated
in 5 cm diameter classes for stem numbers. Thus, a relative error index (%)
is defined by a sum of the absolute differences between the actual and predicted
stem numbers of the diameter classes divided by the total stem number N:
where, and
n_{i} are the predicted and observed stem number of diameter class i,
M_{d} is the number of diameter classes. In addition, the relative error
index was calculated when the age is divided into equal 10 years classes.
Growth data: The diameter analysis is based on measurements in pine (Pinus sylvestris) stands at Lithuania. The data were provided by the Lithuanian National Forest Inventory. We included full calliperings of permanent sample plots. Over 20 years period (19761996) in the evenaged uncut stand sample plots were remeasured at the most five times. The following variables were measured: age (t), number of trees per hectare, diameter at breast height (d), trees position (coordinates x, y), height (h) and descriptive variables such as alive or dead trees were also recorded. Approximately 20% of the sample trees were randomly selected for the height measurement. The measurements have been conducted in 30 occasions of permanent treatment plots and the initial planting densities are unknown. The age of stands ranges from 12 to 103 years. The diameter at breast height varies from 2.2 to 51.5 cm. Height was measured to the nearest 0.1 m with a digital height meter. Diameter was measured to the nearest 0.1 cm. For model estimation observations on 900 pines were used. The observed data sets of study plots are shown in Fig. 1 and 2. Figure 1 and 2 show the variation of diameter and height subject to age.
RESULTS AND DISCUSSION
Deterministic heightage models: Using the observed data set presented
in Fig. 1 and 2, were calculated the parameter
estimations of the exogenous curves h_{1}(t), h_{2}(t) defined
by Eq. 23. Notice that the original observed
data set was arranged by averaging the values observed in equidistant times.
Estimation of models 23 is achieved using Nonlinear Weighted Least Square method.
The values of the weighted least squares estimators (standard errors) are
=0.0176 (0.0009), 1 0.6340
(0.0169) for the Mitcherlich model and for
the Gompertz model.
Visual examination of the residuals versus predicted heights provided a random
pattern around zero with approximately constant variance both for the Mitcherlich
model and the Gompertz model (Fig. 3a, b).
With the exception of some possible outliers the Mitcherlich and Gompertz models
provide a good representation of the height data.
The distribution of a normal probability plot that is nearly linear suggests
normal distribution of the standardized residuals. Figure 4a
and b do not indicate any serious violation of the assumption
of normality for standardized residuals. Typically, normal probability plots
are not perfect straight. For the Gompertz and Mitcherlich heightage models
the pvalues of the ShapiroWilk (1965) statistic, W,
are 0.0553 and 0.0772, respectively.
The exogenous heightage curves h_{1}(t), h_{2}(t), are presented in Fig. 5. For comparison, estimates for precision of the models were carried out based on the coefficient of determination (R^{2}) and the relative error in prediction (RE%). The expressions of these statistics are defined by:
where, and
are
the observed, predicted and mean values of the tree height, respectively; n
is the total number of observations used to fit the model and p is the number
of model parameters. As was expected, both Gompertz and Mitcherlich exogenous
curves have about the same explanatory power, as the coefficient of determination
takes values 0.9640 and 0.9596, respectively. The relative error takes values
6.96 and 7.49%, respectively.
Stochastic agediameter models: Using the observed data set presented
in Fig. 1 and 2 were calculated the parameter
estimations of the stochastic diameter growth model defined by Eq.
1. We shall assume that the stochastic diameter growth process is observed
without error at a given collection of time instances this
justifies the notation of a discretely observed diffusion process. First, the
original observed data set is arranged by averaging the values observed in equidistant
times. The time increments between consecutive arranged data set will be defined
Δ_{i} = t_{i}t_{i1} = 1 for i = 1,2,...,n. In
this study, the estimates of parameters of the stochastic nonhomogeneous model
with an exogenous factor g(t) are defined by a technique that is based on Maximum
Likelihood Estimates (MLE). The estimate of parameters of the stochastic homogeneous
model (α_{1} = 0) is compound of the Least Squares Estimate (LSE)
of the deterministic part (drift) and the MLE of the diffusion coefficient σ.
Hence, we estimate σ by keeping fixed the previously obtained drift parameter
estimates The
MLEs of the stochastic nonhomogeneous model are defined by equations (A1)(A10).
The values of the estimators (standard errors) are presented in Table
1.

Fig. 1: 
The scatter graph of tree diameter at breast
height against tree age for total set of sample trees n = 900 used for parameterization
of growth equations 

Fig. 2: 
The scatter graph of tree height against tree age for total
set of sample trees n = 900 used for parameterization of growth equations 

Fig. 3: 
Scatter plots of standardized residuals vs. predicted values:
(a) the Gompertz exogenous model and (b) the Mitcherlich exogenous model 

Fig. 4: 
Normal probability plots of standardized residuals: (a) the
Gompertz heightage model and (b) the Mitcherlich heightage model 

Fig. 5: 
Plot of the exogenous curves h1(t), h2(t) 
Figure 6ac show the mean and standard
deviation trajectories of the stochastic process D(t), tε[t_{0};
T] of tree diameter. These functions are obtained by replacing the parameters
in Eq. 8 and 9 by their estimators given
in Table 1. All curves monotonically evolve to the steady
state values. As we can see in Fig. 6a and b,
the mean and standard deviation curves of tree diameter are very similar for
both nonhomogeneous (Mitcherlich, Gompertz) models. The mean and standard deviation
curves for the homogeneous model (Fig. 6c) describe a similar
shape for trees less than 30 years age and subsequently get enlarged values
than the nonhomogeneous ones.
Figure 7a c show the estimated univariate
transition probability density function (EDF) of tree diameter defined by Eq.
5. These density functions indicate that the EDF of tree diameter is steeper
for the young stands and less steep for the mature stands. Figure
7ac don’t fix marked difference between the EDFs
using nonhomogeneous and homogeneous models.
Table 1: 
Parameter estimations 


Fig. 6: 
Plot of the mean and standard deviation dynamic of tree diameter
with the parameterization data sets: (a) Mitcherlich exogenous factor; (b)
Gompertzian exogenous factor and (c); homogeneous model; mean (continuous
curve), mean±SD (noncontinuous curve) 

Fig. 7: 
Plot of the estimated univariate transition
probability density function: (a) for the Mitcherlich exogenous factor;
(b) for the Gompertzian exogenous factor and (c) no exogenous; estimated
surface of the agedependent density (Eq. 5) of tree diameter (left); the
density of tree diameter at the age t = 20, 60, 100 years (right) 
For the evaluation of goodnessoffit of our presented lognormal shape univariate
agedependent transition probability density function (5) we use the ShapiroWilk
statistic and normal probability plot. The normal probability plots of the pseudoresiduals
using the estimates of parameters presented in Table 1 are
shown in Fig. 8ac. From Fig.
8ab, it is possible to conclude that both EDFs for Mithcherlich
and Gompertz exogenous factors fit not too bad. For the EDF with the Mitcherlich
shape exogenous factor computed the ShapiroWilk statistic W yield a value 0.9792
(pvalue 0.0010). For the EDF with the Gompertz shape exogenous factor computed
the ShapiroWilk statistic W yield a value 0.9795 (pvalue 0.0013). Finally,
for the EDF of homogeneous model computed the ShapiroWilk statistic W yield
a value 0.9792 (pvalue 0.0008). These results lead us to a conclusion that
the observed data set is compatible with the EDF (5) in all cases. It is worth
remarking that the ShapiroWilk statistic provides a generally superior omnibus
measure of nonnormality. Moreover, the fitting data set was sufficiently large
n = 900.
Finally, the relative error index was used in the comparisons as a measure of goodness of fit of the EDFs for the nonhomogeneous and homogeneous models. The values of the REI% measure Eq. 12 calculated for each EDF of stochastic nonhomogeneous (Mitcherlich, Gompertz) and homogeneous models were 23.00, 23.00, 21.44%, respectively. If we look at the relative error index from the age, the relative error index varies from 7 to 73% (Fig. 9). The relative error index is at its minimum at the age of 55 years. Taking into account that most of the stands covered in this study were within 1280 years, the relative error index is a peaking function for ages greater than 80 years.
Application: The development of simple and accurate standspecific volume
model based on easily obtainable tree and stand characteristics is a main problem
of forest mensuration. Traditionally, the mean tree volume is
estimated as an average of sample tree volumes:
where, V(d, h) is an individual tree volume equation on diameter and height.
Much greater accurateness is obtained by substituting (smoothing) a density
function p(d) of tree diameter and integrating by all diameters d>0. If the
tree volume regression function V(d, h) additionally depends on age (V(d, h,
t)) and the density p(d) function additionally depends on age and height (p(d,
h, t)), then Eq. 15 can be rewritten as follows:
The integral form Eq. 16 describes the mean tree volume
as an explicit function of height and age and can provide additional information
about volume dynamic. The commonly used functional dependence for volume (V(d,
h, t)) calculation takes the form of the power function and
parameters δ_{0}, δ_{1}, δ_{2}, δ_{3}
to be estimated. The estimators and their standard deviations (in parenthesis)
are =
9.5282 (0.0127), =
1.9183 (0.0072), =
0.9807 (0.0104), =
00.0268 (0.0042).

Fig. 8: 
Normal probability plots of pseudoresiduals:
(a) Mitcherlich exogenous factor; (b) Gompertzian exogenous factor and (c)
homogeneous model 

Fig. 9: 
Relative error index: Mitcherlich exogenous
factor; Gompertzian exogenous factor and homogeneous model 
To derive the ageheightdependent density function p(d, h, t) of tree diameter for the nonhomogeneous stochastic model (1), we define Eq. 6 in the following form:
Substituting Eq. 17 into Eq. 5 gives the lognormal density function which depends on height and age as follows:
Figure 10a and b show the estimated lognormal
density function of tree diameter which depends on height and age, defined by
Eq. 18. These density functions are represented at heights
15 and 30 m. The higher is the height for a fixed age, the more skewed toward
large diameters and the wider are densities curves. The ability to model age
and height relationship in the diameter density function is often useful, especially
in modeling diameter dynamic.

Fig. 10: 
Plot of the estimated lognormal density function Eq.
18 at height 15 m (a) and at height 30 m (b); estimated surface of the
agedependent density of tree diameter (left); the density of tree diameter
at the age t = 20, 60, 100 years (right) 
The purpose of this contribution was to present a specific modeling approach
based on the methodology of stochastic differential equations. To deal with
the growth models in a numerical fashion, probabilistic means were adopted to
give an understanding of the problems of the modeling of mean diameter, standard
deviation of diameter and mean volume. Equations for predicting mean diameter
(d) and standard deviation (s) of diameter are expressed in the general forms:
Thus the mean diameter Eq. 19, standard deviation of diameter
Eq. 20 and mean volume Eq. 16 are modeled
as a densitydependent set of curves. Figure 11ac
show the mean diameter of a tree, the standard deviation of tree diameter and
the mean volume of a tree subject to height and age. These graphics demonstrate
that the growth of mean diameter, standard deviation of diameter and mean volume
is a peaking function over a single inflection point at diameter, increasing
with age and height.
Using probabilistic mean diameter and volume growth models Eq.
19, 16 and heightage growth curves h = h(t) = h_{1}(t)
or h = h(t) = h_{2}(t) Eq. 2 and 3
we can define the current (the mean) annual diameter and volume increments of
an average tree in the following form:
Relationships between the current annual diameter Eq. 21
and volume Eq. 22 increments against the height of a tree
are illustrated in Fig. 12a and b. As
we see in Fig. 12a and b, the height exerts
a strong influence on current annual diameter and volume increments. The effect
of a height on current annual diameter and volume increments becomes negligible
above 100 year age.
Figure 13ad shows the current and mean
annual diameter and volume increments against the age and the mean diameter
of a tree using the Mitcherlich heightage growth curve h = h(t) = h_{2}(t)
(Eq. 3). From Fig. 13a and c
we see that the culmination of volume increment is reached even later than that
of diameter increment. The peak in current and mean annual diameter increments
occurred at 21 and 81 years of age (Fig. 13ac),
respectively and current and mean annual volume increments peaked at 26 and
125 years of age (Fig. 13ac), respectively.
If an objective of forest management is to maximize the produced stem volume,
the trees should be retained until they attain their maximum mean annual volume
increment at the age 125 years (Fig. 13c) or at the mean
diameter 42.2 cm (Fig. 13d). The mean annual diameter increment
is greatest at the mean diameter 13.3 cm (Fig. 13b).

Fig. 11: 
Plot of the mean diameter, standard deviation
of diameter and mean volume of a tree: the heightagedependent response
surface of mean diameter (a); standard deviation of diameter (b); mean volume
of a tree (c); the agedependent mean diameter, standard deviation of diameter
and mean volume of a tree (left); at heights 15 and 30 m (right) 

Fig. 12: 
Relationship between the current annual diameter and volume
increments of an average tree against the height of a tree: (a) current
diameter increment and (b) current volume increment at the height 20, 10
and 5 m 

Fig. 13: 
Annual diameter and volume increments: (a) annual diameter
increments against the age; (b) annual diameter increments against the mean
diameter; (c) annual volume increments against the age and (d) annual volume
increments against the mean diameter 
The interrelations of the current annual volume increment and mean annual volume
increment curves of a stand and the position of their point of intersection
are of particular interest to forest management. Enlarged understanding and
statistical inference in stand current and mean annual volume increment models
require an adequate representation of the prediction of tree mortality (survival).
CONCLUSION
Given the importance of stochastic analysis in modern forestry, we consider the case where the governing tree diameter dynamic is defined by an elementary stochastic differential equation. A theoretical prerequisite of our presented approach was the stochastic Gompertz diameter growth law driven by onedimensional standard Wiener process. The results obtained here have shown that it is possible to relate nonlinear stochastic diameter growth law and diameter distribution law. For a realistic representation of diameter and height growth, was used Gompertzian and Mitcherlich growth models.
Thus, the proposed method could be continued in terms of properly modifying the drift and diffusion functions of the stochastic diameter growth process and choosing exogenous factors.
The accuracy of the ageheightdependent diameter distribution (Eq. 5) depends on the amount of information available from the stand. Our methodology extends some way to inclusion of the basalarea or/and density of a stand as an exogenous factor or as an independent variable.
APPENDIX
The maximum likelihood estimates: Here, section we collect some results which were used in order to estimate the model parameters. To write the maximum likelihood function explicitly is possible because the transition probability density function of the diameter stochastic process D(t), tε[t_{0}; T] is explicitly solved by Eq. 5.
The resulting maximum likelihood function is defined by:
Where:
U_{β} is 2xn matrix defined by
Thanks to quadratic form in α the maximum likelihood estimators of α
and σ^{2} are given by (Gutierrez et al.,
2008):
Where:
While the likelihood estimators of α and σ^{2} for stochastic
Gompertz process (1) is well established by Eq. A7, A8, estimating parameter
β is not straightforward. In this paper, the estimation approach we follow
is to first estimate by likelihood estimation procedure the parameter β
of the ordinary differential Eq. 1 (σ = 0), which represents
the deterministic part of the stochastic Gompertz process (1). The maximum likelihood
estimator of the parameter β is defined in the following form:
where, y_{i} = ln d_{i}.