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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:


183



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


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


184



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


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.


185




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.


values of θ

5. The PDF of the BLFR distribution based on FGM


in Figure (5) when 𝛽1 = 𝛽2 > 1 and when 𝛽1 = 𝛽2 < 1 with fixed value of 𝛼𝑗 and θ.


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


values of θ

Graphical Description of the Bivariate Weibull

Distribution 1. The PDF of the BW distribution based on Gaussian


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.


186



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


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.


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


function based on Plackett copula for four different

levels of dependence, assuming the parameters 𝛽1 =𝛽2 = 1.5 𝑎𝑛𝑑 𝛾1 = 𝛾2 = 1.3


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.


187




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



of dependence, assuming the parameters

2.1,5.0 2121


values of θ

Graphical Description of the Bivariate Exponential

Distribution 1. The PDF of the BE distribution based on Gaussian


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


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


in Figure (15) when 1, 21 and when 1, 21

with fixed value of θ.


188



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


levels of dependence, assuming the parameters

1 21, 1.5.


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


values of θ

Graphical Description of the Bivariate Rayleigh

Distribution 1. The PDF of the BR distribution based on Gaussian


in Figure (19) when 121 and when 121

with fixed value of ρ.


189



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


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


in Figure (21) when 121 and when 121

with fixed value of .


values of the parameters (a) β1 = β2 = 1.5, θ = 0.9 and (b) β1 = β2 = 0.9, θ = 0.9



levels of dependence, assuming the parameters

1 2 1.5.


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.


190




values of the parameters (a)β1 = β2 = 1.5, θ = 0.9 and (b) β1 = β2 = 0.9, θ = 0.8



of dependence, assuming the parameters .5.121


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:


191



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


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


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


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)


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


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(),,,,(


192




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


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


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


)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


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


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


is given by:


193



)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


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


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


u v u v

u v u v u v u v

(49)


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


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)


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


))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


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


and j and equating each result to zero, we get:


194



(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


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


)}-(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


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


u v u v

u v u v u v u v


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


)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


))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


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


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)


195




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


)}-(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


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


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


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


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


196




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


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


sizes, n=15, 35, 50, 100,150 and 200, for Gaussian and

placket copula while we consider n=35, 50, 100, 150 and


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



BLFR distribution is generated.


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).


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



BE distribution is generated.


with respect to 21, , and correlation parameters of

copulas.

3) The above steps are repeated 1000 times.


197



For 1000 replications, the means estimate of the parameters




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


sizes, n=15, 35, 50, 100, 150 and 200, for Gaussian and

placket copula while we consider n=35, 50, 100,150 and


computed to estimate the parameters of the BRD using the

following steps:

1) For given value of the parameters ),( *2*

1 and



BR distribution is generated.


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



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


198



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.

REFERENCE [1] Farlie, Dennis JG. "The performance of some correlation

coefficients for a general bivariate distribution." Biometrika 47.3/4

(1960): 307-323. [2] Genest, Christian, and Jock MacKay. "The joy of copulas:

bivariate distributions with uniform marginals." The American

Statistician 40.4 (1986): 280-283. [3] Gumbel, E. J. "Statistics of extremes. 1958." Columbia Univ.

press, New York (1958).

[4] Hougaard, Philip. "A class of multivariate failure time distributions." Biometrika (1986): 671-678.

[5] Johnson, Richard A., James William Evans, and David W. Green.

Some bivariate distributions for modeling the strength properties of lumber. US Department of Agriculture, Forest Service, Forest

Products Laboratory, 1999.

[6] L. J. Bain, Analysis for the linear failure-rate life-testing distribution, Technometrics, 16, 4 (1974), 551-559.

[7] Lee, Lung-Fei. "Generalized econometric models with

selectivity." Econometrica: Journal of the Econometric Society (1983): 507-512.

[8] Lu, Jye-Chyl, and Gouri K. Bhattacharyya. "Some new

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[9] Morgenstern, Dietrich. "Einfache beispiele zweidimensionaler

verteilungen." Mitteilungsblatt für Mathematische Statistik 8.1 (1956): 234-235.

[10] NELSEN, Roger B. An introduction to copulas. Springer Science

& Business Media, 2007. [11] NELSEN, Roger B. Properties and applications of copulas: A brief

survey. In: Proceedings of the First Brazilian Conference on

Statistical Modeling in Insurance and Finance,(Dhaene, J., Kolev, N., Morettin, PA (Eds.)), University Press USP: Sao Paulo. 2003.

p. 10-28.

[12] Plackett, Robin L. "A class of bivariate distributions." Journal of the American Statistical Association 60.310 (1965): 516-522.

[13] Zaindin, Mazen, and Ammar M. Sarhan. "Parameters estimation

of the modified Weibull distribution." Applied Mathematical Sciences 3.11 (2009): 541-550.

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/

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