Iraqi Statisticians Journal / Vol. 1, Issue 1, 2024: 24-42
Iraqi Statisticians Journal
https://isj.edu.iq/index.php/rjes
Estimate a nonparametric copula density function based on probit
and wavelet transforms
Fatimah Hashim Falhi1,*, Munaf Yousif Hmood2
1,
Department of Statistics, College of Administration and Economics, University of Basrah, Basrah, Iraq
fatima.falhi@uobasrah.edu.iq
2
Department of Statistics, College of Administration and Economics, University of Baghdad. Baghdad, Iraq
munaf.yousif@coadec.uobaghdad.edu.iq
ARTICLE INFO
Article history:
Received
Revised
Accepted
Available online
4 March 2024
10 March 2024
15 March 2024
15 March 2024
Keywords:
Wavelets,
Copulas
Wavelet Transforms
Probit transform
Multire solutions
Boundary effects.
ABSTRACT
This study employs wavelet transforms to address the issue of boundary effects.
Additionally, it utilizes probit transform techniques, which are based on probit
functions, to estimate the copula density function. This estimation is dependent on the
empirical distribution function of the variables. The density is estimated within a
transformed domain. Recent research indicates that the early implementations of this
strategy may have been more efficient. Nevertheless, in this work, we implemented
two novel methodologies utilizing probit transform and wavelet transform. We then
proceeded to evaluate and contrast these methodologies using three specific criteria:
root mean square error (RMSE), Akaike information criterion (AIC), and loglikelihood (LogL). The wavelet transform method works better than the probit
transform method at all three levels of correlation, as shown by a simulated study with
four types of copulas, five sample sizes, and three levels of correlation. Research has
demonstrated that probit transformation methods are most appropriate for linkages
involving large and medium sample sizes, as indicated by Frank, Joe, and Tawn
Copula. On the other hand, for copula functions for all sample sizes, the wavelet
transform method was found to be ideal in cases with low correlation values.
1. Introduction
The nonparametric estimation technique is
a common and flexible tool for analyzing data
and modeling relationships between variables.
The nonparametric estimation is different from
the parametric estimation in that it does not
take a fixed form or a specific form, but is
obtained according to the information derived
from the data. All information regarding the
phenomena under research is assumed to be
regularly distributed in parametric models.
Under tight assumptions and circumstances, if
the random variables are not normally
distributed, we cannot use standard correlation
measurements like Kendall's or Spearman's.
Separating random variables' effects is
extremely challenging, especially when
evaluating the degree of positive and negative
dependence. As a result, researchers use
nonparametric approaches such as the kernel
density function to detect dependencies,
especially in multivariate distributions.
The problem in the modeling of
multivariate functions is the presence of
dependency between the observations of the
variables of the examined phenomena, which
*
Corresponding author.
E-mail address: fatima.falhi@uobasrah.edu.iq
This work is licensed under https://creativecommons.org/licenses/by-nc-sa/4.0/
Fatimah Hashim Falhi, Munaf Yousif Hmood/Iraqi Statisticians Journal / Vol. 1, Issue 1, 2024: 24-42
can lead to a variety of issues, including
boundary effects. In this situation, It is
impossible to get the exact estimation for these
functions. A suitable statistical tool must be
used to characterize the dependence structure
between the variables of the examined
phenomenon, particularly when the effect
extends over a long or medium period of time
and the data distribution is unknown.
Nonparametric approaches are employed to
estimate the copula functions in this research.
Many studies have been published by
researchers to help develop ideas for modeling
dependency measures in many fields,
especially the challenges encountered during
the analysis, such as problems of association
between study variables and problems of
boundary effects. [1] developed the theory of
nonparametric estimation of the copula
function of a random variable based on the
empirical Copula and measuring the sample
dependency by means of the empirical copula,
and obtained a consistent empirical copula
function. [2] clarified and reviewed some
parametric,
nonparametric,
and
semiparametric methods and suggested methods
for estimating the probability density function
and choosing the appropriate method for
estimating
smoothing
parameter
and
comparing the mentioned methods in
determining the best estimator for the
probability density function using the
simulation method.
[3] used the copula theory in modelling the
survival function of the bivariate variable
Weibull distribution and bivariate standard
normal distribution cut off at zero point and
using simulation experiments for comparison
between the estimation of the survival
function by using six different copulas [4]
presented a paper for inference copula models,
based on the rank method. Working in detail
on a small imaginary numeric example,
illustrate the different steps for checking the
dependence between two random variables
and modeling it using copulas. It also
introduces simple graphical tools and
numerical techniques for selecting a suitable
model, estimating its parameters, and checking
its suitability. An application of the
methodology to hydrological data is then
presented. [5] investigated kernel methods for
obtaining smooth and flexible estimates of the
bivariate correlation cumulative distribution
function, also discuss the selection of
bandwidth parameters. [6] presented a
proposal for a new copula by applying the
Plackett copula through a mathematical
modification that was made on that copula and
comparing the Plackett copula with the
proposed copula using simulations. [7]
introduced the probit transformation of
estimating the density of the kernel on the unit
interval and he proposed a correct and simple
method by combining the concept of
transformation with estimating the local
likelihood density, resulting in workable
density estimations that are free of boundary
issues in most cases. [8] investigated the
probit transformation of the nonparametric
kernel estimation of the copula density. He
proposed a kernel type copula density based
on the idea of transforming the margin of
copula density to normal distributions using
the probit function and estimating the density
in the transformed domain without boundary
bias problems. Thus, obtaining an estimation
of the copula density via the backtransformation, and it was then demonstrated
that when this method is combined with
methods of estimating the local polynomial
density. [9] presented a method for estimating
the copula density using different kernel
density methods, including mirror reflection
method, beta kernel method and kernel
transformation method, and then comparing
the three methods using simulation
experiments, the results showed that the
transformation kernel estimator is the best
among the three methods, and it is proved that
the copulas are highly explicitly for high
dependency, especially of the Gaussian
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Fatimah Hashim Falhi, Munaf Yousif Hmood/Iraqi Statisticians Journal / Vol. 1, Issue 1, 2024: 24-42
type.[10] presented a R package called
Kdevine to estimate the density of the
multivariate kernel with vine copulas.[11]
studied reliability structural analysis methods
with multidimensional correlation and when
conducting a structural reliability analysis and
calculating the probability of structural failure.
The techniques that helped analyze structural
reliability in light of the correlation problem,
include the third-moment technique, the
fourth-moment technique, and the D-Vine
copula technique. These techniques were
based on the first-order reliability method in
its basic techniques when transforming the
studied random variables into independent
standard normal random variables, and
iterative algorithms were used to find the
probability point of most failures.
These studies were confined to nonparametric
kernel functions using a fixed-value
smoothing coefficient or a symmetric diagonal
matrix.
In addition to many researchers have been
studies wavelets. [12] studied the wavelet
properties of the sunspot series. [13] employed
variable kernel functions to estimate the risk
for censored data. [14] used wavelets to
estimate the return stock rate of the private
banking sector. [15] studied multivariate
fractional Brownian motion using discrete
wavelets.
The purpose of this research is to estimate
the copula density by nonparametric methods
through probit transformation depending on
the Kernel copula function for the purpose of
correcting the boundary effects and wavelet
transformation using multi-resolution analysis
and comparison. Probit transformation is one
of the methods used in boundary correction,
and it is the most commonly used method,
because this method suffers from biases at
boundary points, we used a smoothing
coefficient in the form of a full positive
matrix.
2. Materials and Methods
2.1 Copula definition
A copula is a function that illustrate
modeling the dependency of random variables.
Sklar's created and initially utilized the copula
[16].
This function has several advantages for
modeling dependencies in multivariate data.
first, consider the joint distribution's
separation into the dependency structure
(copula) and the basic marginal distributions.
And which can be viewed as a
mathematical tool that is used to represent the
relationship structure between two or more
random variables. Many articles and studies
have been written about nonparametric
estimation of copulas. The use of
nonparametric methods is more flexible than
standard parametric methods, as no
assumptions are required.
According to Sklar theorem 1959, every
joint cumulative distribution function F of
continuous random quantities
can be
written as
, for all
, where
and
are continuous
marginal distributions and
is a unique corresponding to this joint
distribution. Therefore, the copula is the joint
cumulative
distribution
function
with
uniformly distributed marginal distributions
on [0, 1] [17][18].
Therefore, every multivariate CDFs with
standard uniform marginal that show the
dependence structure of random variables X
and Y, and their marginal cumulative
distribution functions are described by
Where U and V are uniformly distributed
variables and
.The probability of
two random variables,
and
, is
described by the joint CDF
.
Where
is called a copula and can be
uniquely determined when u and v are
continuous [19].
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Fatimah Hashim Falhi, Munaf Yousif Hmood/Iraqi Statisticians Journal / Vol. 1, Issue 1, 2024: 24-42
The following is the formula for a Gaussian
∫
√
Represents the standard normal
distribution function, while
represents
Joe copula is provided by
∫
the inverse of standard normal distribution
function.
A Frank copula is given by [21].
)(
)
/
(
.
copula: [20]
as well as its density
Where
. It is distinguished by upper tail dependency. moreover ,
[22].
Tawn copula is
) .
{(
/}
)
)
((
, we recover the Gumbel copula.
it will be asymmetric in its components.
2.2 Kernel and probit estimation:
There are numerous nonparametric
methods for estimating the dependence
structure between two random variables,
such as polynomial approximation copulas
and kernel smoothing copulas [8].
̂
| |
⁄
2.2.1 Kernel density function estimation:
The d-dimensional multivariate kernel
density estimator in its general form is
[23][24].
⁄
∑
∑
| | ⁄ (
Where H is positive and symmetric definite
bandwidth matrix and K is kernel function,
| |
and | |
There
are
several
nonparametric
techniques to estimate the dependence
structure between two random variables,
such as empirical [1].
polynomial
approximation copula [17] and kernel
smoothing copulas [25].
⁄
)
In the classical statistics texts, a kernel is
a nonparametric method for estimating the
probability density function (pdf) of a
continuous
random
variable.
Any
probability density can be used for the
kernel [26]
In this study we use kernel type copula
estimators because this method is the most
commonly used in nonparametric estimation
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Fatimah Hashim Falhi, Munaf Yousif Hmood/Iraqi Statisticians Journal / Vol. 1, Issue 1, 2024: 24-42
of copulas, Although its flexible [7]. But is
not appropriate for the unit squared copula
densities, essentially because it is heavily
influenced by boundary bias issues for
estimation function. In addition, most
common copulas permit unbounded
densities, and kernel methods are not
The standard kernel estimator for c, denoted by ̂
̂
|
|
⁄
∑
(
The use of kernel techniques to estimate an
unknown bivariate copula density we will
see that the boundedness of a copula
density's support necessitates the use of
more advanced techniques than the one
considered. U, V ~U[0, 1] are random
variables with the joint distribution C and
the corresponding density c: [0, 1]2 →R. We
assume that the copula C has i.i.d variables
}, and
{
our goal is to estimate the density c [7].
2.3 probit Transformation Estimation
Method (PTE) :
Data transformations are commonplace,
and widely used to enhance the application
and performance of classical estimating
methods, this procedure, deals almost
skewed data, heavy tails, or bounded
supsport.
Several studies have investigated the
transformation density estimation technique
in the context of kernel density estimation,
and
Where
is the standard normal cumulative
distribution function and
its quantile
function or the probit transformation [7].
Given that both U and V are uniform
distribution [0,1], S and T have standard
normal distributions, but this does not imply
that the vector (S, T) is bivariate normal. If
the joint CDF of (S,T) is the Gaussian ,then
is the Gaussian copula because copulas
⁄
consistent in that case. Therefore, many
researchers study and provide solutions to
the boundary bias, including [27] [28] [8].
(
))
and they have presented a number of
transformation families and transformation
selection criteria. These studies created
parametric families of transformations that
approximate normality in a range of nonnormal distribution. Although our essential
goal of simple density estimation does not
necessitate normality, Transformations can
serve a variety of purposes in statistical
analysis [29].
To solve the problems that caused
boundary bias by transforming the data so
that its distribution is supported on the full
R2.In other words, this method can be
correct the boundaries in a natural way, and
this method is characterized by being able to
deal with boundary copula densities [25].
The difficulty in the copula density
estimation of (U, V) is primarily due to the
constrained nature of its support
.Now define
are invariant for increasing transformations.
[18] has unconstrained support R2, and
estimating its density
cannot be affected
by boundary problems. Furthermore, due to
its normal margins, one can expect
to be
well-behaved and easy to estimate. Under
mild assumptions,
and its partial
derivatives up to the second order are found
to be bounded on R2, in this case copula
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density is unbounded. If FST refer to copula
normal distribution , then we can write
C, and the variables (S,T) are standard
Sklar's theorem as equation below :
(
)
When differentiate FST with respect to s and t , we get the joint density of (s,t)
(
)
Where is standard normal density Inverting this equation yields.
For any
, therefore, any estimator ̂
density estimate on the interior of I .
̂
̂
Where the symbol
is refer to the
transformation idea. When appropriate,
̂ can alternatively be defined by
continuity at the limits of . This
transformation-based estimator has a
number of appealing qualities. Because
∫∫ ̂
Then, through transformation in variables
automatically generates a Copula
is not defined for
cannot allocate any probability
outside . Also, if
is a true density
function, in the sense that
and
and
∫∫
̂
̂
,
According the bivariate kernel density estimator, which we shall denote by ̂
the copula:
(
)
) (
(
for all
The first basic idea we should use the
standard kernel density estimator such as
̂ .Specifically, we use the estimate as:
̂
∑
when apply to
)
⁄
(
)
|
|
Where K is a bivariate kernel function and
is symmetric positive –definite matrix, and
{
}
not Available, and
as well. Instead, one must
Is the transform domain sample but
make use of
,( ̂
(̂ ) ̂
( ̂ ))
That pseudo-transformed sample as a result, the feasible form ̂
is
⁄
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Fatimah Hashim Falhi, Munaf Yousif Hmood/Iraqi Statisticians Journal / Vol. 1, Issue 1, 2024: 24-42
̂
⁄
∑
̂
/
̂
.
|⁄
|
Based on equation (11), this leads to a "probit transform kernel copula density estimator". [7][10]
̂
⁄
̂
∑ (
/)
.
̂
|⁄
|
As a result, the asymptotic equation for the
parameter of probit transformation is also
obtained. The bias and variance of this
̂
((
)
((
)
(
Where
∫
The variance is
(̂
)+
)+
)
,
*
(
*
[
{
)]-
| |
method for copula density estimator are in
the following form, respectively
(
)
(
}
⁄
(
| |
Where
∫
Then the variance of probit transformation copula density as below
( ̂
)
| |
⁄
When we using standard normal distribution
of kernel density and normal distribution for
density function then
[
⁄
(
)
)
)
)
| |
⁄
. Where
d represents a number of variables
Observe that
]
̂
3. Wavelet Copula Density Estimation (WCDE):
3.1Wavelets:
Wavelets are an extension of Fourier
analysis in that both seek to express
complex functions using the sum of simple
ones. Wavelet theory, on the other hand,
came considerably later than Fourier
analysis. [30] [31]
Wavelets
have
accomplished
impressive acceptance in earth sciences [32]
[33]. Wavelets have been used successfully
in a variety of application, including
numerical analysis, engineering, signal and
image processing,
statistics, and
geophysics. We will use mathematical
creation of wavelets discrete type
transformations, first, provide a detailed of
the space
from the standpoint of the
multi-resolution analysis.
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Fatimah Hashim Falhi, Munaf Yousif Hmood/Iraqi Statisticians Journal / Vol. 1, Issue 1, 2024: 24-42
Multi-resolution is a method for describing
the building of spaces and providing an
analytical explanation of the components
and bases of these spaces. Let us first
construct the square-integrable function,
often known as the space of Lebesgue
measurable functions, which is written as
and defined as [34]
|
{
} [35].
∫ |
A wavelet is a mathematical function tool
used to divide a given function into
compounds of different frequencies and
explore each configuration using the
appropriate solution for each measurement.
These tiny waves display information and
data in time and frequency domains. The
continuity of their signal is limited in two
variables: Unlike the sine function, which
extends between
, the wavelet
function is irregular and asymmetric.
Wavelet is defined mathematically as a real
value function on the real axis that fluctuates
up and down consistently around zero. [36]
[37] in other words it is defined as a signal
(
)
√
Where a and b are dilation and translation
parameters,
refer to mother wavelet
refer to daughter wavelet
There are two types of wavelet
transforms: continuous wavelet transforms
and discrete wavelet transforms.
of limited time length (continuity) with an
average value of zero. The wavelet
transform is based on the pressure of the
wavelet to be processed with two functions:
the first is the mother wavelet function Ψ(x)
to obtain a set of coefficients characterized
by the wavelet coefficients or detailed
coefficients D(s,t), and the second is the
scaling function ∅(x), also called the father's
function, to obtain the approximate
coefficients A(s,t) [38].
Then , we approximate the signal using
wavelets and find a group of wavelet
subgroups that are constructed from
expansion or compression and shifting of the
original wavelet and represent the signal or
data to be analyzed. In other words, the
process is the transformation of large-scale
measurements into precise measurements by
aggregating these data or signals. The main
result of the transformation process is the
mother wavelet function defined as: [39]
[40]
To approximate the probability density
function, the probability density function is
decomposed into a set of infinite functions
(daughter wavelets) in the time domain on
an orthonormal basis by a scaling function
(father wavelet) and a wavelet function
(mother wavelet) [39].
The approximation is defined as:
and
⁄
⁄
(
)
(
)
In this study, we use mother and father Daubechies wavelets [40].
3.2.Wavelet – Copula Estimation:
In this section, it will be referred to as
step uses the decomposed input variables to
"wavelet- copula", and the procedure can be
estimate the copula density function. Since
easily performed in two steps:
modeling dependence by copula is sensitive
The first step involves using wavelet
to the marginal model, a major innovation of
analysis to decompose variables. The second
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Fatimah Hashim Falhi, Munaf Yousif Hmood/Iraqi Statisticians Journal / Vol. 1, Issue 1, 2024: 24-42
dependence structure. In general, the
wavelet analysis of the second - order
function is
a hat allows you to
analyze
this
mapping
infinitely
simultaneously number of resolution levels
j = 0, 1, . . . .
The decomposition at any level
is
given by
At every level
, the decomposition
is given by
the procedure is the combination of wavelet
analysis with copula models.
Copula density estimates are constructed
using wavelet analysis. This process is easy
to implement using out-of-the-box wavelet
tools and is based on algorithms that
automatically deal with boundary effects.
Pseudo-samples
, measured on
arbitrary divisions of the unit square, are a
more promising approach [41].
Wavelet-based estimation of copula
density helps explain the underlying
so that
∑
is a trend (or approximation) and
∑ (∑
)
∑
is a collection of three sorts of details:
vertical edges, horizontal edges, and oblique
(corner of the square). In this form, the
with
and
,
coefficients
and
are unique for each choice of
∑
. For all j
the functions
N and
and
,
,
and
are defined as follow:
in terms of a certain scaling function, a
corresponding wavelet, and their locationscale transformations provided by .
⁄
(
⁄
and
(
for any
, and
. The functions
φ and ψ (the father and mother wavelet
functions respectively) are defined by Many
technical limitations have to be achieved. To
ensure that the family of position scales they
create constitutes an orthonormal system
)
)
of , the set of square-integrable functions.
The selection of each pair (φ, ψ) yields a
separate multiresolution analysis with the
required degree of regularity. In this study
assumed to have compact support [0, L] as
is the case for the widely utilized and
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provides an overview of this viewpoint.
[40][41]
A wavelet representation is
distinguished by the fact that the trend at
level
Consistent with the trend at
The actual copula
Assume that
random sample
distribution
.
(
)
was detected with
(
)
by setting
is a
the unknown
empirical are
)
∑
∑
(∑
from
The
level , highlighted by horizontal, vertical,
and diagonal features corresponding to level
.in other words
represented by
related to F and G
as in (4) for each
. The set
{
}
is the orthonormal basis of
for any
arbitrary
. Given a copula density c,
it may be expanded as (3) with
∫ ∫
According to Eq. (15), the change in variables
yields
{
∫ ∫
where
is the expectation based on the
original observations' common distribution
∑
distributions
)
(
W
here
and
are the ranks of
and
respectively .
Let
and
be the corresponding
wavelet for a given scaling function. Both
functions are considered real-valued and
compactly support [0, L] for some L > 0.
For
each
j
N,
define
̃
and
(
)
(
)}
If
are unknown, a nonparametric is
generated by substituting F and G with their
empirical distribution function,
.
The estimator is therefore rank-based, i.e.
∑
A wavelet-based estimate of c is then given by:
∑
̃
̃
where the smoothing index of the technique
is denoted by the number . It is worth
noting that, is not always the copula density,
̃ just as an empirical copula is not a
copula. [1]
̃
In particular, it can be
negative in the section of the domain, so that
(35)
it cannot be merged into 1. When you want
an estimate of the intrinsic copula density, it
can be obtained by truncating and
normalizing ̃
From a numerical standpoint, it is crucial to
notice that the sum over k in (18) is finite
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since the wavelet is supported by compact
support. Consequently, in reality. Only
c terms must be computed in the
special situation when the copula density
must be estimated at a single point
. For these reasons, the
procedure's performance is determined by
the level
selected. The latter should be
determined in the most efficient method
possible [39].
5. Discussion and results:
(1) We simulate five different random
samples (n= 32,64,128,256, 512) with
replication (r=1000).
(2) Generate X, Y variables from a
uniform distribution.
(3) The marginal distributions of the
random variables X and Y (F and G) are
uniformly simulated.
(4) Finding the probit transformation of
the observations of the variables that
were generated in step 2.
(5) Determine the number of vanishing
moments at 4 degree.
(6) For the dependence structure, we
consider four copula function (Gaussian ,
Frank ,Tawn, and Joe ) ,with Kendall’s
tau τ = 0.7,0.5,0.3. as shown in Tables
from 1 to 12.
Tables from 1 into 12 represent the
estimated root mean squares error of the
copula density functions for nonparametric
estimation methods and Akaike criteria and
logarithm maximum likelihood criteria
(LogL) at a correlation level tau =
0.7,0.5,0.3
respectively
with
1000
repetitions for each experiment that were
used to determine the performance of the
best estimation method it was found that the
best estimation method for the copula
density function in the case of strong and
weak correlations and for all sample sizes
and for four copulas(Gaussian, Frank, Tawn,
and Joe).The
method was probit
transformation for all sample sizes and for
all four copulas is the best at tawn and Joe
copulas when tau is strong but at Frank
copula at small sample size. While the
method (WCDE) is the Best at Gaussian
copula in all level correlation ant at all
sample size.
In medium and weak correlation, it was the
method (WCDE) at all Frank, Tawn and Joe
copulas function.
The 3D plot of the real copula functions
(Gaussian, Frank, Tawn, and Joe) are
illustrated in Figures (1, 2, 3, and 4) below,
in addition to the preface shapes for each of
them using the (PTE, WCDE) methods. It
can be noted, through 3D figures, that the
distribution of the observations of the copula
function estimated by the (WCDE) method
was accurate at the edges while it was less
accurate at the center for all functions. It is
also evident from the three-dimensional
figures that the probability density function
of the real (Gaussian) copula function is
characterized by the similar concentration of
observations at the center and at the edges,
with the withdrawal of observations towards
the tail and its relatively little expansion at
the center. Through the three-dimensional
figure, the (WCDE) smoothing of the
Gaussian function was more flat at the
center and more congruent at the tails
(extremities) when compared to the real
probability density function. Besides, when
estimating the copula function (Frank, tawn,
and Joe), the smoothing of the probability
density functions was less flat at the center,
but it was more withdrawn towards the tails
despite the presence of a great match
between the smoothed and the real
functions. Additionally, despite having
observed that the smoothed and real
functions had a significant match, the
smoothing of the probability density
functions while estimating the copula
function (Frank, Tawn, and Joe) was less
flat. In general, it can be said that the
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Fatimah Hashim Falhi, Munaf Yousif Hmood/Iraqi Statisticians Journal / Vol. 1, Issue 1, 2024: 24-42
smoothing when estimating the copula
function (Gaussian) is slightly better than
the smoothing when estimating the copula
functions (Frank,Tawn, Joe).
Table 1: Root-mean square error,(AIC)criterion and logarithm likelihood criteria for Gaussian copula when
Gaussian
Sample size
32
64
128
256
512
Method
RMSE
AIC
LOGL
PTE
WCDE
PTE
WCDE
PTE
WCDE
PTE
WCDE
PTE
WCDE
0.29933
0.18843
0.23146
0.16175
0.21907
0.13695
0.22168
0.0767
0.18511
0.03925
-38.599
-57.4329
-97.1761
-116.785
-215.913
-414.451
-374.616
-494.185
-771.349
-1314.73
22.71667
29.8099
49.99241
59.63232
109.6572
208.0832
194.7401
248.3973
387.8594
658.6745
Table 2: Root-mean square error,(AIC)criterion and logarithm likelihood criteria for Frank
Frank
Sample size
32
64
128
256
512
Method
PTE
WCDE
PTE
WCDE
PTE
WCDE
PTE
WCDE
PTE
WCDE
RMSE
AIC
copula when
LOGL
0.15321
0.16953
0.15021
0.16593
0.14841
0.22525
0.14168
0.05901
0.14168
-81.6971
-73.1424
-95.0123
-122.257
-280.421
-260.514
-458.297
-485.975
-485.297
41.67683
37.44939
48.9454
62.16542
141.4846
131.355
230.3296
244.9677
244.3296
0.05179
-1260.37
631.4375
Table 3: Root-mean square error,(AIC)criterion and logarithm likelihood criteria for Tawn copula when
Tawn
Method
RMSE
AIC
LOGL
Sample size
PTE
0.15321
-81.6971
41.67683
32
WCDE
0.19771
-70.7703
40.02418
PTE
0.15021
-95.0123
48.9454
64
WCDE
0.18102
-83.0877
42.97897
PTE
0.14841
-280.421
141.4846
128
WCDE
0.17395
-240.125
127.4758
PTE
0.14168
-485.297
244.3296
256
WCDE
0.17055
-414.966
215.1595
PTE
0.14168
-1000.297
501.3296
512
WCDE
0.14552
-485.56
244.807
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Fatimah Hashim Falhi, Munaf Yousif Hmood/Iraqi Statisticians Journal / Vol. 1, Issue 1, 2024: 24-42
Table 4: Root-mean square error,(AIC)criterion and logarithm likelihood criteria for Joe copula when
Joe
Method
RMSE
AIC
LOGL
Sample size
PTE
0.17041
-86.6256
43.92094
32
WCDE
0.59031
-60.2817
30.79977
PTE
0.15969
-117.411
59.98481
64
WCDE
0.42713
-63.8449
32.46232
PTE
0.15494
-214.156
108.7846
128
WCDE
0.38647
-67.469
36.21485
PTE
0.14883
-453.003
228.3562
256
WCDE
0.37677
-248.321
126.1468
PTE
0.14246
-872.756
438.4459
512
WCDE
0.20222
-435.536
220.0326
Table 5: Root-mean square error,(AIC)criterion and logarithm likelihood criteria for Gaussian copula when
Gaussian
Method
RMSE
AIC
LOGL
Sample size
PTE
0.64513
-14.0259
9.10046
32
WCDE
0.55886
-18.9267
11.26035
PTE
0.51196
-50.1634
27.14061
64
WCDE
0.46729
-66.2573
34.80296
PTE
0.49618
-102.509
53.73536
128
WCDE
0.45667
-113.008
58.77383
PTE
0.44895
-180.244
95.50428
256
WCDE
0.35513
-216.31
111.0381
PTE
0.42957
-342.784
174.5219
512
WCDE
0.25761
-358.82
182.4047
Table 6: Root-mean square error,(AIC)criterion and logarithm likelihood criteria for Frank copula when
Frank
Method
RMSE
AIC
LOGL
Sample size
PTE
0.42059
-16.5343
10.4112
32
WCDE
0.48065
-25.2846
14.28811
PTE
0.41134
-81.704
42.66972
64
WCDE
0.44904
-59.8056
31.32155
PTE
0.39342
-145.273
74.18602
128
WCDE
0.36845
-148.747
76.70922
PTE
0.38824
-221.718
113.3838
256
WCDE
0.27853
-249.719
127.3506
PTE
0.38815
-484.543
245.4771
512
WCDE
0.24437
-644.141
324.3424
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Table 7: Root-mean square error,(AIC)criterion and logarithm likelihood criteria for Tawn copula when
Tawn
Method
RMSE
AIC
LOGL
Sample size
PTE
0.42059
-16.5343
10.4112
32
WCDE
0.54244
-8.03816
6.34288
PTE
0.41134
-59.704
31.66972
64
WCDE
0.49021
-45.6272
25.02905
PTE
0.39342
-148.273
76.18602
128
WCDE
0.48712
-121.104
62.55991
PTE
0.38824
-195.718
100.3838
256
WCDE
0.18847
-221.834
113.3944
PTE
0.38815
-484.543
245.4771
512
WCDE
0.15348
-986.863
494.9124
Table 8: Root-mean square error,(AIC)criterion and logarithm likelihood criteria for Joe copula when
Joe
Method
RMSE
AIC
LOGL
Sample size
PTE
0.42297
-20.1695
11.94385
32
WCDE
0.53243
-19.6327
11.63855
PTE
0.41975
-55.5584
29.87232
64
WCDE
0.48461
-45.4424
25.01778
PTE
0.47773
-138.883
71.77869
128
WCDE
0.41914
-154.231
79.1561
PTE
0.45859
-217.66
111.6068
256
WCDE
0.37908
-250.979
127.8033
PTE
0.42731
-399.153
202.6058
512
WCDE
0.22915
-445.204
225.2306
Table 9: Root-mean square error,(AIC)criterion and logarithm likelihood criteria for Gaussian copula when
Gaussian
Method
RMSE
AIC
LOGL
Sample size
PTE
0.90599
-8.0097
6.17241
32
WCDE
0.71163
-10.79708
7.51427
PTE
0.80242
-25.8767
15.42052
64
WCDE
0.55352
-48.6918
26.3745
PTE
0.73139
-27.7824
17.43187
128
WCDE
0.42452
-109.671
56.80845
PTE
0.71448
-86.0686
46.49385
256
WCDE
0.41359
-271.17
137.6521
PTE
0.66743
-112.706
60.65348
512
WCDE
0.34671
-694.429
349.3544
Table 10: Root-mean square error,(AIC)criterion and logarithm likelihood criteria for Frank copula when
Frank
Method
RMSE
AIC
LOGL
Sample size
PTE
0.90599
-10.0097
7.17241
32
WCDE
0.7231
-10.2541
7.32794
PTE
0.80242
-25.8767
15.42052
64
WCDE
0.66538
-26.0737
15.78511
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Fatimah Hashim Falhi, Munaf Yousif Hmood/Iraqi Statisticians Journal / Vol. 1, Issue 1, 2024: 24-42
128
256
512
PTE
WCDE
PTE
WCDE
PTE
WCDE
0.73139
0.59807
0.71448
0.44442
0.66743
-27.7824
-65.7989
-86.0686
-253.352
-112.706
17.43187
35.52795
46.49385
128.7044
60.65348
0.3717
-415.229
209.8027
Table 11: Root-mean square error,(AIC)criterion and logarithm likelihood criteria for Tawn copula when
Tawn
Method
RMSE
AIC
LOGL
Sample size
PTE
0.7151
-9.30299
6.94363
32
WCDE
0.66441
-11.8305
8.07624
PTE
0.71314
-21.1549
13.43301
64
WCDE
0.64268
-21.6373
13.47802
PTE
0.6932
-56.5897
31.11079
128
WCDE
0.60952
-63.3696
33.94843
PTE
0.69094
-104.666
55.57158
256
WCDE
0.46535
-223.006
113.7815
PTE
0.67853
-171.691
90.04039
512
WCDE
0.15607
-854.443
428.785
Table 12: Root-mean square error,(AIC)criterion and logarithm likelihood criteria for Joe copula when
Joe
Method
RMSE
AIC
LOGL
Sample size
PTE
0.71841
-5.35054
5.12134
32
WCDE
0.62913
-12.0969
8.09585
PTE
0.68521
-21.1653
13.1342
64
WCDE
0.49163
-42.0206
23.06409
PTE
0.6379
-42.2662
24.05792
128
WCDE
0.41235
-113.394
58.62762
PTE
0.72761
-115.09
60.9177
256
WCDE
0.39105
-409.191
206.0382
PTE
0.71878
-232.741
119.9393
512
WCDE
0.39068
-334.387
169.9307
The figures 1,2,3 and 4 are explain the behavior for all four copula
Figure (1) three dimension Gaussian copula density when ( n=128, tau=0.7)
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Fatimah Hashim Falhi, Munaf Yousif Hmood/Iraqi Statisticians Journal / Vol. 1, Issue 1, 2024: 24-42
Figure (2) three dimension for Frank copula density when (n=128, tau=0.7)
Figure (3) three dimension for Tawn copula density when (n=128, tau=0.7)
Figure (4) three dimension for Joe copula density when (n=128, tau=0.7)
A copula functions were also drawn for the
data that were generated at several levels of
correlation. There are many drawing
methods to describe, interpret and analyze
the nature of the associative functions, but
the circular form, which is based on a
normal distribution, and the threedimensional form were chosen because they
are considered one of the most common and
used shapes in this field. The normal of the
probability density functions of the assumed
copula at the correlation level (0.7) can be
more clearly understood through threedimensional drawings.
Figure (1) above represents the assumed and
estimated three-dimensional shapes of the
Gaussian function when tau = 0.7 and n =
128. It is clear from it that the Gaussian
function is characterized by similar
dependency at the center and at the edges,
and that the observations of the probability
density function estimated by the (WCDE)
method are characterized by flatness.
Clearly at the center, but at the edges, the
smoothing was identical to the assumed
copula function.
Figure (2) above represents the estimates of
the Frank function when (n = 128 and
tau=0.7) and it shows that the Frank
function is characterized by similar
dependency at the center and at the edges,
noting that the difference in the distribution
of observations between the Gaussian and
Frank functions It is that the observations at
39
Fatimah Hashim Falhi, Munaf Yousif Hmood/Iraqi Statisticians Journal / Vol. 1, Issue 1, 2024: 24-42
the center in the Frank function are less flat
than in the Gaussian function
As for smoothing using the WCDE method,
we notice that the distribution of
observations fluctuates at the edges, but it is
better at the center.
Figure (3)above represents the assumed and
estimated probability density function of the
copula (Tawn) at the high level of
correlation and the sample size (128), and it
is clear from it that the copula function
(Tawn) is characterized by a large
concentration of observations at the right
side. the (WCDE) method are characterized
by flatness. Clearly at the center, but at the
edges, the smoothing was identical to the
assumed copula function.
Figure (4) above represents the probability
density function for the Joe association
when ( tau=0.7 and n=128), and it is clear
from it (that the assumed Joe copula
function has a right tail and that the
concentration of observations was clearly on
the left side, while the distribution of
observations in the middle appears flat)
Estimation using the PTE method: It is clear
that there is instability in the flatness of the
observations at the center, and that the
flatness of the observations at the right tail
and the left edge was more identical. As for
the (WCDE) method, the performance was
not good at the center, which was
characterized by instability because the
observations were too flat, or at the right
tail, where the concentration of observations
was greater, but the concentration of
observations at the left end was more similar
to the assumed form of the association.
6.Conclusion:
This study introduced copula estimation
using probit and wavelet transforms,
specifically employing Daubechies wavelets
of four degrees. The simulation results, were
obtained by employing four copulas
(Gaussian, Frank, Tawn, and Joe), for five
different sample sizes (n = 32, 64, 128, 256,
512) and evaluated based on three criteria
(RMSE, AIC, and LOGL), provide a
statistical measure for selecting the copula
that exhibits the best performance when
wavelets are used to estimate the copula
density function at high, medium, and low
correlation levels (tau = 0.7, 0.5, 0.3).
1- For all the copula functions that have
been studied for all nonparametric
estimation methods referred to in the
theoretical part and for all sample sizes and
at correlation levels, the value of the square
root of the mean square error (RMSE)
decreases as the sample size increases, while
the (LogL) criterion is as maximum as
possible, As for the Akaike criteria as
minimum as possible.
2-Estimation and identification of Copula
density functions based on rank-dependent
wavelets.
3- Presented the root mean square error and
developed a linear wavelet estimator.
4-Wavelet algorithms are quick to compute,
and simple to update and adapt to your
model.
5-The suggested linear wavelet density
estimator's numerical performance was
shown on simulated datasets.
6-Comparisons of generated data for various
sample sizes were also explained. However,
wavelet-based copula function estimators for
the underlying dependency structure, do not
cover the basic criteria of parametric
models.
7- The method of estimating the copula
density function using (PTE) is the best
method for the Frank and Joe copulas
function.
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