INTRODUCTION
The moisture retention characteristic MRC, which is the
relation between water content (the volumetric water content Θ) and
pressure head (the soil capillary pressure h), can be measured for natural
soils. It is referred to as the soil water retention curve. This relationship
is primarily based on the soil pore structure and the pore size distribution.
The MRC Θ(h) is typical for a given soil having its particular status
of consolidation, geometrical arrangement of particles and aggregation
and other chemical and biological feature. The retention curve graphically
displays a continuously differentiable (smooth) Sshaped curve between
the saturated and residual water contents (Θ_{s} and Θ_{r},
respectively). Knowledge of the MRC is indispensable in describing soil
water processes (flow). The MRC can be determined either directly in the
field or in the laboratory on undisturbed core samples. MRC data from
Vereecken et al. (1989) are used. These data sampled the soil horizons
of forty important Belgian soil series and measured their MRC.
Several of parametric models have been proposed to describe
MRC and all these models are nonlinear regression models (NLRmodels).
Most of the proposed models are curvefitting equations and they are able
to describe the typical Sshaped behavior of MRC and represent NLRmodels
which can be fitted to give data of MRC. The modelparameters can be estimated
by applying algorithms minimizing a least squares (LS) object function
(Bates and Watts, 1988).
Mathematical statistics have developed several methods
for model selection in nonlinear regression. In this study, the researcher
studies problems of model selection for MRC from a statistical and mathematical
viewpoint. This study is carried out through the comparison between the
used MRCmodel proposed by van Genuchten (1980), a MRC model developed
by King (1965) and its variants. A main task of the analysis of experimental
data is the estimation of the parameters in a NLRmodel (Bates and Watts,
1988; Ratkowsky, 1983; Ratkowsky, 1990; Seber and Wild, 2005). Usually
it is not known whether a proposed regression model describes the unknown
true regression function sufficiently well. But the results of the statistical
analysis may heavily depend on the chosen model.
Modelers probably have one of the three purposes in mind
when they wish to fit a NLRmodel to set of data. First, they are interested
in obtaining a good fit to the data. That means, they are primarily interested
in the representation of the relationship between the independent and
dependent variables by the chosen model. Secondly, they focus on the prediction
of values of the dependent (response) variable for given values of the
independent (regressor) variable. Moreover in some situations the modelers
wish to make inference based upon interpretation of the parameter estimates
of the corresponding model. The current study concerns with the second
and the third aspects.
A more general NLRmodel may be written:
where, y_{ij} is the values of the response variable
Y at fixed values x_{i} of exploratory variables (nonrandom design
points). Here, the real valued function f(.,β) is expectation function
and it is known up to a Pdimensional vector of parameters β = (β_{1},
β_{2},...,β_{p}) ∈R^{P} and this
function is twice continuously differentiable in β. Moreover, the
random perturbations ε_{ij} (the measurement errors) are
uncorrelated random variables with zero mean and unknown variance, σ^{2}
which do not depend on x_{i}. Given the date
the estimation of the regression function f reduces to
estimate its parameters. The most popular approach to estimate β
= (β_{1}, β_{2}, ..., β_{p}) is
to employ the ordinary least squares estimator
, which is a minimizer of the sum of squares
The linearity of the regression model (1) may be produced
if the expected response f(x, β) is a linear function of the parameter
vector β = (β_{1}, β_{2},...,β_{p}).
The solution
of the corresponding minimum problem
with respect to a linear regression model (LRmodel)
will be given by an explicit algebraic expression. This solution, which
is the least squares estimator
of β, is a linear function of the data vector of y_{ij},
asymptotically normally distributed, unbiased and it has a minimum variance
(Bates and Watts, 1988; Haines et al., 2004).
For the NLRmodels the situation is different. There
is no explicit expression for .
One gets the least squares estimator
for β only by an iterative procedure starting from some assumed value
of β. In general the least squares estimators of the parameters of
a NLRmodel are biased, nonnormally distributed (skewed) with variance
exceeding the minimum possible variances in the corresponding linear model
(Ratkowsky, 1983). The extend of bias, nonnormality and excess of variances
differs widely from model to model. The researcher looks for MRC models
which, with respect to their estimation behavior, are closely related
to LRmodels.
NLRmodels differ in their estimation properties from
linear regression models. Given the usual assumption of independent and
identically distributed measurement errors, the parameter (LS) estimators
for linear models are linear, unbiased, at least asymptotically normally
distributed, minimum variance estimators. On the other hand, NLRmodels
tend to do so only when the sample size becomes very large.
In this study, the purpose is the exploration and the
comparison of the properties of estimators for some different MRC models
in situations having sample sizes typically obtained in practice. The
MRC models with closetolinear estimation behavior are looked. The extend
of nonlinear estimation behavior (e.g., bias) depends upon the nonlinearity
of a model/data set combination. Therefore, this work can be carried out
in the first step by using simulation studies with respect to the considering
models and in the second step by computing some measure of nonlinearity.
The aim of the presented study is to select an optimal
MRC model, to determine the cause of the nonlinearity in this selected
model and finally to diminish (or avoid) this defect through a suggested
reparameterization.
THE CLASS OF FUNCTIONS USED FOR MODEL SELECTION
van Genuchten (1980) presented a widely used class of
functions for parametrizing measured soil water characteristics
where α, n and m are positive parameters defining
the MRC`s shape and this model is denoted by (VG1). This function is a
MRC model of five parameters α, n, m beside, Θ_{s},
Θ_{r}. Moreover, King (1965) suggested the following model
with five real parameters Θ_{s}, h_{0},
b, ε and γ where b having negative values and this model denoted
by (K1). The Table 1 indicates to special cases (VG2),
(VG3), (VG4) and (VG5) of van Genuchten model (VG1) and also the model
(K2) as a special case of the model (K1) of King, which are models with
fewer parameters:
Each of the models (VG2), (VG3) and (VG4) has four parameters
Θ_{s}, Θ_{r}, α, n but (VG5) is the model
with only three parameters Θ_{s}, α, n. Also, the model
(K2) has four parameters and Θ_{s}, h_{0}, b and
γ.
Table 1: 
Some variants of van Genuchten model (VG1) and King model
(K1) 

USING METHODS TO SELECT A NLRMODEL FOR MRC
Two methods to select a NLRmodel for MRC are used in
this study. These methods are described in the following two subsections.
Description of the Simulation Experiments
Simulation studies (experiments) are probably the most direct and
best way to enable the modeler to study the sampling properties of the
LS estimator. Data are generated using a set of predetermined values of
the parameters, allowing only the values of the measurement error to change
randomly or pseudorandomly form set to set. By this means, many sets
of simulated data produced and each set provides a LS estimates of the
parameters of the model under consideration. These estimated parameters
may be examined for their bias, variance and other distributional properties.
Throughout this study, the reported results of simulation
are based on 100 sets of simulated data. Recall that the NLRmodel is
given by (1), where the ε_{ij} are uncorrelated identically
distributed random variables. The measurement errors are generated under
the considering assumptions (independent identically normally distributed
measurement errors with zero mean and constant variance σ^{2}).
If one assumes such a type of distribution of the measurement error, then
there is a possibility to choose an appropriate value of the variance.
The considering selection of σ is based on MRC data from Vereecken
et al. (1989) (Table 2). The variance is model
independent estimated using the repeated measurements (pure error, intra
sample estimates). The calculated value
is a typical mean value with respect to the other data sets of Vereecken
with repeated measurements. Now, a simulation study for two different
soils (sand and loam) is illustrated using this estimated value
and true parameters for the model (VG2).
Table 3 contains on the true parameters
of the model (VG2) for two different soils sand and loam (Carsel and Parrish,
1988).
Every simulated data set has the structure of Vereecken
MRC data. A simulated value is the sum of the expected value corresponding
to the value of pressure and a random error, which is normally distributed
with zero mean and variance .
Table 2 shows the measured MRC data of a Belgian soil
(Bates and Watts, 1988), the theoretically expected values of sand and
the first simulated data set.
Each set of simulated data is then fitted by least squares.
That means, the vector of parameters β of the considering models
(VG1), (VG2), (VG3), (VG4), (VG5), (K1) and (K2) is estimated.
Table 2: 
Measured MRCdata, theoretically expected (MODEL) of sand
and the first simulated MRCdata for sand (SIMUL_1) 

Table 3: 
True parameters of van Genuchten model (VG2) 

Table 4: 
Estimated vector of parameters of the model (VG1) corresponding
to the first simulated data set for sand 

Different algorithms are used, for instance the Levenberg/Marquardt
method (Bates and Watts, 1988). In the case of model (VG1) and (VG2),
the true (simulated) vector of parameters is considered as an initial
estimate. Table 4 contains the estimated vector of parameters
of the model (VG1) corresponding to the first simulated data set for sand.
For every model under consideration, 100 estimates are
produced. Taking each parameter separately, univariate statistics are
calculated and the corresponding distribution is graphed. In addition,
it is possible to examine the multivariate behavior of the LS estimator
by calculating bivariate statistics and using two or three dimensional
plots. This means that the researcher can see how close to linear in its
behavior the LS estimator is. An acceptable model leads to LS estimations
for its parameters with small biases, with distributions close to normal
distribution and variances close to the minimal possible variances.
Description of the Measures of Nonlinearity
Another possible way to analyze the nonlinear behavior of a model
data set combination is the calculation of so called measures of nonlinearity
(e.g., curvature, bias and skewness). Using differential geometryconcepts,
the measures of nonlinearity based on the notion of curvature were developed
(Bates and Watts, 1988; ElShehawy, 2001; ElShehawy and Karawia, 2006).
These measures are independent of scale changes in both the data and parameters.
They can be used to compare different models with different parameterizations
combined with different data set.
The Nvector η(β) with the components
defines a Pdimensional surface, the so called expectation
surface or solution locus in the ndimensional response space. The LS
estimate
corresponds to that pointη (
) on this surface which is closest to the measured Ndimensional vector
of response y. The parameter vector β is assumed in the neighborhood
of its LS estimate
.
If the quadratic term in the second order Taylor approximation
of η(β) is neglected, one will have for β in the vicinity
of
the linear approximation
where
is the (N, P) Jacobimatrix
at β =
(Ratkowsky, 1983). The range of the matrix
is the tangent plane to the expectation surface at the point
and the linear approximation (8) amounts to approximating the expectation surface
in a neighborhood of
by this plane. Using (8) the following form is got
From some statistical properties of the linear models
(Bates and Watts, 1988; Haines et al., 2004), the following ellipsoid
with center
is considered as an approximation of an 100 (1a)% confidence region for
β = (β_{1}, β_{2}, ..., β_{p})
∈R^{P}:
where
is the residual mean square, is the upper a quantile for the Fdistribution
with P, NP degrees of freedom.
From (7), (8) and (10), the expectation surface η(β)
lies approximately within the intersection of tangent plane and a sphere
with center η(
) and radius .
A LRmodel has a linear solution locus, which means a
hyperplane for p≥3 and (9) holds exactly. In addition, lines (parameter
lines) on the solution locus representing constant values of β_{r},
r = 1, 2, ..., P, are straight, parallel and equally spaced for equal
increments of β_{r}.
For a nonlinear solution locus the situation is different.
The solution locus is a curved surface and the parameter lines on this
surface (or the projections of these lines onto the tangent plane) are,
in general, neither straight, parallel nor equispaced.
The extent of the curvature of the solution locus has
been called intrinsic nonlinearity, since this nonlinearity cannot be
altered by reparametrization. It is an inner geometrical property of the
surface. The extent of the curvature of the parameter lines, their lack
of parallelism and equispacedness has been called parametereffects nonlinearity,
since it is determined by the way in which the parameters appear in the
model, that means, it depends on the parameterizations (Bates and Watts,
1988).
The validity of the tangent plane approximation (8) will
depend on the magnitude of the quadratic term in the Taylor expansion
of η(β) relative to the linear term. In making this comparison
it is helpful to split the quadratic term into two orthogonal components,
the respective projections onto the tangent plane and the component normal
to the tangent plane. These components were compared with the linear term
and got two measures of nonlinearity, the parametereffects curvature
and the intrinsic curvature. Standardizing the model and the data leads
to scale independent quantities. Both measures depend on the direction
(β  ).
Root Mean Square (RMS) curvatures are used, which are the square root
of the average over all directions of squared curvatures. These measures
are denoted by c^{IN} (intrinsic curvature) and c^{PE}
(parametereffects curvature). The symbol 1/
refers to the inverse of the radius of the (standardized) sphere (10).
A convenient scale of reference can be established by
comparing the RMS curvature with that of the (scaled) confidence disk
(10) at a specified level (1a), a>0. That means, we compare the radius
of curvature 1/c (c = c^{IN} or c = c^{PE}) with the radius
of the confidence disk .
An RMS curvature will be considered as small if it is much less than the
curvature of the (1a) confidence disc, that is if c<<1/,
where F = F (P, NP; a).
Following some earlier several research study s, an expectation
surface with radius of curvature 1/c is considered and the deviation of
the tangent plane from the surface or the deviation of the parameter line
from the straight line at a distance
is determined. This deviation, expressed as a percentage of the radius
of the confidence disk, is
, so that:
• 
A value of
causes the surface to deviate by 10% of the radius of the confidence
at the edge of the confidence disk; 
• 
A value of
causes the surface to deviate by 27% of the radius of the confidence
at the edge of the confidence disk; 
• 
A value of
causes the surface to deviate by 100% of the radius of the confidence
at the edge of the confidence disk and so on. 
If c^{IN} is replaced by c^{PE} then
a corresponding rule about the deviation of a parameter line from a straight
line will be get.
In almost all cases known from several works the intrinsic
curvature is very much smaller than the parametereffects nonlinearity.
Since c^{PE} depends on the parameterization it is possible to
reduce the parametereffects nonlinearity using an appropriate reparametrization
for the model under consideration.
Another practicable way to study the nonlinear behavior
of a model data set combination is the calculation of estimation for the
bias in the LS estimates. A corresponding formula for bias was presented
by Box (1971) and ElShehawy (2001). The approximate bias for each component
of the estimate
is calculated separately.
Although the bias can be used as a measure of the extent
to which parameter estimates may exceed or fall short of the true parameter
values yet it cannot be used to compare parameters in two different parameterizations
(Ratkowsky, 1990). This comparison is possible with another measure of
nonlinearity, the measure of skewness (of the distribution of the estimated
parameter) due to Hougaard (1985) and ElShehawy (2001). Following Ratkowsky
(1990), it is possible to use a ruleofthumb for asserting whether the
estimator
of the rth component β_{r} of the parameter vector
β, as assessed by the measure of skewness Sk_{r}, is closetolinear
(nearly symmetrically distributed) or contains considerable nonlinearity:
• 
If Sk_{r} < 0.1, the estimator
of β_{r} will be very closetolinear in behavior; 
• 
If 0.1 ≤ Sk_{r} <0.25, the estimator will be reasonably
closetolinear in behavior; 
• 
If Sk_{r} ≥ 0.25, the skewness is very apparent and 
• 
If Sk_{r} > 1, indicates considerable nonlinear behavior. 
RESULTS
Simulation Experiments
Table 5 shows the mean value of the residual sum
of squares
for all models under consideration with respected to the simulation of
sand.
If the retention models with respect to the ability to
fit the simulated data are compared, the models with five parameters (VG1)
and (K1) will be most flexible. The difference between these best fitting
models and the models with four parameters is marginal, only the model
(VG5) with three parameters is not flexible enough. These properties of
the retention models neither depend on the type of the simulated soil
nor on the used optimization algorithm.
Consider the distribution of the estimated parameters
of the models (VG1) and (VG4). Some descriptive statistics of the 100
estimated parameters of the model (VG1) for the simulated data sets of
sand are given in Table 6.
Table 5: 
Residual sum of squares for the simulated data sets of sand 

Table 6: 
Descriptive statistics for the distribution of the estimated
parameters for the model (VG1) and the simulated data sets of sand 


Fig. 1: 
The shape of the distributions of the estimated parameters
Θ_{r} ( (a), (b), (c) figures) and m ((d), (e), (f) figures)
of the model (VG1) and the simulated data sets of sand (Histogram with density,
PoxPlot and QQ Plot) 

Fig. 2: 
The scatterplot matrix for the estimated parameters of the
model (VG1) and the simulated data sets of sand 
The symbol Sign. refer to the pvalues for the goodness
of fit test (KolmogorovSmirnov test) for the fit of a normal distribution.
Only the distribution of the linear parameter Θ_{r} is similar
to a normal distribution (pvalue>0.05). In case of all other parameters
the hypotheses of normal distribution has to be rejected (since pvalue<0.05).
These parameters are nonlinear parameters. The estimation of these parameters
is biased and the distribution of the estimations is nonsymmetric with
a great variability. Especially, the parameters n and m are skewed (right
tail) with some very extreme values. Figure 1 describes
the shape of the distributions and the similarity to the normal distribution
of the linear parameter Θ_{r} and of the nonlinear parameter
m.
Consider dependencies between the estimated parameters
of the model with five parameters (VG1). The scatterplot matrix in Fig.
2 shows strong relationships especially between the nonlinear parameters
a, n, m. In the Fig. 2, Beta 1, Beta 2, Beta 3, Beta
4 and Beta 5 denote to Θ_{r}, Θ_{s}, α,
n and m, respectively.
Table 7: 
Descriptive statistics for the distribution of the estimated
parameters for the model (VG4) and the simulated data sets of sand 


Fig. 3: 
The shape of the distributions of the estimated parameters
Θ_{r} ((a), (b), (c) figures) and n ((d), (e), (f) figures)
of the model (VG4) and the simulated data sets of sand (Histogram
with density, PoxPlot and QQ Plot) 
Spearman`s rank correlation coefficient (a measure on
monotone dependence (Johnson and Bhattacharyya, 1992) for a pair α,
m is equal to 0.917. Within the three dimensional space the estimated
values α, n, m describe a nonrandom point could. The points are concentrated
along space curves. The corresponding constellations of parameters allow
an equal fit of the same data. That means, with the model (VG1) over fitting
is possible, estimated parameters may depend on each other and therefore
it is impossible to identify a (simulated or real) soil using the estimated
vector of parameters.
If the model (VG4) with four parameters and m = 1 is
considered, the results for the simulated data sets of sand are given
in Table 7.
It is obvious, that the estimation properties of the
parameters of the model (VG4) are much more better than the corresponding
properties of the model (VG1). The strong nonlinear parameter m of the
model (VG1) is fixed. Although the model (VG4) is flexible yet it is also
stiff enough (Fig. 3).
The scatterplot matrix in Fig. 4 shows
strong relationships especially between the nonlinear parameters α
and n.
The parameter n of the model (VG4) has the worst estimation
properties in comparison with the other parameters Θ^{r},
Θ^{s} and α. The distribution of the estimated values
of this parameter has a great variability and is skewed (Fig.
3).
If the interest is considered in the model (VG4), an attempt to improve
these problematic qualities will be occurred by a reparametrization. As
an example, consider the replacement of the parameter n by .
This reparametrization of the model (VG4) is denoted by the model (RVG4).
The distributional properties of this new parameter
are studied. Table 8 contains descriptive statistics
for the parameter
for the model (RVG 4) and the simulated data sets of sand.
Table 8 and Fig. 5
show these the distributional properties in comparison with the corresponding
properties of the model (VG4). The results with respect to the other parameters
do not change.

Fig. 4: 
The scatterplot matrix for the estimated parameters of the
model (VG4) and the simulated data sets of sand 

Fig. 5: 
The shape of the distributions of the estimated parameter
of the model (RVG4) and the simulated data sets of sand (Histogram with
density, PoxPlot and QQ Plot) 
The statistical properties of the model (VG4) are better
than the estimation properties of the other models of the van Genuchten
type with four parameter (VG2) and (VG3). These observed properties do
not depend on the simulated soil.
Consider the model (K1) of King with five parameters
and its variant (K2), a similar situation will be obtained. The model
(K1) is very flexible but not stiff enough. Fix the strong nonlinear parameter
ε, the model (K2) with four parameters will be obtained. The distributional
properties of the estimated parameters of this model are much better than
the properties of the corresponding model (K1).
Some Calculations for Measures of Nonlinearity
Throughout this subsection some results of nonlinearity measures are
discussed. These results relate to the current and previously studied
models in combination with some simulated data sets of sand and loam.
Firstly, consider the calculated RMS curvature measures
for the models (VG1), (VG2), (VG3), (VG4), (VG5), (K1) and (K2). Table
9 indicates to the scaled RMS curvatures (the intrinsic curvature
and the parametereffects curvature) relative to a 95% confidence disk
radius (i.e., a = 0.05) for the studies models under consideration and
the first simulated data set SIMUL_1 of sand.
Table 8: 
Descriptive statistics for the distribution of the estimated
parameters for the model (RVG4) and the simulated data sets of sand 

Table 9: 
Scaled RMS curvatures relative to 95% confidence disk radius
for the considered models and the first simulated data set of sand 

Table 10: 
Calculated parametereffects curvature in the direction corresponding
to the parameters of the van Genuchten models in combination with the first
simulated data set of sand 

The intrinsic curvature only depends on the geometry
(the shape) of the model data set combination. This measure is independent
of the chosen parameterization of the model.
Referring to Table 9, it can be seen
that the model (VG4) with four parameters possess the best value with
respect to the parametereffects curvature in comparison with the other
models. This table shows also the extreme value of RMS parametereffects
curvature with respect to the model with five parameters (VG1).
Consider different data sets, different values of the
curvature are obtained and the order of models is in general stable. If
the number of repeated measurements increases, the curvature will be reduced.
A value of causes
a moderately deviation of 10% between the expectation surface and the
approximating tangent plane at the radius of confidence disk. The values
of the parametereffects curvature are much more greater. This means that
there are a great difference between the parameter lines on the expectation
surface and the parameter lines on the tangent plane at the edge of the
confidence disk. For example, the parametereffects value 1.00 corresponds
to a deviation of a parameter curve from a straight line at the edge of
100%.
Considering for the single parameters of the studied model, which cause the
corresponding value of the parametereffects curvature, it is possible to calculate
a value of
in the direction of a rth component of parameter vector of the model (Bates
and Watts, 1988). Table 10 contains the calculated parametereffects
curvature in the directions corresponding to the parameters α, n, m of
the van Genuchten models (VG1), (VG2), (VG3), (VG4), (VG5) in combination with
the first simulated data set SIMUL_1 of sand.
It is obvious, that the corresponding values for the
parameters Θ_{r} and Θ_{s} are zero, since these
parameters having linear behavior in the models of van Genuchten. Similarly,
the linear parameter Θ_{s} in the models of King has a value
of zero. On the other hand, the parametereffects curvature has the largest
values for the parameters n and m, which having strong nonlinear behavior.
Therefore, these are strong nonlinear parameters with estimation properties,
which greatly differ from the properties of linear or near linear parameters.
Since these values depend on the parameterization, it is possible to reduce
this nonlinear behavior by a reparametrization.
Table 11: 
Bias and skewness for the parameter n and its reparameterization
in the models (VG4) and (RVG4) respectively in combination with the first
simulated data set of sand. 

Considering the reparametrization (RVG4) of the model (VG4), a value of
for the RMS curvature is got for the same simulated data set SIMUL_1 of sand
for the RMS curvature but the corresponding values for the intrinsic curvature
do not change. A value 0.53 of the parametereffects curvature for the direction
of the parameter
is obtained. The parameter
in the model (RVG4) has properties, which are better than the corresponding
properties with respect to the parameter n in the model (VG4). Moreover, the
value of the parametereffects curvature for (RVG4) is smaller than the corresponding
value for the model (VG4).
The values of the RMS curvatures for the models (K1)
and (K2) are smaller than the corresponding values for the van Genuchten
model (VG1) with five parameters, but as a rule the model (VG4) with four
parameters has the best value of the parametereffects curvature. On the
other hand, the King model (K2) with four parameters has more favorable
qualities in comparison with the model (K1) of King with five parameters.
The results of the calculations of the bias and the skewness
measures for the parameter n and its reparametrization
in the models (VG4) and (RVG4) respectively in combination with the simulated
data set SIMUL_1 of sand are given in Table 11.
Referring to Fig. 3, 5
and Table 11, it can be seen that the distribution
of the estimated values for the parameter n is skewed with heavy right
tail and the distribution of
is near symmetric. The mean value of the estimations of
corresponds to the true value, but the mean value of the estimated values
of n is greater than the true one.
Using data sets of different simulations (or soils),
similar results with respect to the used measures of nonlinearity will
be obtained as a rule.
CONCLUSIONS
Model choice of NLRmodels depends on the main aim of
the analysis. A retention model, which considers as a NLRmodel, has to
be able to describe the typical Sshaped retention curve. If one is interested
in the best fitting model, very flexible models with more parameters should
be used. As a criterion ordinary or in case of heteroscedasticity, weighted
least squares can be used.
In case of a retention curve, the models (VG1) and (K1)
with five parameters of van Genuchten and King respectively are optimal
with respect to the goodness of fit.
If the stable estimation of the parameters of a model
is of prime importance, the distributional properties of the model parameters
should be studies. These can be done by simulation studies or the calculation
of measures of nonlinearity. In this case one looks for models with parameters
which have near linear estimation behavior. These estimation properties
are dependent on the inner nonlinearity of model data set combination,
which is independent of the parameterization and the nonlinearity of model
data set combination, which is dependent on the used parameterization.
Near linear behavior means unbiased, near symmetrically distributed estimations
with a near minimal variability in comparison with an approximating linear
model.
Using the list of the studied models for the retention
function in combination with the considered data sets (simulated or real),
the van Genuchten model (VG4) with four parameters is as an optimal model.
At the same time, this model has a strong nonlinear parameter n. A possible
appropriate reparametrization with respect to this parameter is the model
(RVG4).
ACKNOWLEDGMENTS
I would like to thank the committee of the journal and
the referees for their constructive comments.