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Copyright © 2017 IJISM, All right reserved
182
International Journal of Innovation in Science and Mathematics
Volume 5, Issue 6, ISSN (Online): 2347–9051
Parameters Estimation of Bivariate Modified Weibull
Lutfiah Ismail Al turk1*, Mervat K. Abd Elaal1, 2 and Shatha H. Ba-Hamdan1 1Statistics Department, Faculty of Sciences, King Abdulaziz University, Jeddah, Kingdom of Saudi Arabia.
2Statistics Department, Faculty of Commerce, Al-Azhar University, Girls Branch, Cairo, Egypt.
Date of publication (dd/mm/yyyy): 22/11/2017
Abstract – The modified Weibull distribution generalizes
four of the most important distributions which are:
exponential, Rayleigh, linear failure rate and Weibull. Those
distributions are the most commonly used for analyzing
lifetime data as they have nice physical interpretations and
several desirable properties. This paper construct a new
bivariate modified Weibull distribution. Several bivariate
special cases (linear failure rate, Weibull, exponential and
Rayleigh) based on three types of copula (Gaussian, Plackett
and Farlie-Gumbel-Morgenstern) are considered. Maximum
likelihood method of estimation of the unknown parameters of
the proposed bivariate distributions are considered. The
Monte Carlo simulation study is used to investigate and
compare the different estimates of different sample sizes for
each bivariate distribution of the various copulas functions.
Keywords – Bivariate Modified Weibull Distribution,
Copula, Linear Failure Rate Distribution, Maximum
Likelihood Estimation Method, Weibull Distribution.
I. INTRODUCTION
Constructing multivariate distribution is known as one of
the classical fields of statistics science. Therefore, it
continues to be an active field of research. Several authors
have developed different procedures to derive multivariate
or bivariate distributions, see for examples, Hougaard [4],
Genest et al. [2], Lu et al. [8], Johnson et al. [5]. Copula is a
way of formalizing dependence structures of random
vectors. In many of application fields, copula models have
become increasingly popular during the last 10 years.
Furthermore, in many cases of statistical modeling, it is
essential to obtain the joint distribution between multiple
random variables. Copula models can be used to obtain the
joint distribution from marginal distributions, but their joint
distribution may not be easy to be obtained from those
marginal distributions without using the copula. In this
paper, we will discuss constructing bivariate distribution of
a special case of the modified weibull distribution. Sarhan
and Zaindin introduced a modified Weibull distribution, his
modification is a generalization of the Weibull distribution.
This distribution has a wide domain of applicability,
particularly in analyzing lifetime data and most used in
reliability and life testing [13]. The Weibull, exponential,
linear failure rate and Rayleigh distributions are the most
widely used distributions in reliability and life testing, it also
has many nice physical interpretations and desirable
characteristics which enable them to be used frequently, for
more details see [6].
"Copula" this word is used for the first time the statistical
or mathematical sense in theory " Sklar " by Sklar (1959)
Nelsen, Roger B, see [10]. Relationships of copulas to other
work is described in Nelsen [10]. Copulas is popular in
many fields, in finance, copulas are used in credit scoring,
risk management, default risk modeling, derivative pricing
and asset allocation and in other areas; see Nelsen [11].
II. THE MODIFIED WEIBULL DISTRIBUTION
The modified Weibull distribution with the parameters
(α, β, γ), such that α, β ≥ 0, γ > 0 and α + β > 0. Here α is scale parameter and β, γ are shape parameters. The CDF for
modified Weibull distribution is given by:
( , , , ) 1 , 0 , 0, , 0x xF t e x (1)
The PDF of the modified Weibull distribution is as
follows:
1 ( , , , ) ( ) , 0 , 0, , 0.x xf x x e x (2)
and the hazard function (HRF) of the modified Weibull
distribution is: 1 ( , , , ) , 0 , 0, , 0.h t x x (3)
Modified Weibull distribution is generalized to the
following distributions:
1. Linear failure rate distribution, LFR (α, β) when γ = 2;
2. Weibull distribution W (β, γ) when α = 0;
3. Exponential distribution, E (α) when β = 0; and
4. Raleigh distribution, R (β) when α = 0, γ = 2. And by
compensation in Equations (1), (2) and (3) the PDF,
CDF and hazard function for each distribution can be
obtained.
III. BIVARIATE COPULA
Copula is a useful tool for understanding relationship
between multivariate variables and important tool for
describing the dependence structure among random
variables, represent different dependencies with different
copulas. Copula is a powerful and flexible method to
analyze and produce large classes of multivariate
distributions. These distributions are necessary to estimate
the parameters that govern interdependent processes.
Theorem (Sklar in two Dimensions) Let H be a joint distribution function with margins F and
G. Then there exists a copula C such that for all x, y in R,
( , ) ( ( ), ( )).H x y C F x G y (4)
If F and G are continuous, then C is unique; otherwise, C
is uniquely determined on Ran F× Ran G. Conversely, if C is a copula and F and G are distribution functions, then the
function H defined by (10) is a joint distribution function
with margins F and G. A bivariate copulas can be defined as follows: let X be a
2-dimensional random variable with parametric univariate
marginal distributions 𝐹𝑋𝑗(𝑥𝑗 , 𝛿𝑗), j = 0, 1. And, let a copula
𝐶 = {𝐶𝜃 , 𝜃 ∈ Θ}, belong to a parametric family. By Sklars theorem the distribution of X can be expressed
as:
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183
International Journal of Innovation in Science and Mathematics
Volume 5, Issue 6, ISSN (Online): 2347–9051
1 21 2 1 1 2 2 ( , ) { ( , ), ( , ), }.X X XF x x C F x F x (5)
And, the density:
1 2
2
1 2 1 2 1 1 2 2
1
( , ; , ) `{ ( , ), ( , ), } ( , )X X j jj
f x x C F x F x f x
(6)
Since 2
1 2
1 2
1 2
( , ) ̀( , )
C u uC u u
u u
Where: 𝑢𝑗 = 𝐹𝑋𝑗(𝑥𝑗 , 𝛿𝑗), j = 0, 1.
In this paper, three families of copulas (Gaussian, Plackett
and Farlie-Gumbel-Morgenstern) are used to derive a
bivariate modified Weibull distribution, bivariate linear
failure rate distribution, bivariate Weibull distribution,
bivariate exponential distribution and bivariate Rayleigh
distribution.
The Gaussian copula function proposed by Lee [7]. The
expression of distribution function for Gaussian copula is: 1 1 ( , , ) ( ( ), ( ), ).GC u v u v
1 1
2 2
1 1 2 2( ) ( ) 2
1 22
1exp{ ( 2 )}
2(1 ). (7)
2 1
v ux x x x
dx dx
where Φρ denotes the bivariate standard normal distribution
function with correlation parameter ρ ∈ (−1, 1) and Φ−1 denotes the inverse of univariate standard normal
distribution function. The density of the bivariate Gaussian
copula is differentiation of 𝐶𝐺(𝑢, 𝑣, 𝜌), such that:
2 2
2
2
1exp { ( 2 )}
1(1 )` ( , , ) . (8)
2 1G
u uv v
C u v
The second proposed copula in this paper is the plackett
copula. It is proposed by Plackett [12]. The expression of
distribution function for Plackett copula is:
2[1 ( 1) ( )] 4 ( 1)1 ( 1) ( )( , ) ( 9 )
2( 1) 2( 1)
P P PP
P
P P
u v u vu vC u v
Where 𝑢, 𝑣 ∈ Π and 𝜃𝑃 ≥ 0 is a dependence parameters. The density function of the bivariate plackett copula is
differentiation of 𝐶𝑃(𝑢, 𝑣, 𝜃𝑃), such that:
2 3\2
[1 ( 1) ( 2 )]` ( , , . ) (10)
{[1 ( 1) ( )] 4 (1 )}
P P
p P
P P P
u v uvC u v
u v uv
No closed form exists for the value of Kendall’s τk. Instead, Spearman’s ρs is given by:
1for ,)1(
ln2
1
1)(
2
P
P
PP
P
PPs
The third proposed copula in this paper is the FGM copula
that discussed by Morgenstern [9], Gumbel [3] and Farlie
[1]. The expression of distribution function for FGM copula
is:
( , , ) (1 ) (1 ) (11)FGM F FC u v uv uv u v
Where 𝑢, 𝑣 ∈ Π and 𝜃𝐹 ∈ [−1,1] is a dependence parameters.
The density function of the bivariate FGM copula is
differentiation of 𝐶𝐹𝐺𝑀(𝑢, 𝑣, 𝜃𝐹), such that:
` ( , , ) 1 (1 2 ) (1 2 ) (12)F G M F FC u v u v
Two useful relationships exist between F and,
respectively, Kendall’s τk And Spearman’s ρs is given by:
. 3
)( and 9
2)( FFs
FFk
3.1. Bivariate Modified Weibull (BMW) Distribution
Based on Copulas From Equation (2) the density function of bivariate
modified Weibull distribution based on Gaussian copula can
be written as: 1 2
1 1 1 1 1 2 2 2 2 21 1
1 1 1 1 1 1 2 2 2 2
2 2
2
2
( , ) [( ) ] [( ) ]
1exp{ ( 2 )}
2(1 )
2 1
x x x xf x x x e x e
u uv v
(13)
The density function of bivariate modified Weibull
distribution based on Plackett copula can be written as: 1 2
1 1 1 1 1 2 2 2 2 21 1
1 1 1 1 1 1 2 2 2 2
2
( , ) [( ) ] [( ) ]
[1 ( 1) ( 2 )]
{[1 ( 1) ( )] 4
x x x x
P P
P
f x x x e x e
u v uv
u v
3\2 (14)
(1 )}P Puv
And the density function of bivariate modified Weibull
distribution based on FGM copula can be written as: 1 2
1 1 1 1 1 2 2 2 2 21 1
1 1 1 1 1 1 2 2 2 2 ( , ) [( ) ] [( ) ]
[ 1 (1 2 ) (1 2
x x x x
F
f x x x e x e
u v
)] (15)
3.2. Bivariate Special Cases of Modified Weibull (BMW) Distributions based on Copulas
The bivariate linear failure rate distribution based on
Gaussian copula can be expressed by: 2 2
1 1 1 1 2 2 2 2
1 1 1 1 1 2 2 2
2 2
2
2
( , ) [( 2 ) ] [( 2 ) ]
1exp{ ( 2 )}
2(1 ) (16)
2 1
x x x xf x x x e x e
u uv v
Whereas, the density function of bivariate linear failure
rate distribution based on Plackett copula can be written as:
And the density function of bivariate linear failure rate
distribution based on FGM copula can be written as: 2 2
1 1 1 1 2 2 2 2
1 1 1 1 1 2 2 2( , ) [( 2 ) ] [( 2 ) ]
[1 (1 2 )(1 2 )] (18)
x x x x
F
f x x x e x e
u v
The density function of bivariate Weibull distribution
based on Gaussian copula can be written as: 1 2
1 1 1 2 2 21 1
1 1 1 1 1 2 2 2
2 2
2
2
( , ) [( ) ] [( ) ]
1exp{ ( 2 )}
2(1 ) (19)
2 1
x xf x x x e x e
u uv v
2 21 1 1 1 2 2 2 2
1 1 1 1 1 2 2 2
2 3\2
( , ) [( 2 ) ] [( 2 ) ]
[1 ( 1)( 2 )] (17)
{[1 ( 1)( )] 4 (1 )}
x x x x
P P
P P P
f x x x e x e
u v uv
u v uv
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184
International Journal of Innovation in Science and Mathematics
Volume 5, Issue 6, ISSN (Online): 2347–9051
While, the density function of bivariate Weibull
distribution based on Plackett copula can be written as:
1 21 1 1 1 2 2 21 1
1 1 1 1 1 2 2 2
2 3\2
( , ) [( ) ] [( ) ]
[1 ( 1)( 2 )] (20)
{[1 ( 1)( )] 4 (1 )}
x x
P P
P P P
f x x x e x e
u v uv
u v uv
Also, the density function of bivariate Weibull
distribution based on FGM copula can be written as:
1 21 1 1 2 2 21 1
1 1 1 1 1 2 2 2( , ) [( ) ] [( ) ]
[1 (1 2 )(1 2 )] (21)
x x
F
f x x x e x e
u v
The density function of bivariate exponential distribution
based on Gaussian copula can be written as:
1 1 2 2
1 1 1 2
2 2
2
2
( , ) ( ) ( )
1exp { ( 2 )}
2(1 ) (22)
2 1
x xf x x e e
u uv v
The density function of bivariate exponential distribution
based on Plackett copula can be written as:
1 1 2 2
1 1 1 2
2 3\2
( , ) ( ) ( )
[1 ( 1)( 2 )] (23)
{[1 ( 1)( )] 4 (1 )}
x x
P P
P P P
f x x e e
u v uv
u v uv
And the density function of bivariate exponential
distribution based on FGM copula can be written as:
1 1 2 2
1 1 1 2( , ) ( ) ( ) [1 (1 –2 ) (1 2 )]x x
Ff x x e e u v (24)
The density function of bivariate Rayleigh distribution
based on Gaussian copula can be written as: 2 2
1 1 2 2
1 1 1 1 2 2
2 2
2
2
( , ) [(2 ) ] [(2 ) ]
1exp{ ( 2 )}
2(1 ) (25)
2 1
x xf x x x e x e
u uv v
Also, the density function of bivariate Rayleigh
distribution based on Plackett copula can be written as:
2 21 1 2 2
1 1 1 1 2 2
2 3\2
( , ) [(2 ) ] [(2 ) ]
[1 ( 1)( 2 )] (26)
{[1 ( 1) ( )] 4 (1 )}
x x
P P
P P P
f x x x e x e
u v uv
u v uv
Then, the density function of bivariate Rayleigh
distribution based on FGM copula can be written as: 2 2
1 1 2 2
1 1 1 1 2 2( , ) [(2 ) ] [(2 ) ]
[1 (1 2 )(1 2 )] (27)
x x
F
f x x x e x e
u v
Graphical Description of the Bivariate Linear Failure
Rate Distribution 1. The PDF of the BLFR distribution based on Gaussian
copula for different values of the parameters are plotted
in Figure (1) when 𝛽1 = 𝛽2 > 1 and when 𝛽1 = 𝛽2 < 1 with fixed value of 𝛼𝑗 and ρ.
Fig. 1. Plots the PDF of the BLFR distribution for different
values of the parameters: (a) α1 = α2 = 0.5, β1 = β2 = 1.2, ρ =
0.7 and (b) α1 = α2 = 0.5, β1 = β2 = 0.2, ρ = 0.7
2. Figure (2) displays the contour plots of the BLFR density function based on Gaussian copula for four different levels of
dependence, assuming the parameters 𝛼1 = 𝛼2 = 0.5, 𝛽1 =𝛽2 = 1.2.
Fig. 2. Contour plots of BLFR distribution for different
values of ρ
3. The PDF of the BLFR distribution based on Plackett copula for different values of the parameters are plotted
in Figure (3) when 𝛽1 = 𝛽2 > 1 and when 𝛽1 = 𝛽2 < 1 with fixed value of 𝛼𝑗 and θ.
Copyright © 2017 IJISM, All right reserved
185
International Journal of Innovation in Science and Mathematics
Volume 5, Issue 6, ISSN (Online): 2347–9051
Fig. 3. Plots the PDF of the BLFR distribution for different
values of the parameters (a) α1 = α2 = 0.5, β1 = β2 = 1.2, θ =
0.2 and (b) α1 = α2 = 0.5, β1 = β2 = 0.2, θ = 0.2
Figure (4) displays the contour plots of the BLFR density
function based on Plackett copula for four different levels of
dependence, assuming the parameters 𝛼1 = 𝛼2 = 0.5, 𝛽1 =𝛽2 = 1.2.
Fig. 4. Contour plots of BLFR distribution for different
values of θ
5. The PDF of the BLFR distribution based on FGM
copula for different values of the parameters are plotted
in Figure (5) when 𝛽1 = 𝛽2 > 1 and when 𝛽1 = 𝛽2 < 1 with fixed value of 𝛼𝑗 and θ.
Fig. 5. Plots the PDF of the BLFR distribution for different
values of the parameters (a) α1 = α2 = 0.5, β1 = β2 = 1.2, θ =
0.2 and (b) α1 = α2 = 0.5, β1 = β2 = 0.2, θ = 0.8
6. Figure (6) displays the contour plots of the LFR density
function based on FGM copula for four different levels
of dependence, assuming the parametersα1 = α2 =0.5, β1 = β2 = 1.2
Fig. 6. Contour plots of BLFR distribution for different
values of θ
Graphical Description of the Bivariate Weibull
Distribution 1. The PDF of the BW distribution based on Gaussian
copula for different values of the parameters are plotted
in Figure (7) when 𝛽1 = 𝛽2 𝑎𝑛𝑑 𝛾1 = 𝛾2 and 𝛽𝑖 < 𝛾𝑖 and when 𝛽1 = 𝛽2 𝑎𝑛𝑑 𝛾1 = 𝛾2 and 𝛽𝑖 > 𝛾𝑖 with fixed value ρ.
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International Journal of Innovation in Science and Mathematics
Volume 5, Issue 6, ISSN (Online): 2347–9051
Fig. 7. Plots the PDF of the BW distribution for different
values of the parameters (a)β1 = β2 = 1.3, γ1 = γ2 = 1.5, ρ =
0.8 and (b) β1 = β2 = 1.6, γ1 = γ2 = 1.1, ρ = 0.8
2. Figure (8) displays the contour plots of the BW density
function based on Gaussian copula for four different
levels of dependence, assuming the parameters:
𝑎) 𝛽1 = 𝛽2 = 1.5 𝑎𝑛𝑑 𝛾1 = 𝛾2 = 1.3 and 𝑎) 𝛽1 =𝛽2 = 1.6 𝑎𝑛𝑑 𝛾1 = 𝛾2 = 1.1.
Fig. 8. Contour plots of BW distribution for different
values of 𝜌
3. The PDF of the BW distribution based on Plackett
copula for different values of the parameters are plotted
in Figure (9) when 𝛽1 = 𝛽2 𝑎𝑛𝑑 𝛾1 = 𝛾2 and 𝛽𝑖 < 𝛾𝑖 and when 𝛽1 = 𝛽2 𝑎𝑛𝑑 𝛾1 = 𝛾2 and 𝛽𝑖 > 𝛾𝑖 with fixed value θ.
Fig. 9. Plots the PDF of the BW distribution for different
values of the parameters (a)β1 = β2 = 1.3, γ1 = γ2 = 1.5, θ =
0.8 and (b) β1 = β2 =1.6, γ1 = γ2 = 1.1, θ = 0.8
4. Figure (10) displays the contour plots of the BW density
function based on Plackett copula for four different
levels of dependence, assuming the parameters 𝛽1 =𝛽2 = 1.5 𝑎𝑛𝑑 𝛾1 = 𝛾2 = 1.3
Fig. 10. Contour plots of BW distribution for different
values of θ
5. The PDF of the BW distribution based on FGM copula for different values of the parameters are plotted in
Figure (11) when 𝛽1 = 𝛽2 𝑎𝑛𝑑 𝛾1 = 𝛾2 and 𝛽𝑖 < 𝛾𝑖 and when 𝛽1 = 𝛽2 𝑎𝑛𝑑 𝛾1 = 𝛾2 and 𝛽𝑖 > 𝛾𝑖 with fixed value θ.
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International Journal of Innovation in Science and Mathematics
Volume 5, Issue 6, ISSN (Online): 2347–9051
Fig. 11. Plots the PDF of the BW distribution for different
values of the parameters (a)β1 = β2 = 1.3, γ1 = γ2 = 1.5, θ =
0.8 and (b)β1 = β2 = 1.6, γ1 = γ2 = 1.1, θ = 0.9
6. Figure (12) displays the contour plots of the BW density
function based on FGM copula for four different levels
of dependence, assuming the parameters
2.1,5.0 2121
Fig. 12. Contour plots of BW distribution for different
values of θ
Graphical Description of the Bivariate Exponential
Distribution 1. The PDF of the BE distribution based on Gaussian
copula for different values of the parameters are plotted
in Figure (13) when 1, 21 and when 1, 21
with fixed value of ρ.
Fig. 13. Plots the PDF of the BE distribution for different
values of the parameters (a) α1 = 2, α2 = 2.5, ρ = 0.7 and (b)
α1 = 0.9, α2 = 0.3, ρ = 0.7
2. Figure (14) displays the contour plots of the BE density
function based on Gaussian copula for four different
levels of dependence, assuming the parameter
121
Fig. 14. Contour plots of BE distribution for different
values of 𝜌
3. The PDF of the BE distribution based on Plackett
copula for different values of the parameters are plotted
in Figure (15) when 1, 21 and when 1, 21
with fixed value of θ.
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International Journal of Innovation in Science and Mathematics
Volume 5, Issue 6, ISSN (Online): 2347–9051
Fig. (15). Plots the PDF of the BE distribution for different
values of the parameters (a)α1 = 2 α2 = 2.5, θ = 0.8 and (b)
α1 = 0.9 α2 = 0.3, θ = 0.8
4. Figure (16) displays the contour plots of the BE density
function based on Plackett copula for four different
levels of dependence, assuming the parameters
1 21, 1.5.
Fig. 16. Contour plots of BE distribution for different
values of θ
5. The PDF of the BE distribution based on FGM copula
for different values of the parameters are plotted in
Figure (17) when 1 2, 1 and when 1 2, 1 with
fixed value of θ.
Fig. 17. Plots the PDF of the BE distribution for different
values of the parameters (a)α1 = 2, α2 = 2.5, θ = 0.7 and (b)
α1 = 0.9, α2 = 0.3, θ = 0.7
6. Figure (18) displays the contour plots of the BE density function based on FGM copula for four different levels
of dependence, assuming the parameters
.5.1,1 21
Fig. 18. Contour plots of BE distribution for different
values of θ
Graphical Description of the Bivariate Rayleigh
Distribution 1. The PDF of the BR distribution based on Gaussian
copula for different values of the parameters are plotted
in Figure (19) when 121 and when 121
with fixed value of ρ.
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International Journal of Innovation in Science and Mathematics
Volume 5, Issue 6, ISSN (Online): 2347–9051
Fig. 19. Plots the PDF of the BR distribution for different
values of the parameters (a)β1 = β2 = 2, ρ = 0.7 and (b) β1 =
β2 = 0.7, ρ = 0.7
2. Figure (20) displays the contour plots of the BR density
function based on Gaussian copula for four different
levels of dependence, assuming the parameter
1 2 1.5.
Fig. 20. Contour plots of BR distribution for different
values of 𝜌
3. The PDF of the BR distribution based on Plackett
copula for different values of the parameters are plotted
in Figure (21) when 121 and when 121
with fixed value of .
Fig. 21. Plots the PDF of the BR distribution for different
values of the parameters (a) β1 = β2 = 1.5, θ = 0.9 and (b) β1 = β2 = 0.9, θ = 0.9
4. Figure (22) displays the contour plots of the BR density
function based on Plackett copula for four different
levels of dependence, assuming the parameters
1 2 1.5.
Fig. 22. Contour plots of BR distribution for different
values of θ
5. The PDF of the BR distribution based on FGM copula
for different values of the parameters are plotted in
Figure (23) when 1, 21 and when 1, 21 with
fixed value of θ.
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International Journal of Innovation in Science and Mathematics
Volume 5, Issue 6, ISSN (Online): 2347–9051
Fig. 23. Plots the PDF of the BR distribution for different
values of the parameters (a)β1 = β2 = 1.5, θ = 0.9 and (b) β1 = β2 = 0.9, θ = 0.8
3. Figure (24) displays the contour plots of the BR density
function based on FGM copula for four different levels
of dependence, assuming the parameters .5.121
Fig. 24. Contour plots of BR distribution for different
values of θ
IV. MAXIMUM LIKELIHOOD ESTIMATION
METHOD
The maximum likelihood estimation (MLE) Method is
usually used to estimate all the parameters simultaneously.
In this paper, we used the MLE method to estimate the
parameters of our studied bivariate distributions. With the
density function of bivariate distribution given as:
. ),()()(),( vucygxfyxh
For a sample for size n, the likelihood function will be
given as:
. ))(),(()()(),()()(11
n
i
iiii
n
i
iiii yGxFcygxfvucygxfL
Compensation can be in the equation this product, it is
often use the fact that the logarithm is an increasing function
so-called log-likelihood function. The one-step procedure
estimates parameters by maximizing the log-likelihood for
the joint distribution, that is:
.))](),((ln)(ln)([ln
)],(ln)(ln)([ln
1
1
n
i
iii
n
i
iii
yGxFcygxf
vucygxfl
Officially, we can define the maximum likelihood
estimator (MLE) as the value θ such that:
. )|()|( xlxl
And, the first derivative of the log-likelihood function is
called Fisher’s score function, and is denoted by:
.)|(
)(
xlu
Note that by solving the system of equations we can find
maximum likelihood estimator by setting the result to zero:
.0)(
u
4.1 Maximum Likelihood Estimation for Bivariate
Modified Weibull Distribution based on Copulas: In this section, the ML method will be used to estimate
the unknown parameters of the BMWD distribution.
4.1.1 Bivariate Modified Weibull Distribution based
on Gaussian Copula: In this subsection, the MLE method of the unknown
parameters of the BMW distribution is discussed. Let
niXXX iii ,...,1 ),,( 21 be a bivariate random sample of
size n from BMW distribution by (25). Then the likelihood
function is given by:
))2()1(2
1exp( )12(
)(exp().(),,,,,,(
1
2
1
2
12
2
2
1 1 1
1
2211
n
i
ii
n
j
n
i
jijjij
n
i
jijjjjj
vvuu
xxxL jj
Therefore, the log likelihood function of equation above
given by:
)2()1(2
1 )12ln(
)()ln(),,,,,,(
1
2
1
2
12
2
2
1 1 1
1
2211
n
i
ii
j
n
i
n
i
jijjijjijjjjj
vvuun
xxxl jj
Differentiating the log likelihood function with respect to
and ,,jjj
and equating each result to zero, we get:
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11 1
1 (28)
j
n n
ji
i ij j j j ji
lx
x
Thus
1
1
1 (29)
n
j j j jini
ji
i
x
x
Similarly, and ,for jj
1
1
1 1
1
- 0 (30)
j
j
j
n
j ji ni
jinij
j j j ji
i
xl
x
x
Then:
1
1 1
1 (31)
j j
j
j n n
ji j ji
i i
x x
1
1 1
1
1
1 1
.( log( ) 1)
. log( ) 0 (32)
j
j
j
n n
j ji j ji
i i
n
jj j j ji
i
n n
j ji ji
i i
x xl
x
x x
3 2 2 2
1 1
2 2
( ) [( )] (1 )
0 (33)(1 )
n n
i i i i
i i
n u v u vl
Equations (32) and (33) will be solved numerically to
obtain the estimates of the parameters and j .
4.1.2 Bivariate Modified Weibull Distribution based
on Plackett Copula In this subsection, the MLE of the unknown parameters
of the BMW distribution is discussed. Let
niXXX iii ,...,1 ),,( 21 be a bivariate random sample of
size n from BMW distribution given by (26). Then the
likelihood function is given by:
)}-(1v4u-)]v1)(u-({[1
)]v2u-v1)(u-([1
)(exp().(),,,,,,(
n
1i3'2
ii
2
ii
iiii
2
1 1 1
1
2211
PPP
PP
j
n
i
jijjij
n
i
jijjjPjjjj xxxL
Therefore, the log likelihood function of equation above
given by:
n
1i
ii
2
ii
n
1i
iiii
2
1 1 1
1
2211
)}-(1v4u-)]v1)(u-(ln{[12
3
)])v2u-v1)(u-([1ln(
)()ln(),,,,,,(
PPP
PP
j
n
i
n
i
jijjijjijjjPjjjj xxxl
Differentiating the log likelihood function with respect to
𝜃𝑃 and equating result to zero, where we got an estimate of
the parameters and , j jj from equations (29), (31) and
(32).
(34) 1])1)(-)[(2(41)-4(
)1(4)]1)(-([12
3
])2()1(1[
)2()12(1
n
1i
n
1i 1
2
1
1
iiPiiiiPiiP
n
i
PPiiiiP
n
i
iiiiPP
n
i
iiiiP
P
vuvuvuvu
vuvu
vuvu
vuvul
Equations (34) will be solved numerically to obtain the
estimate of the parameter 𝜃𝑃. 4.1.3 Bivariate modified Weibull distribution based
on FGM copula In this subsection, the MLE of the unknown parameters of
the BMW distribution is discussed. Let
niXXX iii ,...,1 ),,( 21 be a bivariate random sample of
size n from BMW distribution given by (27). Then the
likelihood function is given by:
)21)(21(1
)(exp().(),,,,,,(
n
1i
2
1 1 1
1
222111
iiF
j
n
i
jijjij
n
i
jijjjF
vu
xxxL jj
Therefore, the log likelihood function of the above
equation is given by:
))2121( ln(1)(
)ln(),,,,,,(
n
1i1
2
1 1
1
222111
iiF
n
i
jijjij
j
n
i
jijjjF
v)(uxx
xl
j
j
By differentiating the equation above with respect to 𝜃𝐹
and equating result to zero F
will be obtained From Equation (36), where we are got the estimates of the
parameters 𝛼𝑗 , 𝛽𝑗 and 𝛾𝑗 from Equations (29), (31) and (32).
1
1
(1 2 )(1 2 )
0
1 (1 2 )(1 2 )
n
i i
i
n
FF i i
i
u vl
u v
(35)
1
1
(1 2 )(1 2 )F n
i i
i
u v
(36)
4.2 Maximum Likelihood Estimation for Bivariate
Linear Failure Rate Distribution based on Copulas In this section, the MLE method will be used to estimate
the unknown parameters of the BLFR distribution. 4.2.1 Bivariate Linear Failure Rate Distribution based
on Gaussian Copula In this subsection, the MLE of the unknown parameters of
the BLFR distribution is discussed. Let n..,,1i ),X,X(X i2i1i be a bivariate random sample of size
n from BLFR distribution given by (28), then the likelihood
function is given by:
n
i
ii
n
j
n
i
jijjij
n
i
jijj
vvuu
xxxL
1
2
1
2
12
2
2
1 1
2
1
2211
)2()1(2
1exp( )12(
)(exp().2(),,,,(
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Therefore, the log likelihood function of equation above
is given by:
)2()1(1
1)12ln(
)2ln(),,,,(
1
2
1
2
12
2
2
1 1 1
2
2211
n
i
ii
j
n
i
n
i
jijjijjijj
vvuun
xxxl
differentiating the log likelihood function with respect to
𝛼𝑗 , 𝛽𝑗 and 𝜌 and equating each result to zero, we get:
(37) 2
1
1 1
n
i
n
i
ji
jijjj
xx
l
Thus
1
1
12
n
j j jini
ji
i
x
x
(38)
Similarly, for βj and ρ
(39) 2
2
1 1
2
n
i
n
i
ji
jijj
ji
j
xx
xl
Then
(40)
2
1
11
2
n
i
ji
j
n
i
ji
j
xx
(41) 0 )1(
)1()][()(
22
1
2
1
223
n
i
ii
n
i
ii vuvunl
Equation (41) will be solved numerically to obtain the
estimate of the parameter ρ.
4.2.2 Bivariate Linear Failure Rate Distribution based
on Plackett Copula In this subsection, the ML estimation of the unknown
parameters of the BLFR distribution is discussed. Let
n...,,1i ),X,X(X i2i1i be a bivariate random sample of size
n from BLFR distribution given by (29). Then the likelihood
function is given by: 2
2
1 1 2 2
11 1
ni i i i
2 3'2i 1 i i i i
( , , , , ) ( 2 ).exp( ( )
[1 ( -1)(u v -2u v )]
{[1 ( -1)(u v )] -4u v (1- )}
n n
P j j ji j ji j ji
ij i
P P
P P P
L x x x
Therefore, the log likelihood function of equation above
given by:
)}-(1v4u-)]v1)(u-(ln{[1
-)])]v2u-v1)(u-([1ln([
)2ln(),,,,(
n
1i
3'2
ii
2
ii
iiii
n
1i
2
1 1 1
2
2211
PPP
PP
j
n
i
n
i
jijjijjijjP xxxl
By differentiating the log likelihood function with respect
to θP and equating result to zero we obtain Equation (42),
where we got the estimates of the parameters jj and
from equations (38) and (40).
1
1
n2
i 1 1
n
i 1
1 (2 1) ( 2 )
[1 ( 1) ( 2 )]
3
2 [1 ( -1)( )] 4 (1 )
4( -1) 4 2( )[( -1)( ) 1] (42)
n
P i i i i
i
n
PP P i i i i
i
n
P i i i i P P
i
P i i P i i i i P i i
u v u vl
u v u v
u v u v
u v u v u v u v
Equation (42) will be solved numerically to obtain the
estimate of the parameter θP.
4.2.3 Bivariate Linear Failure Rate Distribution based
on FGM Copula In this subsection, the MLE of the unknown parameters of
the BLFR distribution is discussed. Let
n...,,1i ),X,X(X i2i1i be a bivariate random sample f size n
from BLFR distribution given by (30). Then the likelihood
function is given by:
)21)(21(1
)(exp().2(),,,,(
n
1i
2
1 1
2
1
2211
iiF
j
n
i
jijjij
n
i
jijjF
vu
xxxL
Therefore, the log likelihood function of the equation
above is given by:
))2121( ln(1
)2ln(),,,,(
n
1i
2
1 1 1
2
2211
iiF
j
n
i
n
i
jijjijjijjF
v)(u
xxxl
By differentiating the log likelihood function with respect
to θF and equating result to zero F
will be obtained from
Equation (44), where we got the estimates of the parameters
jj and from equations (38) and (40).
(1 2 )(1 2 ) 0 (43)
1 (1 2 )(1 2 )
i i
F F i i
u vl
u v
1
1 (44)
(1 2 )(1 2 )
F n
i i
i
u v
4.3 Maximum Likelihood Estimation for Bivariate
Weibull Distribution Based on Copulas In this section, the ML method will be used to estimate
the unknown parameters of the BW distribution.
4.3.1 Bivariate Weibull Distribution based on
Gaussian Copula In this subsection, the ML estimation of the unknown
parameters of the BW distribution is discussed. Let
niXXX iii ,...,1 ),,( 21 be a bivariate random sample of
size n from BW distribution given by Equation (31). Then
the likelihood function is given by:
))2()1(1
1exp( )12(
.)(exp().(),,,,(
1
2
1
2
12
2
2
1 1 1
1
2211
n
i
ii
n
j
n
i
jij
n
i
jijj
vvuu
xxL jj
Therefore, the log likelihood function of equation above
is given by:
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)2()1(1
1)12ln(
)2ln(),,,,(
1
2
1
2
12
2
2
1 1 1
1
2211
n
i
ii
j
n
i
n
i
jijjijj
vvuun
xxl jj
differentiating the log likelihood function with respect to
and, jj and equating each result to zero, we get:
1
1j
n
ji
ij j
lx
(45)
Thus
1
1
j
j n
ji
i
x
(46)
Similarly, and for j
1 1 1
1ln( ) . ln( ) 0 j
n n n
ji j ji ji
i i ij j
lx x x
(47)
3 2 2 2
1 1
2 2
( ) [( )] (1 )
0(1 )
n n
i i i i
i i
n u v u vl
(48)
Equations (47) and (48) will be solved numerically to
obtain the estimate j and 𝜌 respectively.
4.3.2 Bivariate Weibull Distribution Based on Plackett
Copula In this subsection, the MLE of the unknown parameters of
the BW distribution is discussed. Let n..,,1i ),X,X(X i2i1i
be a bivariate random sample of size n from BW distribution
given by (32). Then the likelihood function is given by:
)}-(1v4u-)]v1)(u-({[1
)]v2u-v1)(u-([1
)(exp().(),,,,(
n
1i3'2
ii
2
ii
iiii
2
1 1 1
1
2211
PPP
PP
j
n
i
jij
n
i
jijjPjj xxL
Therefore, the log likelihood function of equation above
is given by:
)}-(1v4u-)]v1)(u-(ln{[1
)])]v2u-v1)(u-([1ln([
)2ln(),,,,(
n
1i
3'2
ii
2
ii
iiii
n
1i
2
1 1 1
1
2211
PPP
PP
j
n
i
n
i
jijjijjPjj xxl
Then, differentiating the log likelihood function with
respect to θP and equating result to zero Equation (49) will
be obtained, where we got the estimates of the parameters
jj and from Equations (46) and (47).
1
1
1 (2 1) ( 2 )
[1 ( 1) ( 2 )]
n
P i i i i
i
n
PP P i i i i
i
u v u vl
u v u v
n2
i 1 1
n
i 1
3
2 [1 ( -1)( )] 4 (1 )
4( -1) 4 2( )[( -1)( ) 1]
n
P i i i i P P
i
P i i P i i i i P i i
u v u v
u v u v u v u v
(49)
Equation (49) will be solved numerically to obtain the
estimate of the parameter θ.
4.3.3 Bivariate Weibull Distribution based on FGM
copula In this subsection, the MLE of the unknown parameters of
the BW distribution is discussed. Let n..,,1i ),X,X(X i2i1i
be a bivariate random sample of size n from BWD given by
(33). Then the likelihood function is given by:
)21)(21(1
)(exp().(),,,,(
n
1i
2
1 1 1
1
2211
iiF
j
n
i
jij
n
i
jijjF
vu
xxL jj
Therefore, the log likelihood function of equation above
is given by:
))2121( ln(1
)2ln(),,,,(
n
1i
2
1 1 1
1
2211
iiF
j
n
i
n
i
jijjijjF
v)(u
xxl jj
After that, differentiating the log likelihood function with
respect to θF and equating result to zero F
will be obtained from Equation (51), where we got the estimates of the
parameters
jj and from equations (46) and (47).
1
1
(1 2 )(1 2 )
0
1 (1 2 )(1 2 )
n
i i
i
n
FF i i
i
u vl
u v
(50)
1
1
(1 2 )(1 2 )
F n
i i
i
u v
(51)
4.4 Maximum Likelihood Estimation for Bivariate
Exponential Distribution Based on Copulas In this section, the MLE method will be used to estimate
the unknown parameters of the BE distribution.
4.4.1 Bivariate Exponential Distribution Based on
Gaussian Copula In this subsection, the MLE of the unknown parameters of
the BE distribution is discussed. Let n..,.,1i ),X,X(X i2i1i
be a bivariate random sample of size n from BE distribution
given by (34). Then the likelihood function is given by:
))2()1(2
1exp(
.)12.())exp((),,(
1
2
1
2
12
22
1 1
21
n
i
ii
n
j
n
i
jjj
vvuu
xL
Therefore, the log likelihood function of equation above
given by:
)2()1(2
1
)12ln(ln),,(
1
2
1
2
12
22
1 1
21
n
i
ii
j
n
i
jijj
vvuu
nxnl
differentiating the log likelihood function with respect to
and j and equating each result to zero, we get:
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(52) 1
n
i
ji
jj
xnl
Thus
(53)
1
n
i
ji
j
x
n
Similarly, for
(54) 0 )1(
)1()][()(
22
1
2
1
223
n
i
ii
n
i
ii vuvunl
Equation (54) will be solved numerically to obtain the
estimate 𝜌. 4.4.2 Bivariate Exponential Distribution Based on
Plackett Copula In this subsection, the ML estimation of the unknown
parameters of the BE distribution is discussed. Let
niXXX iii ,...,1 ),,( 21 be a bivariate random sample of size
n from BE distribution given by (35). Then the likelihood
function is given by:
)}-(1v4u-)]v1)(u-({[1
)]v2u-v1)(u-([1
))exp((),,(
n
1i3'2
ii
2
ii
iiii
2
1 1
21
PPP
PP
j
n
i
jjjP xL
Therefore, the log likelihood function of equation above
given by:
)}-(1v4u-)]v1)(u-(ln{[1
)])v2u-v1)(u-([1ln(
ln),,(
n
1i
3'2
ii
2
ii
n
1i
iiii
2
1 1
21
PPP
PP
j
n
i
jijjP xnl
differentiating the log likelihood function with respect to θP
and equating result to zero. Equation (55) will be obtained,
where we got the estimates of the parameters
j by
Equation (53).
1
1
1 (2 1) ( 2 )
[1 ( 1) ( 2 )]
n
P i i i i
i
n
PP P i i i i
i
u v u vl
u v u v
n2
i 1 1
n
i 1
3
2 [1 ( -1)( )] 4 (1 )
4( -1) 4 2( )[( -1)( ) 1] (55)
n
P i i i i P P
i
P i i P i i i i P i i
u v u v
u v u v u v u v
Equation (55) will be solved numerically to obtain the
estimate of parameter θP.
4.4.3 Bivariate Exponential Distribution Based on
FGM Copula In this subsection, the MLE of the unknown parameters of
the BE distribution is discussed. Let n..,.,1i ),X,X(X i2i1i
be a bivariate random sample of size n from BE distribution
given by (36). Then the likelihood function is given by:
)21)(21(1 ))exp((),,(n
1i
2
1 1
21
iiF
j
n
i
jjjF vuxL
Therefore, the log likelihood function of equation above is
given by:
))21)(21( ln(1 )ln(),,(n
1i
2
1 1
21
iiF
j
n
i
jijjF vuxnl
differentiating the log likelihood function with respect to θF
and equating result to zero F
will be obtained from
Equation (57), where we got the estimates of the parameters
j from Equation (53).
1
1
(1 2 )(1 2 )
0
1 (1 2 )(1 2 )
n
i i
i
n
FF i i
i
u vl
u v
(56)
1
1
(1 2 )(1 2 )
F n
i i
i
u v
(57)
4.5 ML Method of Estimation Parameter of Bivariate
Rayleigh Distribution Based on Copulas In this section, the ML method will be used to estimate
the unknown parameters of the BR distribution. 4.5.1 Bivariate Rayleigh Distribution Based on
Gaussian Copula In this subsection, the ML estimation of the unknown
parameters of the BR distribution is discussed. Let
n..,.,1i ),X,X(X i2i1i be a bivariate random sample of size
n from BR distribution given by (37). Then the likelihood
function is given by:
))2()1(2
1exp(
)12.()exp(2),,(
1
2
1
2
12
22
1 1
2
21
n
i
ii
n
j
n
i
jjjij
vvuu
xxL
Therefore, the log likelihood function of equation above
given by:
)2()1(2
1)12ln(
ln2ln),,(
1
2
1
2
12
2
2
1 1
2
1
21
n
i
ii
j
n
i
jij
n
i
jij
vvuun
xxnl
differentiating the log likelihood function with respect to
and j and equating each result to zero, we get:
2
1
.n
ji
ij j
l nx
(58)
Thus
2
1
. j n
ji
i
n
x
(59)
Similarly, for ρ
3 2 2 2
1 1
2 2
( ) [( )] (1 )
0 (1 )
n n
i i i i
i i
n u v u vl
(60)
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International Journal of Innovation in Science and Mathematics
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Equation (60) will be solved numerically to obtain the
estimate 𝜌.
4.5.2 Bivariate Rayleigh distribution based on Plackett
copula In this subsection, the MLE of the unknown parameters of
the BR distribution is discussed. Let n..,.,1i ),X,X(X i2i1i
be a bivariate random sample of size n from BR distribution
given by (38). Then the likelihood function is given by:
)}-(1v4u-)]v1)(u-({[1
)]v2u-v1)(u-([1
)exp(2),,(
n
1i3'2
ii
2
ii
iiii
2
1 1
2
21
PPP
PP
j
n
i
jjjijP xxL
Therefore, the log likelihood function of equation above
given by:
)}-(1v4u-)]v1)(u-(ln{[1 )])v2u-v(u
1)-([1ln( ln2ln),,(
n
1i
3'2
ii
2
iiiiii
n
1i
2
1 1
2
11
21
PPP
PP
j
n
i
jij
n
i
ji
n
i
jP xxl
differentiating the log likelihood function with respect to P
and equating result to zero, But we got an estimate of the
parameters
j in equations (59).
(61) 1])1)(-)[(2(41)-4(
)1(4)]1)(-([12
3
])2()1(1[
)2()12(1
n
1i
n
1i 1
2
1
1
iiPiiiiPiiP
n
i
PPiiiiP
n
i
iiiiPP
n
i
iiiiP
P
vuvuvuvu
vuvu
vuvu
vuvul
Equation (61) will be solved numerically to obtain the
estimate 𝜃𝑃. 4.5.3 Bivariate Rayleigh Distribution Based on FGM
Copula In this subsection, the ML estimation of the unknown
parameters of the BR distribution is discussed. Let
n..,.,1i ),X,X(X i2i1i be a bivariate random sample of size
n from BR distribution given by (39). Then the likelihood
function is given by:
2 n
2
1 2
1 1 i 1
( , , ) 2 exp( ) 1 (1 2 )(1 2 ) n
F j ji j j F i i
j i
L x x u v
Therefore, the log likelihood function of equation above
given by:
n
1i
2
1 1
2
11
21
))2121( ln(1
ln2ln),,(
iiF
j
n
i
jij
n
i
ji
n
i
jF
v)(u
xxl
differentiating the log likelihood function with respect to F
and equating result to zero, where we are got an estimate of
the parameters
j in equations (59).
1
1
(1 2 )(1 2 )
0
1 (1 2 )(1 2 )
n
i i
i
n
FF i i
i
u vl
u v
(62)
1
1
(1 2 )(1 2 )
F n
i i
i
u v
(63)
V. SIMULATION STUDY
In this section a numerical study is provided to illustrate
the various theoretical results.
5.1 Simulation Study of Bivariate Linear Failure Rate
Distribution: Simulation study has been performed for different sample
sizes, keeping 3.121 , 2.021 and Gaussian
copula parameter 7.0 , plackett copula parameter
8.0P and FGM copula parameter 2.0F , with sample
sizes, n=15, 35, 50, 100,150 and 200, for Gaussian and
plackett copula while we consider n= 35, 50, 100, 150 and
200, for FGM copula. In this case, the ML estimators are
computed to estimate the parameters of the BLFR
distribution using the following steps:
1) For given value of the parameters ),,,( *2*
2
*
1
*
1 and
correlation parameters ),,( ***
FP , a sample of size n from
BLFR distribution is generated.
2) The ML estimates of the parameters are computed by maximizing the log-likelihood function in Equation (41)
with respect to 2211 ,,, and correlation parameters of
copulas.
3) The above steps are repeated 1000 times. For 1000 replications, the means estimate of the parameters
along with the MSE are computed and the results are
reported in Table (1).
The results in Table (1), (2) and (3) for the ML estimates
of the unknown parameters for each bivariate distribution in
each types of copulas are quite satisfactory. It is observed
that when the sample size increases, the MSE decrease for
all the parameters, as expected.
Table (1): The MSE under BLFR distribution with
= 0.2 and 2=β1= 1.3 and β2=α1Gaussian copula when α
ρ=0.7
MSE Sample
size
2
2
1
1 n
2320.0 233.2. 23.0.0 2320.. 2313.. 15
2322.0 2333.0 233.3. 2333.. 233.23 35
2322.. 2321.0 232... 2321021 23202. 50
0.0023 0.0283 0.0436 0.0269 0.0448 100
23223. 2323.. 2320.0 2323.2. 232000 150
2322320 2323.0 2320020 232300 2320023 200
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Volume 5, Issue 6, ISSN (Online): 2347–9051
Table (2): The MSE under BLFR distribution with
P= 0.2 and θ2=β1= 1.3 and β2=α1Plackett copula when α
=0.8
MSE Sample
size
P
2
2
1
1 n
23.000 033.0. 23..00 330.2. 2300320 15
23.2.0 2330.20 233..3 2330020 233..0 35
233..0 2321.0 233203 232..1 23321. 50
23211. 232000 23202. 232.3. 232020 100
232... 2323.0 232... 232300 232.03 150
0.0324 2323.. 2320.. 23230. 2320.. 200
Table (3): The MSE under BLFR distribution with
F= 0.2 and θ2=β1= 1.3 and β2=α1when α FGM copula
=0.2
MSE Sample
size
F
2
2
1
1 n
23..00 233.00 233020 233303 233.10 35
2302.23 232.0. 2332.0 232.20 23201220 50
232030 23200. 23202. 232.2. 232..0 100
2321.23 232300 232... 23230. 232.00 150
232... 2323.1 232000 2323.. 2320.1 200
5.3 Simulation Study of Bivariate Weibull Distribution
Simulation study has been performed for different sample
sizes, keeping ,3.1,5.1 21 1.0,2.0 21 and Gaussian
copula parameter 5.0 , Plackett copula parameter
3.0P and FGM copula parameter 03.0F , with sample
sizes, n=15, 35, 50, 100,150 and 200, for Gaussian and
placket copula while we consider n=35, 50, 100, 150 and
200, for FGM copula. In this case, the ML estimators are
computed to estimate the parameters of the BLFR
distribution using the following steps:
1) For given value of the parameters ),,,( *2*
2
*
1
*
1 and
correlation parameters ),,( ***
FP , a sample of size n from
BLFR distribution is generated.
2) The ML estimates of the parameters are computed by maximizing the log-likelihood function in Equation (41)
with respect to 2211 ,,, and correlation parameters of
copulas.
3) The above steps are repeated 1000 times. For 1000 replications, the means estimate of the
parameters along with the MSE are computed and the results
are reported in Table (2).
The results in Table (4), (5) and (6) for the ML estimates
of the unknown parameters are summarized, the results
show that the performance of the ML estimates is quite
satisfactory. Also, it is observed that when the sample size
increases, the MSE decrease for all the parameters, as
expected.
Table (4): The MSE under BW distribution with
𝛒0.1 and 2=0.2, γ1=1.3, γ2=1.5, β1=when β Gaussian copula=0.5
MSE Sample size
2
2
1
1 n
0.4258 0.0006 0.2287 0.0028 0.3224 15 0.0166 0.0002 0.06107 0.0009 0.0818 35 0.0115 0.0001 0.03806 0.0005 0.0509 50 0.0057 0.04572 0.0168 0.0002 0.0252 100 0.0036 0.02937 0.0113 0.0001 0.0165 150 0.0027 0.02214 0.0082 0.0001 0.0119 200
Table (5): The MSE under BW distribution with
P0.1 and θ=2γ0.2, =1γ1.3, =2β1.5, =1βwhen Plackett copula
=0.3
MSE Sample
size
P
2
2
1
1 n
0.4312 0.0007 0.2142 0.0029 0.3465 15
0.0428 0.0002 0.0627 0.0008 0.0977 35
0.0236 0.0001 0.0438 0.0006 0.0589 50
0.0088 0.04631 0.0174 0.0002 0.0262 100
0.0051 0.02784 0.0115 0.0001 0.0156 150
0.0042 0.02084 0.0089 0.0001 0.0112 200
with FGM distribution (6): The MSE under BWTable
=0.03 F0.1 and θ2=0.2, γ1=1.3, γ2=1.5, β1=when β copula
MSE Sample
size
F
2
2
1
1 n
0.3757 0.00025 0.0657 0.00084 0.0806 35
0.2071 0.00015 0.0407 0.00056 0.0538 50
0.0941 0.0461 0.0180 0.0002 0.0227 100
0.0644 0.03037 0.0114 0.00018 0.01505 150
0.0456 0.02227 0.0089 0.00013 0.01108 200
5.4 Simulation Study of Bivariate Exponential
Distribution Simulation study has been performed for different sample
sizes, keeping 5.1,1 21 and Gaussian copula parameter
5.0 , Plackett copula parameter 5.0P and FGM copula
parameter 53.0F , with sample sizes, n=15, 35, 50, 100,
150 and 200, for Gaussian and placket copula while we
consider n=35, 50, 100,150 and 200, for FGM copula. In this
case, the ML estimators are computed to estimate the
parameters of the BLFR distribution using the following
steps:
1) For given value of the parameters ),( *2*
1 and
correlation parameters ),,( ***
FP , a sample of size n from
BE distribution is generated.
2) The ML estimates of the parameters are computed by maximizing the log-likelihood function in Equation (41)
with respect to 21, , and correlation parameters of
copulas.
3) The above steps are repeated 1000 times.
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197
International Journal of Innovation in Science and Mathematics
Volume 5, Issue 6, ISSN (Online): 2347–9051
For 1000 replications, the means estimate of the parameters
along with the MSE are computed and the results are
reported in Table (3).
The results in Table (7), (8) and (9) for the ML estimates
of the unknown parameters for each bivariate distribution-z
in each types of copulas are quite satisfactory. It is observed
that when the sample size increases, the MSE decrease for
all the parameters, as expected.
Table (7): The MSE under BE distribution with
=0.5 𝝆=1.5 and 2=1, α1Gaussian copula when α MSE Sample
size
2
1 n
232.203 23000. 233200 30
2323..0 233..0 232.0.0 .0
23220.1 23330.. 232..2. 02
2322.. 2332.0 23200. 322
230230. 232002. 2323.0 150
23220. 23201. 232300 200
Table (8): The MSE under BE distribution with
=0.5P=1.5 and θ 2=1, α1Plackett copula when α
MSE Sample
size
P
2
1 n
333013 230... 233033 30
233.01 233..0 2320.0 .0
232.203 23300. 232... 02
0.0258 0.1059 0.0231 322
2323.0 23322. 2323.2. 150
232300 2320.. 23230. 200
with FGM distribution Table (9): The MSE under BE
3=0.5F=1.5 and θ 2=1, α1copula when α
MSE Sample
size
F
2
1 n
1.8831 0.1344 0.04773 .0
0.2274 0.1182 0.0364 02
0.08702 0.1008 0.0215 322
0.0583 0.0964 0.0174 150
0.0417 0.09509 0.0154 200
5.5 Simulation Study of Bivariate Rayleigh
Distribution Simulation study have been performed for different
sample sizes, keeping ,2.1,8.0 21 and Gaussian
copula parameter 5.0 , Plackett copula parameter
5.0P and FGM copula parameter 53.0F , with sample
sizes, n=15, 35, 50, 100, 150 and 200, for Gaussian and
placket copula while we consider n=35, 50, 100,150 and
200, for FGM copula. In this case, the ML estimators are
computed to estimate the parameters of the BRD using the
following steps:
1) For given value of the parameters ),( *2*
1 and
correlation parameters ),,( ***
FP , a sample of size n from
BR distribution is generated.
2) The ML estimates of the parameters are computed by maximizing the log-likelihood function in Equation (41)
with respect to 21, and correlation parameters of copulas.
3) The above steps are repeated 1000 times. For 1000 replications, the means estimate of the parameters
along with the MSE are computed and the results are
reported in Table (4).
The results in Table (10), (11) and (12) for the ML
estimates of the unknown parameters are summarized, it
clear that the performance of the MLE method is quite
satisfactory. Also, it is observed that when the sample size
increases, the MSE decreases for all the parameters, as
expected except for the case of the second and third copula
types. At the sample size 15 in the case of plackett copula and sample size 35 in the case of FGM copula, MSE is
considered significant.
Table (10): The MSE under BR distribution with
=0.5 𝝆=1.2 and 2β =0.8,1Gaussian copula when β MSE Sample
size
2
1 n
232.20 23.0.3 2320.3 30
2323.. 23.00. 2320.0 .0
23220. 23.121 232300 02
0.0048 23.1.. 232300 322
2322.3 23.12. 2323.0 150
23220. 23.00. 232301 200
Table (11): The MSE under BR distribution with
=0.5P=1.2 and θ 2=0.8, β1Plackett copula when β
MSE Sample
size
P
2
1 n
1.1269 0.3863 0.0616 30
0.1326 0.3703 0.0273 .0
0.0705 0.3675 0.0213 02
0.0258 0.36407 0.0149 322
0.0145 0.3624 0.01309 150
232300 23.100 23230. 200
with FGM distribution MSE under BR Table (12): The
3=0.5F=1.2 and θ 2=0.8, β1copula when β
MSE Sample size
F
2
1 n
1077.9 0.3469 0.0259 35
2300.. 23.010 23202.. 02
232..20 23.00. 2323.. 322
2320.. 23.010 2323.0 150
232.3. 23.0.0 232300 200
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198
International Journal of Innovation in Science and Mathematics
Volume 5, Issue 6, ISSN (Online): 2347–9051
V. CONCLUSION
The bivariate Linear failure rate, Weibull, exponential and
Rayleigh distributions based on three types of copulas
(Gaussian, Plackett and FGM) with two-dimension are
introduced as a flexible bivariate lifetime models. The MLE method is used to estimate the parameters of (BLFR, BW,
BE, and BR) distributions. Monte Carlo simulation
indicated that the performance of the MLE method are Fully
satisfactory. The simulations are performed for different
sample sizes with one set of the marginals parameters. The
MSE is used to measure the performance for the estimators
when the sample size increases.
The ML estimates in each case of BLRF distributions
based on Gaussian, Plackett and FGM copula are compared.
It is observed that when the sample size increases, the MSE
decreases for all the parameters. But in the case of BLFR
distribution based on FGM copula, it was noted that when
the sample size is too small, the MSE is too large for the
estimate of correlation parameter.
Also, by comparing between ML estimates in each case
of BW distributions based on Gaussian, Plackett and FGM
copula, it is observed that when the sample size increases,
the MSE decrease for all the parameters. But in the case of
BW distribution based on FGM copula, it was noted that
when the sample size is too small, the estimated value of the
correlation parameter is not logical and the MSE is too large.
In addition when Comparing between ML estimates in
each case of BE distributions based on Gaussian, Plackett
and FGM copula, it is observed that when the sample size
increases, the MSE decreases for all the parameters in the
case of BE distribution based on Gaussian copula. But in the
case of BE distribution based on Plackett and FGM copula,
it was noted that when the sample size is small, the MSE is
too large for the estimate of correlation parameter.
Finally, by comparing between ML estimates in each case
of BR distributions based on Gaussian, Plackett and FGM
copula: It is observed that when the sample size increases,
the MSE decreases for all the parameters. But in the case of
BR distribution based on FGM copula, it was noted that
when the sample size is too small, the MSE is too large for
the estimate of correlation parameter.
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AUTHOR’S PROFILE Lutfiah Ismail Al turk is currently working as Associate Professor of
Mathematical Statistics in Statistics Department at Faculty of Sciences, King AbdulAziz University, Saudi Arabia. Lutfiah Ismail Al turk obtained
her B.Sc degree in Statistics and Computer Science from Faculty of
Sciences, King AbdulAziz University in 1993 and M.Sc (Mathematical statistics) degree from Statistics Department, Faculty of Sciences, King
AbdulAziz University in 1999. She received her Ph.D in Mathematical
Statistics from university of Surrey, UK in 2007. Her current research interests include Software reliability modeling and Statistical Machine
Learning.
Email: [email protected] URL: http://lturk.kau.edu.sa
Address: P.O. Box 42713 Jeddah 21551. Kingdom of Saudi Arabia.
mailto:[email protected]://lturk.kau.edu.sa/