Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (1) 2021 

 

 

 85 
 

 

  

 

 

Estimate for Survival and Related Functions of Weighted Rayleigh 

Distribution. 

 

  

 

 

 

 

 

 

  

 

 

 

Abstract 

     In this paper, we introduce a new class of Weighted Rayleigh Distribution based on two 

parameters, one is the scale parameter and the other is the shape parameter introduced in 

Rayleigh distribution. The main properties of this class are derived and investigated[13]. The 

moment method and least square method are used to obtain estimators of parameters of this 

distribution. The probability density function,   survival function, cumulative distribution and 

hazard function are derived and found. Real data sets are collected to investigate two methods 

that depend on in this study. A comparison is made between two methods of estimation and 

clarifies that MLE method is better than the OLS method by using the mean squares error.  

 

Keyword: weighted Rayleigh distribution, Maximum likelihood method and lest square 

method.  

1. Introduction  

        The Rayleigh distribution is one of the important continuous distributions; it encounters 

much attention in the literature in benefit from any other distribution in lifetime sample 

modeling and data analysis. Many researchers developed various generalizations of Rayleigh 

probability density function to increase the flexibility in lifetime sample modeling [1]. We 

Ibn Al Haitham Journal for Pure and Applied Science 

Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index 

 

Doi: 10.30526/34.1.2558 

Article history: Received18, December,2019, Accepted29,Janury,2020, Published in January 2021 

 

Saad A. Zain Iden Hassan Hussein 

 
Department of Mathematics, College 

of Science for Women, University of 

Baghdad, Baghdad, Iraq. 

Iden_alkanani@yahoo.com 

 

Department of mathematics, College 

of Education for Pure Science, Ibn Al 

Haitham, University of Baghdad, 

Baghdad, Iraq. 

ssadadnan.2019@gmail.com 

 

mailto:Iden_alkanani@yahoo.com
mailto:ssadadnan.2019@gmail.com


  

 86  

 

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (1) 2021 

 

Introduced a new class of density function depending on the shape parameter in the normal 

distribution, which is known as weighted normal distribution or (skew-normal distribution) [2]. 

We used the idea of Azzalini to find the shape parameter to an exponential distribution which is 

known as weighted exponential distribution, as well as putthe general mathematical formula to 

treat the weighted statistical distributions which are as follow:   

                                                                                                          

𝑓𝑤(𝑋) =
1

𝑝[𝑥2 < 𝛼𝑥1]
𝑓(𝑥1)𝐹(𝛼𝑥1)                                      (1) 

 

Where: 

 𝑓𝑤(𝑋) Weighted probability density function.  

𝑓(𝑥1) Standard probability density function for 𝑟. 𝑣(𝑥1) 

𝐹(𝛼𝑥1) Cumulative distribution functions with respect to weighted parameter 𝛼  for               

standard distribution. 

𝑃𝑟(𝑥2 < 𝛼𝑥1) Probability for 𝑟. 𝑣(𝑥2) with respect to the 𝑟. 𝑣(𝑥1) and weighted       function 

(𝛼).  

[3] Studied weighed Weibull distribution by using the idea of Azzalini and introduced the basic 

properties for this model [4]. Studied the skew-ness parameter of a gamma distribution by 

using the idea of Azzalini that resulted a new class of weighted gamma distribution.[5] 

proposed the extension of the weighted Weibull distribution and the main properties of this 

class are investigated and derived [6]. Estimated Linley’s approximation method for weighted 

exponential distribution by using a Monte Carlo simulation study [7]. Introduced two shape 

parameters to the existing weighted exponential distribution to develop the weighted Beta 

exponential distribution using the log it of beta function[8] proposed a model named 

exponentiated weighted exponential distribution and some of the basic statistical properties of 

the proposed model are studied and provided[9]. Studied the Nonparametric method such as the 

empirical method, kernel method, and modified shrinkage provided in weighted Weibull 

distribution [10]. Focused on Bayes estimation of weighted exponential distribution with fuzzy 

data[11]. Derived two parameters inverted weighted Exponential distribution and its various 

statistical properties thatwere established [12]. presented a new generalization weighted 

Weibull distribution using topple one family of distribution [13]. Proposed a new class of 

weighted Rayleigh distribution by using the idea of Azzalini and introduce the main 

characteristics of this distribution.This paper aimsto introduce a new weighted Rayleigh 

distribution with its properties which discussed maintained in[13]. Applying this new 

distribution on real data to estimate the parameters by using two methods and calculate the 

death density function, survival function, hazard function for these two methods.                             

The rest of this article is as follows: in section two, the new weighted Rayleigh   distribution 

and its properties is presented, section three xplains estimation methods, in section four is 

devoted to the real data application and section five gives the conclusion.  

                                                              

2- Weighted Rayleigh Distribution   



  

 87  

 

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (1) 2021 

 
     In this section, we will provide the probability density function of new weighted Rayleigh 

distribution which is as follows: 
 

𝑓(𝑥; 𝛼, 𝜃) =
𝛼2 + 1

𝛼2
𝜃𝑥 𝑒

−
𝜃
2
𝑥2

[1 − 𝑒
−

𝜃
2
𝑥2𝛼2

]   ,   𝑥 > 0[13]                                           (2) 

The cumulative distribution of new distribution is as follow: 

𝐹(𝑥;  𝛼, 𝜃) = 1 − [
(𝛼2+1)𝑒

−
𝜃
2
𝑥2

−𝑒
−

𝜃
2
𝑥2(𝛼2+1)

𝛼2
]                                                                 (3) 

The survival function and hazard function is as follows: 

𝑆(𝑥) =
(𝛼2 + 1)𝑒

−
𝜃
2
𝑥2

− 𝑒
−

𝜃
2
𝑥2(𝛼2+1)

𝛼2
                                                                            (4) 

ℎ(𝑥) =
𝜃𝑥(𝛼2 + 1) [1 − 𝑒

−
𝜃
2
𝑥2𝑑2

]

[(𝛼2 + 1) − 𝑒
−

𝜃
2
𝑥2𝛼2

]

                                                                                 (5) 

The 𝑟𝑡ℎ moment of new distribution is as follows: 

𝐸(𝑥𝑟) =
𝛼2 + 1

𝛼2
(
2

𝜃
)

𝑟
2
𝛤 (

𝑟

2
+ 1) [1 −

1

(𝛼2 + 1)
𝑟
2
+1

]                                                (6) 

The mean and the variance of  the new distribution is:   

M= E (𝑋)=
𝛼2+1

𝛼2
[

𝜋

2𝜃
]

1

2
[1 −

1

(𝛼2+1)
3

2⁄
]                                                                             (7) 

 The variance: 

𝜎2 = 𝑣𝑎𝑟(𝑋) =
4𝛼4(𝛼2+2)−𝜋[(𝛼2+1)

3
2−1]

2

 

2𝜃𝛼4(𝛼2+1)
                                                               (8) 

The Moment generated function of this distribution is as follows: 

𝑚. 𝑔. 𝑓 =
𝛼2 + 1

𝛼2
∑

𝑡𝑟

𝑟!

∞

𝑟=0

   (
2

𝛳
)

𝑟
2
Γ (

𝑟

2
+ 1) [1 −

1

(𝛼2 + 1)
𝑟

2⁄ +1
  ]                          (9) 

The Factorial Moment Generating function is:    

𝑀𝑥(𝑡) =
𝛼2 + 1

𝛼2
 ∑

(ℓ𝑛𝑡)𝑟

𝑟!

∞

𝑟=0

(
2

𝜃
)

𝑟
2
Γ (

𝑟

2
+ 1) [1 −

1

(𝛼2 + 1)
𝑟
2
+1

]                 (10) 

The skewness and the kurtosis of this distribution is :  



  

 88  

 

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (1) 2021 

 

𝐶. 𝑆 =  
1

𝛼2
 𝛤 (

5

2
) [

(𝛼2 + 1)
5
2 − 1

(𝛼2 + 1)
3
2

]                                                                      (11) 

𝐶. 𝑘 =
2[(𝛼2 + 1)3 − 1]

𝛼2(𝛼2 + 1)2
− 3                                                                                (12) 

The characteristic function of this distribution is as follows : 

𝑄𝑥 (𝑥) =
𝛼2 + 1

𝛼2
∑

(𝑖𝑡)𝑟

𝑟!

∞

𝑟=0

(
2

𝜃
)
𝑟

Γ (
𝑟

2
+ 1) [1 −

1

(𝛼2 + 1)
𝑟
2
+1

]                   (13) 

 

3. Estimation Methods 

       In this section, estimate the parameters of weighted Rayleigh distribution by employing two 

methods (Maximum likelihood estimator method and Ordinary least square estimator method).  

                                                                                                      

3.1: Maximum Likelihood Method 

      MLE is a famous and classical method used to find the estimators of parameters that 

maximize the likelihood function. The probability density function is:  

 

𝑓(x; α, θ) =  
α2 + 1

α2
 θx e

−
θ
2
x2

[1 − e
−

θ
2
x2α2

]                                                      (14) 

𝐿 (𝛼, 𝜃; 𝑥) =
(𝛼2 + 1)𝑛

𝛼2𝑛
𝜃𝑛 ∏ 𝑥𝑖

𝑛

𝑖=1

 𝑒
−

θ
2
x2

∏ [1 − 𝑒
−

𝜃
2
𝑥2𝛼2

]

𝑛

𝑖=1

                         (15) 

𝐿𝑛𝐿 = 𝑛𝑙𝑛 (𝛼2 + 1) − 2𝑛 𝑙𝑛(𝛼) + 𝑛𝑙𝑛𝜃 + ∑ 𝐿𝑛 𝑥𝑖 

𝑛

𝑖=1

−
Ө

2
∑ 𝑥𝑖

2

𝑛

𝑖=1

 

+ ∑ 𝐿𝑛 [1 − 𝑒
−

𝜃
2
𝑥𝑖

2𝛼2
]

𝑛

𝑖=1

                                                                                           (16) 

𝑑𝑙𝑛𝑙

𝑑𝜃
=

𝑛

𝜃
−

1

2
∑ 𝑥𝑖

2

𝑛

𝑖=1

+ ∑
   
1
2

𝑥𝑖
2 𝛼2𝑒

−
𝜃
2
𝑥𝑖

2𝛼2

[1 − 𝑒
−

𝜃
2
𝑥𝑖

2𝛼2
]

= 𝑔1(𝜃)

𝑛

𝑖=1

                                    (17)  

𝑑𝑙𝑛𝑙

𝑑𝛼
=

2𝑛𝛼

(𝛼2 + 1)
−

2𝑛

𝛼
+ ∑

   𝜃𝛼 𝑥𝑖
2 𝑒

−
𝜃
2 

𝑥𝑖
2𝛼2

[1 − 𝑒
−

𝜃
2
 𝑥𝑖

2𝛼2
]

= 𝑔2

𝑛

𝑖=1

(𝛼)                                 (18) 

 
𝑑𝑔1(𝜃)

𝑑𝜃
= −

𝑛

𝜃2
− ∑

1
4

𝑥𝑖
4𝛼4𝑒−𝜃𝑥𝑖

2𝛼2

[1 − 𝑒
−

𝜃
2
𝑥𝑖

2𝛼2
]
2                                                                 (19)    

𝑛

𝑖=1

 



  

 89  

 

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (1) 2021 

 

𝑑𝑔1(𝜃)

𝑑𝛼
= ∑

−
1
2

𝜃𝛼3𝑥𝑖
4𝑒

−
𝜃
2
𝑥𝑖

2𝛼2
+ 𝛼𝑥𝑖

2𝑒
−

𝜃
2
𝑥𝑖

2𝛼2
− 𝛼𝑥𝑖

2𝑒−𝜃𝑥𝑖
2𝛼2

[1 − 𝑒
−

𝜃
2
𝑥𝑖

2𝛼2
]
2

𝑛

𝑖=1

                 (20)  

d𝑔2(α)

d𝜃
= ∑

−
1
2

θα3xi
4 e

−
θ
2
xi
2α2

+ αxi
2e

−
θ
2
xi
2α2

− αxi
2e−θxi

2α2

[1 − e
−

θ
2
xi
2α2

]
2

n

i=1

                    (21) 

𝑑𝑔2(α)

d𝛼
=

2𝑛(1 − 𝛼2)

(𝛼2 + 1 )2
+

2𝑛

𝛼2
+ ∑

𝜃𝑥𝑖
2𝑒

−
𝜃
2
𝑥𝑖

2𝛼2
− 𝜃𝑥𝑖

2𝑒−𝜃𝑥𝑖
2𝛼2 − 𝜃2𝛼2𝑥𝑖

4𝑒
−

𝜃
2
𝑥𝑖

2𝛼2

[1 − 𝑒
−

𝜃
2
𝑥𝑖

2𝛼2
]
2

𝑛

𝑖=1

         (22) 

The above equations are nonlinear and must use multivariate Newton-Raphson method which 

is as follow: 

        (23)              [
θk+1
αk+1

] = [
θk
αk

] − j−1 [
𝑔1(𝜃)
𝑔2(α)

] 

Where the Jacobean matrix is as follows: 

𝐽 =

[
 
 
 
𝑑𝑔1(𝜃)

𝑑𝜃

𝑑𝑔1(𝜃)

𝑑𝛼
𝑑𝑔

2
(α)

𝑑𝜃

𝑑𝑔
2
(α)

𝑑𝛼 ]
 
 
 
                                                                                        (24) 

Where J is Jacobean is a symmetric and square matrix. 

Then we find the stopping Rule for getting convergence which is as follows: 

|[
θk+1
αk+1

] − [
θk
αk

]| ≤ |
𝜖𝜃
𝜖𝛼

|                                                                                        (25) 

                                                                                                

3.2. The Ordinary Least Square Method 

      The least squares method is a statistical procedure to find the best fit for a set of data points 

by minimizing the sum of the offsets or residuals of points from the plotted curve. Least 

squares regression is used to predict the behavior of dependent variables. 

𝑌𝑖 = 𝛣0 + 𝛣1𝑥 + ε                                                                                                        (26)                                                        

Where: Β0 represents the intercept term 

              Β1 represents the slop term 

ε               represents the error term 

     The idea of this method is to minimize the sum of the squared difference between observed 

sample values and the estimate expected values by linear approximation:  



  

 90  

 

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (1) 2021 

 

    

ε = 𝑌𝑖 − �̂�0 − �̂�1𝑥                                                                                                           (27)  

 

∑ ε
i

2
n

i=1

= ∑[𝑦𝑖 − 𝐸(ŷ)
2]2

n

i=1

 

            

∑ ε
i

2
n

i=1

= ∑  [Yi − Β̂0 − Β̂1x]
2

n

i=1

   

    Now, we apply this method to minimize the square difference between the estimate 

cumulative distribution function and the empirical cumulative distribution function, where the 

empirical (CDF) is:                                                                                   

F(x) = 
𝑖−0.5

𝑛
                                                                                                                          (28)  

 

∑ ε
i

2
n

i=1

= ∑[F̂(xi) − F(xi)]
2

n

i=1

     

 

∑ ε
i

2
n

i=1

= ∑ [1 − [
(α2 + 1)e−

θ
2
xi
2
− e−

θ
2
xi
2(α2+1)

α2
] −

i − 0.5

n
]

n

i=1

2

    

 

∑ ε
i

2
n

i=1

= ∑ [1 −
(i − 0.5)

n
− [

(α2 + 1)e−
θ
2
xi
2
− e−

θ
2
xi
2(α2+1)

α2
]]

2
n

i=1

 

 

∑ ε
i

2
n

i=1

= ∑ [
n − i + 0.5

n
−

(α2 + 1)e−
θ
2
xi
2
− e−

θ
2
xi
2
(α2+1)

α2
]

n

i=1

          

 

Let      yi =
n − i + 0.5

n
                                                                                             (29) 

 

d ∑ ε
i

2n
i=1

dθ
= 2 ∑ [yi −

(α2 + 1)e−
θ
2
xi
2
− e−

θ
2
xi
2(α2+1)

α2
]

∗

n

i=1

 



  

 91  

 

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (1) 2021 

 

          ∗ [0 − (
1

α2
[−

1

2
xi

2(α2 + 1)e−
θ
2
xi
2
+

1

2
xi

2(α2 + 1)e−
θ
2
xi
2(α2+1)])]     

 

𝑑 ∑ ε
𝑖

2𝑛
𝑖=1

𝑑𝜃
 = 2 ∑ [𝑦𝑖 −

(𝛼2 + 1)𝑒−
𝜃
2
𝑥𝑖

2
− 𝑒−

𝜃
2
𝑥𝑖

2(𝛼2+1)

𝛼2
]

𝑛

𝑖=1

∗  (− [
𝑥𝑖

2(𝛼2 + 1)

2𝛼2
(−𝑒−

𝜃
2
𝑥𝑖

2
+ 𝑒−

𝜃
2
𝑥𝑖

2(𝛼2+1))])  

 

 

𝑑 ∑ ε
𝑖

2𝑛
𝑖=1

𝑑𝜃
= ∑ [𝑦𝑖 −

(𝛼2 + 1)𝑒−
𝜃
2
𝑥𝑖

2
− 𝑒−

𝜃
2
𝑥𝑖

2(𝛼2+1)

𝛼2
] ∗

𝑛

𝑖=1

𝑥𝑖
2(𝛼2 + 1)

𝑥2
 

                       ∗ [𝑒−
𝜃
2
𝑥𝑖

2
− 𝑒−

𝜃
2
𝑥𝑖

2(𝛼2+1)] 

                                               

𝑑 ∑ ε
𝑖

2𝑛
𝑖=1

𝑑𝜃
  = ∑  

𝑥𝑖
2(𝛼2 + 1)

𝛼2
∗ [𝑦𝑖 −

(𝛼2 + 1)𝑒−
𝜃
2
𝑥𝑖

2
− 𝑒−

𝜃
2
𝑥𝑖

2(𝛼2+1)

𝛼2
]

𝑛

𝑖=1

∗ [𝑒−
𝜃
2
𝑥𝑖

2
− 𝑒−

𝜃
2
𝑥𝑖

2(𝛼2+1)] 

 

 𝑔1(𝜃) = ∑  
xi

2(α2 + 1)

α2
 [yi −

(α2 + 1)e−
θ
2
xi
2

− e−
θ
2
xi
2
(α2+1)

α2
]

n

i=1

 ∗ [e−
θ
2
xi
2

− e−
θ
2
xi
2
(α2+1)]         (30) 

 

d ∑ ε
𝑖

2

dα
= 2 ∑ [yi −

(α2 + 1)e−
θ
2
xi
2
− e−

θ
2
xi
2(α2+1)

α2
]     

n

i=1

 

               ∗ (− [[
1

α2
[2αe−

θ
2
xi
2
+ θαx2e−

θ
2
xi
2(α2+1)] + [(α2 + 1)e−

θ
2
xi
2
− e−

θ
2
xi
2(α2+1)]] (− 2

𝛼3
))   

 

           
d ∑ ε

𝑖

2

dα
= 2 ∑ [yi −

(α2 + 1)e−
θ
2
xi
2
− e−

θ
2
xi
2(α2+1)

α2
]

n

i=1

 

                       ∗ (
2

α3
[(α2 + 1)e−

θ
2
xi
2
− e−

θ
2
xi
2(α2+1)] −

1

α2
[2αe−

θ
2
xi
2
+ θαxi

2e−
θ
2
xi
2(α2+1)]) 

 



  

 92  

 

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (1) 2021 

 

d ∑ ε
𝑖

2

dα
= 2 ∑ [yi −

(α2 + 1)e−
θ
2
xi
2
− e−

θ
2
xi
2(α2+1)

α2
]

n

i=1

[
2

α3
α2e−

θ
2
xi
2
+

2

α3
e−

θ
2
xi
2
−

2

α3
e−

θ
2
xi
2(α2+1) −

2α2

α3
e−

θ
2
xi
2

−
θα2xi

2

α3
e−

θ
2
xi
2(α2+1)]  

             

𝑔2(𝛼) = ∑ [yi −
(α2 + 1)e−

θ
2
xi
2

− e−
θ
2
xi
2
(α2+1)

α2
] [

4𝑒−
𝜃
2
𝑥𝑖

2

− 4𝑒−
𝜃
2
𝑥𝑖

2
(𝛼2+1) − 2𝜃𝛼2𝑥𝑖

2𝑒−
𝜃
2
𝑥𝑖

2
(𝛼2+1)

𝛼3
]

n

i=1

(31) 

                

 

dg1
dθ

= ∑
xi

4(α2 + 1)

2α2
  ([yi −

(α2 + 1)e−
θ
2
xi
2

− e−
θ
2
xi
2
(α2+1)

α2
] [e−

θ
2
xi
2
(α2+1) − e−

θ
2
xi
2

]

n

i=1

+
(α2 + 1)

α2
[e−

θ
2
xi
2

− e−
θ
2
xi
2
(α2+1)]

2

)                                                                          (32) 

𝑑𝑔1
𝑑𝛼

= ∑   [yi −
(α2 + 1)e−

θ
2
xi

2

− e−
θ
2
xi

2(α2+1)

α2
] (

𝜃𝛼2(𝛼2 + 1)𝑥𝑖
4𝑒−

θ
2
xi

2(𝛼2+1) − 2𝑥2 [𝑒−
θ
2
xi

2

− 𝑒−
θ
2
xi

2(𝛼2+1)]

𝛼3
)

         

n

i=1

 

              +
xi
2(α2+1)

α2
 [e−

θ
2
xi
2
− e−

θ
2
xi
2(α2+1)] [

2𝑒
−

θ
2
xi
2
−2𝑒

−
θ
2
xi
2(𝛼2+1)

−𝜃𝛼2𝑥2𝑒
−

θ
2
xi
2(𝛼2+1)

𝛼3
]                              (33)  

 

𝑑𝑔2
𝑑𝜃

= ∑ [yi −
(α2 + 1)e−

θ
2
xi
2

− e−
θ
2
xi
2
(α2+1)

α2
]

n

i=1

(
𝜃𝛼2(𝛼2 + 1)xi

4e−
θ
2
xi
2
(α2+1) − 2𝑥𝑖

2 [e−
θ
2
xi
2

− e−
θ
2
xi
2
(α2+1)]

𝛼3
)

+
xi

2(α2 + 1)

α2
[
2𝑒−

𝜃
2
𝑥𝑖

2

− 2𝑒−
𝜃
2
𝑥𝑖

2
(𝛼2+1) − 𝜃𝛼2𝑥𝑖

2𝑒−
𝜃
2
𝑥𝑖

2
(𝛼2+1)

𝛼3
] [e−

θ
2
xi
2

− e−
θ
2
xi
2
(α2+1)]    (34)        

 

𝑑𝑔2
𝑑𝛼

= ∑ [yi −
(α2 + 1)e−

θ
2
xi
2
− e−

θ
2
xi
2(α2+1)

α2
]

n

i=1

    

         (
2θ2𝛼2𝑥𝑖

4e−
θ
2
xi
2
(α2+1) − 3 [4e−

θ
2
xi
2

− 4e−
θ
2
xi
2
(α2+1) − 2θ2𝛼2𝑥𝑖

2e−
θ
2
xi
2
(α2+1)]

𝛼4
)           (35) 

       + [
4𝑒−

𝜃
2
𝑥𝑖

2
− 4𝑒−

𝜃
2
𝑥𝑖

2(𝛼2+1) − 2𝜃𝛼2𝑥𝑖
2𝑒−

𝜃
2
𝑥𝑖

2(𝛼2+1)

𝛼3
] [

2𝑒−
𝜃
2
𝑥𝑖

2
− 2𝑒−

𝜃
2
𝑥𝑖

2(𝛼2+1) + 𝛼𝑒−
𝜃
2
𝑥𝑖

2(𝛼2+1)

𝛼3
] 

 



  

 93  

 

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (1) 2021 

 

[
θk+1
αk+1

] = [
θk
αk

] − j−1 [
𝑔1(𝜃)
𝑔2(𝛼)

]                                                                                                                 (36) 

 

𝐽−1 = [

𝑑𝑔1
𝑑𝜃

𝑑𝑔1
𝑑𝛼

𝑑𝑔2
𝑑𝜃

𝑑𝑔2
𝑑𝛼

]

−1

                                                                                                                                   (37) 

 

Symmetric and square, where: The Jacobean matrix is 

 

𝑑𝑔1
𝑑𝛼

=
𝑑𝑔2
𝑑𝜃

 

Then we can terminate and stop when: 

|[
θk+1
αk+1

] − [
θk
αk

]| ≤ |
ε
𝜃

ε
𝛼

|                                                                                                                                   (38) 

 

  4. Real Data Application  

        In this section, real data for brain cancer disease is analyzed because it is widespread and 

deadly in Iraq; the time from infection to death was registered for period from 1/1/2018 to 

31/12/2018 at Republic of Iraq / Ministry of Health/Medical City. The data of size (111) 

patients are considered as complete data set. 

                                                                                                                            

X= { 23  10  14  4  14  11  20  15  20  9  7  15  10  16  12  7  11  15  16  13  14  15  12 9  14  14 

 21  28  16  16  10  5  5  13  17  9  9  9  9  22  10  13  18  22  8  19  20  10  6 12  10  20  5  12  

10  26  8  9  21  12  16  12  14  14  19  17  28  6  10  5  20  6  8  11 14  17  9  18  24  9  10  9  10 

 14  14  8  16 8 8 7 13 11 5 14 24 7 11 15 2 18 10 11 15 20 28 14 19 9 15 7 9 }.                          

                                                                    

After applying the chi square goodness of fit to test the hypothesis: 

𝐻0 = 𝑡ℎ𝑒 𝑑𝑎𝑡𝑎 𝑖𝑠 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑒𝑑 𝑎𝑠 𝑤𝑒𝑖𝑔ℎ𝑡𝑒𝑑 𝑅𝑎𝑦𝑙𝑒𝑖𝑔ℎ 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛. 

𝐻1 = 𝑡ℎ𝑒 𝑑𝑎𝑡𝑎 𝑛𝑜𝑡 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑒𝑑 𝑎𝑠 𝑤𝑒𝑖𝑔ℎ𝑡𝑒𝑑 𝑅𝑎𝑦𝑙𝑒𝑖𝑔ℎ 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛. 

Where the chi square goodness of fit statistic depends on the differences of the theoretical 

frequencies under the assumed distribution and observed differences from data and we applying 

the formula:                                                                                     

  

𝑋2 = ∑
(𝑂𝑖−𝐸𝑖)

2

𝐸𝑖

𝑁

𝑖=1

                                                                                                                                          (39) 



  

 94  

 

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (1) 2021 

 

 

 

 

We apply the chi square goodness of fit to the data to get: 

The calculate chi square test = 4.2963 

The calculate chi square test = 35.17  

Calculate chi- square < tabulate chi-square. 

We accept  𝐻0 which means the data are distributed as weighted Rayleigh distribution.  

We apply the equation (23) and (25) in Maximum likelihood method by using Math Lab 

program version (2014) to find and estimate the values of 𝜃  and 𝛼.̂ 

     Finally, we apply the equation (36) and (38) in Ordinary least square method by using 

MATHLAB program version (2014) to find and estimate the values of  𝜃 and 𝛼,̂ then we get:    

𝛼.̂𝑀𝐿𝐸 =1.9881, 𝜃𝑀𝐿𝐸 = 0.0118 

𝛼.̂𝑂𝐿𝑆 =5.9299,  �̂�𝑂𝐿𝑆= 0.0218 

4.1. Maximum Likelihood Method 

    The value of estimate parameters (�̂�, 𝜃)  will be (1.9881, 0.0118) respectively. The estimated 
Probability Density Function, Survival Function, Cumulative Distribution and Hazard 

Functions will be as illustrated in Table 1. 

 

 

 

 

 
Table 1. Estimate the values of 𝑓(t),  �̂�(t), �̂�(t), ℎ̂(t) for MLE method. 

X �̂�(t) �̂�(t) �̂�(t) �̂�(t) 

2 0.998678 0.002583 0.001322 0.002586 

4 0.981538 0.016821 0.018462 0.017137 

5 0.959154 0.028277 0.040846 0.029481 

5 0.959154 0.028277 0.040846 0.029481 

5 0.959154 0.028277 0.040846 0.029481 

5 0.959154 0.028277 0.040846 0.029481 

5 0.959154 0.028277 0.040846 0.029481 

6 0.924601 0.040877 0.075399 0.04421 

6 0.924601 0.040877 0.075399 0.04421 

6 0.924601 0.040877 0.075399 0.04421 

7 0.877585 0.052931 0.122415 0.060314 

7 0.877585 0.052931 0.122415 0.060314 

7 0.877585 0.052931 0.122415 0.060314 

7 0.877585 0.052931 0.122415 0.060314 

7 0.877585 0.052931 0.122415 0.060314 

8 0.819409 0.062997 0.180591 0.076881 

8 0.819409 0.062997 0.180591 0.076881 



  

 95  

 

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (1) 2021 

 
8 0.819409 0.062997 0.180591 0.076881 

8 0.819409 0.062997 0.180591 0.076881 

8 0.819409 0.062997 0.180591 0.076881 

8 0.819409 0.062997 0.180591 0.076881 

9 0.752565 0.070164 0.247435 0.093233 

9 0.752565 0.070164 0.247435 0.093233 

9 0.752565 0.070164 0.247435 0.093233 

9 0.752565 0.070164 0.247435 0.093233 

9 0.752565 0.070164 0.247435 0.093233 

9 0.752565 0.070164 0.247435 0.093233 

9 0.752565 0.070164 0.247435 0.093233 

9 0.752565 0.070164 0.247435 0.093233 

9 0.752565 0.070164 0.247435 0.093233 

9 0.752565 0.070164 0.247435 0.093233 

9 0.752565 0.070164 0.247435 0.093233 

9 0.752565 0.070164 0.247435 0.093233 

10 0.680162 0.074106 0.319838 0.108953 

10 0.680162 0.074106 0.319838 0.108953 

10 0.680162 0.074106 0.319838 0.108953 

10 0.680162 0.074106 0.319838 0.108953 

10 0.680162 0.074106 0.319838 0.108953 

10 0.680162 0.074106 0.319838 0.108953 

10 0.680162 0.074106 0.319838 0.108953 

10 0.680162 0.074106 0.319838 0.108953 

10 0.680162 0.074106 0.319838 0.108953 

10 0.680162 0.074106 0.319838 0.108953 

10 0.680162 0.074106 0.319838 0.108953 

11 0.605378 0.074984 0.394622 0.123864 

11 0.605378 0.074984 0.394622 0.123864 

11 0.605378 0.074984 0.394622 0.123864 

11 0.605378 0.074984 0.394622 0.123864 

11 0.605378 0.074984 0.394622 0.123864 

11 0.605378 0.074984 0.394622 0.123864 

12 0.53106 0.073266 0.46894 0.137963 

12 0.53106 0.073266 0.46894 0.137963 

12 0.53106 0.073266 0.46894 0.137963 

12 0.53106 0.073266 0.46894 0.137963 

12 0.53106 0.073266 0.46894 0.137963 

12 0.53106 0.073266 0.46894 0.137963 

13 0.459511 0.06955 0.540489 0.151356 

13 0.459511 0.06955 0.540489 0.151356 

13 0.459511 0.06955 0.540489 0.151356 

13 0.459511 0.06955 0.540489 0.151356 

14 0.392426 0.064435 0.607574 0.164197 

14 0.392426 0.064435 0.607574 0.164197 

14 0.392426 0.064435 0.607574 0.164197 

14 0.392426 0.064435 0.607574 0.164197 

14 0.392426 0.064435 0.607574 0.164197 

14 0.392426 0.064435 0.607574 0.164197 

14 0.392426 0.064435 0.607574 0.164197 

14 0.392426 0.064435 0.607574 0.164197 

14 0.392426 0.064435 0.607574 0.164197 

14 0.392426 0.064435 0.607574 0.164197 

14 0.392426 0.064435 0.607574 0.164197 

14 0.392426 0.064435 0.607574 0.164197 



  

 96  

 

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (1) 2021 

 
15 0.330929 0.058456 0.669071 0.176641 

15 0.330929 0.058456 0.669071 0.176641 

15 0.330929 0.058456 0.669071 0.176641 

15 0.330929 0.058456 0.669071 0.176641 

15 0.330929 0.058456 0.669071 0.176641 

15 0.330929 0.058456 0.669071 0.176641 

15 0.330929 0.058456 0.669071 0.176641 

16 0.275656 0.05205 0.724344 0.188823 

16 0.275656 0.05205 0.724344 0.188823 

16 0.275656 0.05205 0.724344 0.188823 

16 0.275656 0.05205 0.724344 0.188823 

16 0.275656 0.05205 0.724344 0.188823 

16 0.275656 0.05205 0.724344 0.188823 

17 0.226855 0.045562 0.773145 0.200842 

17 0.226855 0.045562 0.773145 0.200842 

17 0.226855 0.045562 0.773145 0.200842 

18 0.184472 0.03925 0.815528 0.212767 

18 0.184472 0.03925 0.815528 0.212767 

18 0.184472 0.03925 0.815528 0.212767 

19 0.148234 0.033299 0.851766 0.224641 

19 0.148234 0.033299 0.851766 0.224641 

19 0.148234 0.033299 0.851766 0.224641 

20 0.11771 0.027837 0.88229 0.236489 

20 0.11771 0.027837 0.88229 0.236489 

20 0.11771 0.027837 0.88229 0.236489 

20 0.11771 0.027837 0.88229 0.236489 

20 0.11771 0.027837 0.88229 0.236489 

20 0.11771 0.027837 0.88229 0.236489 

21 0.092372 0.022938 0.907628 0.248324 

21 0.092372 0.022938 0.907628 0.248324 

22 0.071635 0.018636 0.928365 0.260153 

22 0.071635 0.018636 0.928365 0.260153 

23 0.0549 0.014932 0.9451 0.27198 

24 0.04158 0.011801 0.95842 0.283806 

24 0.04158 0.011801 0.95842 0.283806 

26 0.02302 0.007078 0.97698 0.307456 

28 0.012156 0.004025 0.987844 0.331107 

28 0.012156 0.004025 0.987844 0.331107 

28 0.012156 0.004025 0.987844 0.331107 

 

     From Table 2. We find that the death density function is increasing with the failure times 

until (0.074984) when t=11, then the values decreasing with failure times from (0.073266) when 

t=12 until the end of failure times .We find that the survival function is decreasing with the 

increasing of failure times .We find that the hazard function is increasing with increasing 

failure times. 

 

4.2: Ordinary Least square method 

 The value of estimated (�̂�, 𝜃) will be (5.9299, 0.0218)  respectively and the estimated 

Probability Density, Survival, Distribution and Hazard Functions will be as illustrated in Table  

 

 



  

 97  

 

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (1) 2021 

 

 

Table 2. Estimate the values of 𝑓(t),  �̂�(t), �̂�(t), ℎ̂(t) for OLS method 

X �̂�(t) �̂�(t) �̂�(t) �̂�(t) 

2 0.97873 0.033599 0.02127 0.034329 
4 0.863998 0.075079 0.136002 0.086898 
5 0.783413 0.085272 0.216587 0.108846 
5 0.783413 0.085272 0.216587 0.108846 
5 0.783413 0.085272 0.216587 0.108846 
5 0.783413 0.085272 0.216587 0.108846 
5 0.783413 0.085272 0.216587 0.108846 
6 0.695008 0.090785 0.304992 0.130624 
6 0.695008 0.090785 0.304992 0.130624 
6 0.695008 0.090785 0.304992 0.130624 
7 0.6033 0.09194 0.3967 0.152395 
7 0.6033 0.09194 0.3967 0.152395 
7 0.6033 0.09194 0.3967 0.152395 
7 0.6033 0.09194 0.3967 0.152395 
7 0.6033 0.09194 0.3967 0.152395 
8 0.512414 0.089245 0.487586 0.174166 
8 0.512414 0.089245 0.487586 0.174166 
8 0.512414 0.089245 0.487586 0.174166 
8 0.512414 0.089245 0.487586 0.174166 
8 0.512414 0.089245 0.487586 0.174166 
8 0.512414 0.089245 0.487586 0.174166 
9 0.425848 0.083439 0.574152 0.195937 
9 0.425848 0.083439 0.574152 0.195937 
9 0.425848 0.083439 0.574152 0.195937 
9 0.425848 0.083439 0.574152 0.195937 
9 0.425848 0.083439 0.574152 0.195937 
9 0.425848 0.083439 0.574152 0.195937 
9 0.425848 0.083439 0.574152 0.195937 
9 0.425848 0.083439 0.574152 0.195937 
9 0.425848 0.083439 0.574152 0.195937 
9 0.425848 0.083439 0.574152 0.195937 
9 0.425848 0.083439 0.574152 0.195937 
9 0.425848 0.083439 0.574152 0.195937 

10 0.346284 0.075389 0.653716 0.217707 
10 0.346284 0.075389 0.653716 0.217707 
10 0.346284 0.075389 0.653716 0.217707 
10 0.346284 0.075389 0.653716 0.217707 
10 0.346284 0.075389 0.653716 0.217707 
10 0.346284 0.075389 0.653716 0.217707 
10 0.346284 0.075389 0.653716 0.217707 
10 0.346284 0.075389 0.653716 0.217707 
10 0.346284 0.075389 0.653716 0.217707 
10 0.346284 0.075389 0.653716 0.217707 
10 0.346284 0.075389 0.653716 0.217707 
11 0.275522 0.065981 0.724478 0.239478 
11 0.275522 0.065981 0.724478 0.239478 
11 0.275522 0.065981 0.724478 0.239478 
11 0.275522 0.065981 0.724478 0.239478 
11 0.275522 0.065981 0.724478 0.239478 
11 0.275522 0.065981 0.724478 0.239478 
12 0.214499 0.056038 0.785501 0.261249 



  

 98  

 

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (1) 2021 

 
12 0.214499 0.056038 0.785501 0.261249 
12 0.214499 0.056038 0.785501 0.261249 
12 0.214499 0.056038 0.785501 0.261249 
12 0.214499 0.056038 0.785501 0.261249 
12 0.214499 0.056038 0.785501 0.261249 
13 0.163395 0.046244 0.836605 0.28302 
13 0.163395 0.046244 0.836605 0.28302 
13 0.163395 0.046244 0.836605 0.28302 
13 0.163395 0.046244 0.836605 0.28302 
14 0.121786 0.037119 0.878214 0.30479 
14 0.121786 0.037119 0.878214 0.30479 
14 0.121786 0.037119 0.878214 0.30479 
14 0.121786 0.037119 0.878214 0.30479 
14 0.121786 0.037119 0.878214 0.30479 
14 0.121786 0.037119 0.878214 0.30479 
14 0.121786 0.037119 0.878214 0.30479 
14 0.121786 0.037119 0.878214 0.30479 
14 0.121786 0.037119 0.878214 0.30479 
14 0.121786 0.037119 0.878214 0.30479 
14 0.121786 0.037119 0.878214 0.30479 
14 0.121786 0.037119 0.878214 0.30479 
15 0.088818 0.029004 0.911182 0.326561 
15 0.088818 0.029004 0.911182 0.326561 
15 0.088818 0.029004 0.911182 0.326561 
15 0.088818 0.029004 0.911182 0.326561 
15 0.088818 0.029004 0.911182 0.326561 
15 0.088818 0.029004 0.911182 0.326561 
15 0.088818 0.029004 0.911182 0.326561 
16 0.06338 0.022077 0.93662 0.348332 
16 0.06338 0.022077 0.93662 0.348332 
16 0.06338 0.022077 0.93662 0.348332 
16 0.06338 0.022077 0.93662 0.348332 
16 0.06338 0.022077 0.93662 0.348332 
16 0.06338 0.022077 0.93662 0.348332 
17 0.044253 0.016378 0.955747 0.370102 
17 0.044253 0.016378 0.955747 0.370102 
17 0.044253 0.016378 0.955747 0.370102 
18 0.030233 0.011848 0.969767 0.391873 
18 0.030233 0.011848 0.969767 0.391873 
18 0.030233 0.011848 0.969767 0.391873 
19 0.02021 0.00836 0.97979 0.413644 
19 0.02021 0.00836 0.97979 0.413644 
19 0.02021 0.00836 0.97979 0.413644 
20 0.013219 0.005756 0.986781 0.435415 
20 0.013219 0.005756 0.986781 0.435415 
20 0.013219 0.005756 0.986781 0.435415 
20 0.013219 0.005756 0.986781 0.435415 
20 0.013219 0.005756 0.986781 0.435415 
20 0.013219 0.005756 0.986781 0.435415 
21 0.00846 0.003868 0.99154 0.457185 
21 0.00846 0.003868 0.99154 0.457185 
22 0.005298 0.002537 0.994702 0.478956 
22 0.005298 0.002537 0.994702 0.478956 
23 0.003246 0.001625 0.996754 0.500727 
24 0.001946 0.001017 0.998054 0.522498 
24 0.001946 0.001017 0.998054 0.522498 
26 0.000655 0.000371 0.999345 0.566039 



  

 99  

 

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (1) 2021 

 
28 0.000202 0.000123 0.999798 0.609581 
28 0.000202 0.000123 0.999798 0.609581 
28 0.000202 0.000123 0.999798 0.609581 

 

      From table (2), we find that the death density function is increasing with the failure times 
until (0.09194) when t=7, then decreasing with failure times from (0.089245) when t=8 until 

the end of failure times .We find that the survival function is decreasing with the increasing of 

failure times. We find that the hazard function is increasing with the increasing failure times. 

 

�̂�𝑀𝐿𝐸 = 𝟏. 𝟗𝟖𝟖𝟏, �̂�𝑀𝐿𝐸 = 𝟎. 𝟎𝟏𝟏𝟖,  �̂�𝑂𝐿𝑆 = 𝟓. 𝟗𝟐𝟗𝟗, �̂�𝑂𝐿𝑆 =0.0218 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

�̂�𝑀𝐿𝐸 = 𝟏. 𝟗𝟖𝟖𝟏, �̂�𝑀𝐿𝐸 = 𝟎. 𝟎𝟏𝟏𝟖,  �̂�𝑂𝐿𝑆 = 𝟓. 𝟗𝟐𝟗𝟗, �̂�𝑂𝐿𝑆 =0.0218 



  

 100  

 

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (1) 2021 

 

 

�̂�𝑀𝐿𝐸 = 𝟏. 𝟗𝟖𝟖𝟏, �̂�𝑀𝐿𝐸 = 𝟎. 𝟎𝟏𝟏𝟖,  �̂�𝑂𝐿𝑆 = 𝟓. 𝟗𝟐𝟗𝟗, �̂�𝑂𝐿𝑆 =0.0218 



  

 101  

 

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (1) 2021 

 

 
 

�̂�𝑀𝐿𝐸 = 𝟏. 𝟗𝟖𝟖𝟏, �̂�𝑀𝐿𝐸 = 𝟎. 𝟎𝟏𝟏𝟖,  �̂�𝑂𝐿𝑆 = 𝟓. 𝟗𝟐𝟗𝟗, �̂�𝑂𝐿𝑆 =0.0218 



  

 102  

 

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (1) 2021 

 

 
 

.Conclusion 5  

1. In the two estimation methods above, the probability estimation values of the survival 

function are decreasing with the increase in failure time values. This shows that there is an 

inverse relationship between the probability survival function and the failure times.                                                                                                            

2. In the two estimation methods, we can watch the estimation value of the probability hazard 

function is increasing with the increase in failure times, and then it is very clear that this 

relationship is direct between the times of failure and the probability hazard function.                               

3. From our reading of the values of the estimated probability density function of the two 

methods used are increasing values until the time of failure (x = 11) in MLE and (x=7)in OLS, 

then the probability density function decreases with increasing failure times. This shows us that 

the relationship between the probability density function and the failure times is not constant 

i.e. according to the values of failure times.                          

4. That clear the values of the cumulative distribution function are increasing with increasing 

the values of failure times. 



  

 103  

 

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (1) 2021 

 

5. Observing that the estimation values of the parameters which are the shape parameter and the 

scale parameter in the weighted Rayleigh distribution.  It is clear that for the �̂� values is one 

close to the other in the MLE, OLS method but on the other hand, the value of  �̂� is close to 

one unit of the other. 

In the table below, the estimate values of the two parameters for weighted Rayleigh distribution 

by MLE and OLS can be show in following Table: 

Table 3. Estimate the values of �̂�, �̂� by MLE and OLS methods 

 

6. Note that, as is known, the values of the probability hazard function depend on the value of 

the shape parameter, thus the probability hazard function is an increasing such that (α > 1) for 

the two estimation methods. 

9. The mean squares error values for the MLE, OLS methods by using the equation is: 

𝑀𝑆𝐸[�̂�(t)] =  
∑ [�̂�(t)−s(t)]2𝑛𝑖=1

𝑛
                                                                             (40) 

We find the MSE for MLE and OLS and there is a clear difference and eventually.  The MLE 

method is better than OLS as below: 

  

MSE 

OLS MLE 

0.0626 0.0006 

 

 

References 

1. Azzalini, A. A Class of Distributions which includes. The normal ones, Board of the 

foundation of the Scandinavian journal of statistics.1985, 12, 2, 171-178.  

2. Gupta, R.D.; Kundu, D. A new class of weighted exponential distributions, A journal of 

the article and Applied statistics.2009, 43, 6, 621-634. 

3. Shahbaz, S.; Shah 76غ baz, M.Q.; Butt, N.S. A class of weighted Weibull Distribution, 

Scorn Electronic Journal.2010, 6, 1, 53-59. 

4. Ramadan, M. M A class of weighed Gamma Distributions its properties, Economic 

Quality Control.2010, 26, 2, 133-144.  

5. Essam, A.A.; Mohamed, A.H.  A weighted three –parameters weibull distribution, 

Journal of Applied Sciences Research.2014, 9, 13, 6627-6635. 

6. Farahani, Z.S.M.; Khorram, E. Bayesian statistical inference for weighted Exponential 

Distribution, Taylor and Francis group. LLc.2014, 43, 6, 1384-1462.  

7. Tawfiq, LNM.; Jabber, AK. Mathematical Modeling of Groundwater Flow. Global 

Journal of Engineering Science and Researches. 2016, 3, 10, 15-22. 

�̂� �̂� The methods 

0.0118 1.9881 MLE 

0.0218 5.9299 OLS 



  

 104  

 

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (1) 2021 

 

8. Oguntunde, p. E. On the Exponentiated Weighted Exponential Distribution and its Basic 

statistical properties. PSCI Publication.2015, 10, 3, 160-167. 

9. Zahrani, B. on the estimation of Reliability of Weighted weibull Distri- bution: A 

comparative study, international journal of statistics and probability.2016, 5, 4, 1427-

7040. 

10. Al-Noor, N.H.; Hussein, L.K. Weighted Exponential distribution: Approximate Bayes 

Estimations with Fuzzy Data, scientific international conference college of science, Al-

Nahrain Unevercity.2017, 0, 1, 174-185. 

11. Oguntunde, P.E.; Ilori, K.A.; Okagbue, H. I. The inverted weighted exponential 

distribution with applications, interracial journal of Advanced and Applied 

Sciences.2018, 5, 11, 46-50. 

12. Tawfiq, LNM; Oraibi. YA. Fast Training Algorithms for Feed Forward Neural Networks. 

Ibn Al-Haitham Journal for Pure and Applied Science. 2017, 26, 1: 1275-280. 

13. Hussein. Iden Hasan and Zain. Saad Adnan. Some Aspects of Weighted Rayleigh 

Distribution. Ibn Al- Haitham Journal for Pure and Applied Science. Date of acceptance 

of publication in the above journal. 2019, 33, that will out in 2020. 

 
 

  

http://codental.uobaghdad.edu.iq/jih/index.php/j/article/view/534