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IHSCICONF 2017                                                                                         Special Issue 
Ibn Al-Haitham Journal for Pure and Applied science       https://doi.org/ 10.30526/2017.IHSCICONF.1811
                                                                                                                    
 

 

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Mathematics|378 
 

Using Approximation Non-Bayesian Computation with 
Fuzzy Data to Estimation Inverse Weibull Parameters 

and Reliability Function 

Nadia Hashim Al-Noor   
nadialnoor@uomustansiriyah.edu.iq 

Shurooq A.K.Al-Sultany 
drnadialnoor@gmail.com 

Dept. of Mathematics/ College of Science/ Mustansiriyah University 
 
 

Abstract 
        In real situations all observations and measurements are not exact numbers but more or 
less non-exact, also called fuzzy. So, in this paper, we use approximate non-Bayesian 
computational methods to estimate inverse Weibull parameters and reliability function with 
fuzzy data. The maximum likelihood and moment estimations are obtained as non-
Bayesian estimation. The maximum likelihood estimators have been derived numerically 
based on two iterative techniques namely “Newton-Raphson” and the “Expectation-
Maximization” techniques. In addition, we provide compared numerically through Monte-
Carlo simulation study to obtained estimates of the parameters and reliability function in 
terms of their mean squared error values and integrated mean squared error values 
respectively. 
 

Keywords: Non-Bayesian Methods; Inverse Weibull distribution; Expectation 
Maximization algorithm; Newton-Raphson algorithm; Moment method; Fuzzy data. 

 

 

 

 

 

 

 

 

 

 



IHSCICONF 2017                                                                                         Special Issue 
Ibn Al-Haitham Journal for Pure and Applied science       https://doi.org/ 10.30526/2017.IHSCICONF.1811
                                                                                                                    
 

 

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 1.Introduction 
Statistical analysis of inverse Weibull distribution is based on exact data. However, in real 
situations all observations and measurements are not exact numbers but more or less non-
exact, also called fuzzy. Thus, in this paper we present a non-Bayesian approach to 
estimate the parameters and reliability function of inverse Weibull distribution with fuzzy 
data. Keller et al. (1985) [4] introduced inverse Weibull distribution as a suitable model to 
describe degradation phenomena of mechanical components of diesel engines. Other names 
for this distribution are complementary Weibull distribution, reciprocal Weibull 
distribution and reverse Weibull distribution [7]. 
A random variable X is said to have a two-parameter inverse Weibull distribution if it has 
the following probability density function (PDF)[8], 

𝑓 𝑥; 𝛽, 𝜆 𝛽𝜆𝑥 𝑒 ; 𝑥 0, 𝛽, 𝜆 0                                                                             … 1  

The cumulative distribution function(CDF), reliability function is given respectively by: 

𝐹 𝑥; 𝛽, 𝜆 𝑒
 
; 𝑥 0, 𝛽, 𝜆 0                                                                                                … 2  

𝑅 𝑥 1 𝑒 ; 𝑥 0, 𝛽, 𝜆 0                                                                                                   … 3  

where 𝛽  is the shape parameter and 𝜆 is the scale parameter.  

Definition [9]: let ℝ , 𝒜, P  be a probability space in which 𝒜 is the σ  field of Borel 
sets in ℝ  and P is a probability measure over ℝ . Then, the probability of a fuzzy event A 
in ℝ  is defined by: 

𝑃 𝐴 𝜇 𝑥 𝑑𝑃         ; 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 ∈ ℝ                                                                                          … 4  

In particular, suppose that P be the probability distribution of a continuous random variable 
𝑋 with PDF 𝑔 𝑥 .The conditional density of 𝑋 given 𝐴 is given by: 

𝑔 𝑥|𝐴
𝜇 𝑥  𝑔 𝑥

𝜇 𝑢 𝑔 𝑢 𝑑𝑢
                                                                                                                    … 5  

 

Fuzzy Data and the Likelihood Function 
Let x x , x  , … , x  be an (i.i.d.) random vector of a random sample of size 𝑛 from 
inverse Weibull distribution with PDF given by (1). If an observations of 𝑥 was known 
exactly, then the complete-data likelihood function is: 

𝐿 𝛽, 𝜆; 𝑥 𝑓 𝑥 , 𝛽, 𝜆 𝛽 𝜆 𝑥  𝑒𝑥𝑝 𝜆𝑥  



IHSCICONF 2017                                                                                         Special Issue 
Ibn Al-Haitham Journal for Pure and Applied science       https://doi.org/ 10.30526/2017.IHSCICONF.1811
                                                                                                                    
 

 

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⇒ 𝐿 𝛽, 𝜆; 𝑥 𝛽  𝜆 𝑥  𝑒𝑥𝑝 𝜆 𝑥                                                                  … 6  

Now consider the problem where x is not observed precisely and only partial information 
about  x  is available in the form of a fuzzy subset x with the Borel measurable membership 
function 𝜇 𝑥 . The observed-data natural log-likelihood function can be obtained, using 
Zadeh's definition of the probability of a fuzzy event in the expression (5) as: 

ℓ 𝛽, 𝜆; 𝑥  𝑙𝑛 𝐿 𝛽, 𝜆; 𝑥 𝑙𝑛 𝛽𝜆 𝑥 𝑒𝑥𝑝 𝜆𝑥 𝜇 𝑥 𝑑𝑥 

⇒ ℓ 𝛽, 𝜆; 𝑥 𝑛 𝑙𝑛 𝛽 𝑛 𝑙𝑛 𝜆 𝑙𝑛 𝑥 𝑒𝑥𝑝 𝜆𝑥 𝜇 𝑥 𝑑𝑥                              . . . 7  

Maximum likelihood Estimations 
Differentiating the natural log-likelihood function ℓ β, λ; x , given by (7), partially with 
respect to β and λ and then equating to zero we have: 

𝜕ℓ 𝛽, 𝜆; 𝑥
𝜕𝛽

𝑛
𝛽

𝜆
𝑥

1
𝑥

 𝑙𝑛 𝑥   𝑒  𝜇 𝑥 𝑑𝑥

1
𝑥

   𝑒  𝜇 𝑥 𝑑𝑥
0                                        … 8  

𝜕ℓ 𝛽, 𝜆; 𝑥
𝜕𝜆

𝑛
𝜆

1
𝑥

    𝑒 𝜇 𝑥 𝑑𝑥

1
𝑥

𝑒 𝜇 𝑥 𝑑𝑥
0                                                                  … 9  

The maximum likelihood estimates (MLEs) of 𝛽 and 𝜆 are the solution of the likelihood 
equations (8) and (9). Since there is no closed form of the solutions, an iterative 
approximation technique can be used to obtain the MLEs. In the following, we consider 
two iterative approximation techniques namely the expectation-maximization (EM) 
algorithm and Newton Raphson (NR) algorithm. 

Expectation-Maximization Algorithm 
Dempster et al. (1977) [2] presented a general approach to iterative computation of 
maximum-likelihood estimates when the observations can be viewed as incomplete data. 
Now, since the observed fuzzy data 𝑥 can be seen as an incomplete specification of a 
complete data vector  x, the EM algorithm is appropriate to find the MLEs of the unknown 
parameters [3][5].  
Form equation (6), the natural log-likelihood function for x becomes: 

ℓ 𝛽, 𝜆; 𝑥 𝑙𝑛 𝐿 𝛽, 𝜆; 𝑥  



IHSCICONF 2017                                                                                         Special Issue 
Ibn Al-Haitham Journal for Pure and Applied science       https://doi.org/ 10.30526/2017.IHSCICONF.1811
                                                                                                                    
 

 

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⇒ ℓ 𝛽, 𝜆; 𝑥 𝑛 𝑙𝑛 𝛽 𝑛 𝑙𝑛 𝜆 𝛽 1 𝑙𝑛 𝑥 𝜆 𝑥                                             … 10  

The EM algorithm is given by the following iterative process: 

1. Set ℎ 0 and initial values of 𝛽 and λ, say 𝛽  and  λ  . 
2. At iteration h 1 , compute the E and M steps. 
3. Repeat step (2) until the convergence occurs, i.e. the absolute difference between two 
successive iterations is less than ε for some pre-fixed  ε 0. when the convergence occurs 
then the present β  and λ  be the MLEs of 𝛽 and 𝜆 via EM algorithm which we 
referred to as β , λ . 

E-step: The E-step of the EM algorithm at iteration h 1  requires to compute the 
following conditional expectations using the expression (5), 

𝐸 , ℓ 𝛽, 𝜆; 𝑥|𝑥

𝑛 𝑙𝑛 𝛽 𝑛 𝑙𝑛 𝜆 𝛽 1 𝐸 , 𝑙𝑛 𝑥 |𝑥

𝜆 𝐸 , 𝑥 |𝑥                                                                                    … 11  

where the conditional expectations of X given x is computed as: 

𝐸 , 𝑙𝑛 𝑋 |𝑥
𝑥  𝑙𝑛 𝑥  𝑒    𝜇 𝑥 𝑑𝑥 

𝑥  𝑒     𝜇 𝑥 𝑑𝑥 
                                         … 12  

𝐸 , 𝑋 |𝑥
𝑥  𝑒   𝜇 𝑥 𝑑𝑥 

𝑥  𝑒   𝜇 𝑥 𝑑𝑥 
                                                     … 13  

M-step: The M-step of EM algorithm involves maximizing equation (11) with respect to β 
and λ. This is easily achieved by differentiating equation (11) partially with respect to β 
and λ and then equating to zero which implies that: 

𝛽
𝑛

∑ 𝐸 , 𝑙𝑛 𝑋 |𝑥 𝜆 ∑ 𝐸 , 𝑋 𝑙𝑛 𝑋 |𝑥  
                       … 14  

𝜆
𝑛

∑ 𝐸 , 𝑋 |𝑥
                                                                                                  … 15  

where 𝐸 , 𝑋 𝑙𝑛 𝑋 |𝑥  computed as: 

𝐸 , 𝑋 𝑙𝑛 𝑋 |𝑥
𝑥 𝑙𝑛 𝑥  𝑒  𝜇 𝑥 𝑑𝑥 

𝑥  𝑒   𝜇 𝑥 𝑑𝑥 
                                  … 16  



IHSCICONF 2017                                                                                         Special Issue 
Ibn Al-Haitham Journal for Pure and Applied science       https://doi.org/ 10.30526/2017.IHSCICONF.1811
                                                                                                                    
 

 

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𝐸 , 𝑙𝑛 𝑋 |𝑥  and 𝐸 , 𝑋 |𝑥  as in (12) and (13). We can rewrite 𝛽  and 𝜆  in 

(14) and (15) as: 

𝛽
1
𝑛

𝐸 𝜆 𝐸                                                                                   … 17  

𝜆
𝑛

∑ 𝐸
                                                                                                                             … 18  

where: 

𝐸 𝐸 , 𝑙𝑛 𝑋 |𝑥  ,  𝐸 𝐸 , 𝑋 |𝑥  and 𝐸 𝐸 , 𝑋 𝑙𝑛 𝑋 |𝑥  

Newton-Raphson Algorithm 
The steps of the NR algorithm are [5][7]: 

1. Set ℎ 0 and initial values of 𝛽 and λ, say 𝛽  and  λ  . 
2. At iteration ℎ 1 , estimate the new value of β and λ, as: 

𝛽
𝜆

 
𝛽
𝜆

⎣
⎢
⎢
⎢
⎡𝜕 ℓ 𝛽, 𝜆; 𝑥

𝜕𝛽
𝜕 ℓ 𝛽, 𝜆; 𝑥

𝜕𝛽𝜕𝜆
𝜕 ℓ 𝛽, 𝜆; 𝑥

𝜕𝛽𝜕𝜆
 
𝜕 ℓ 𝛽, 𝜆; 𝑥

𝜕𝜆 ⎦
⎥
⎥
⎥
⎤

 ⎣
⎢
⎢
⎢
⎡𝜕ℓ 𝛽, 𝜆; 𝑥

𝜕𝛽
𝜕ℓ 𝛽, 𝜆; 𝑥

𝜕𝜆 ⎦
⎥
⎥
⎥
⎤

 

              … 19  

where  the first–order derivatives of the natural log-likelihood with respect to β and λ , 
required for proceeding with the NR algorithm, are obtained as in (8) and (9) and the 
second–order derivatives are obtained as follows. 
𝜕 ℓ 𝛽, 𝜆; 𝑥

𝜕𝛽
𝑛

𝛽
𝜆

𝑥
  

𝜆
𝑥

𝑒  𝑙𝑛 𝑥  𝜇 𝑥 𝑑𝑥

1
𝑥

 𝑒   𝜇 𝑥 𝑑𝑥

1
𝑥

2𝜆
𝑥

 𝑙𝑛 𝑥   𝑒 𝜇 𝑥 𝑑𝑥

1
𝑥

   𝑒 𝜇 𝑥 𝑑𝑥

𝜆
𝑥

1
𝑥

 𝑙𝑛 𝑥   𝑒 𝜇 𝑥 𝑑𝑥

1
𝑥

   𝑒 𝜇 𝑥 𝑑𝑥
                              … 20  

𝜕 ℓ 𝛽, 𝜆; 𝑥
𝜕𝜆

𝑛
𝜆

1
𝑥

   𝑒 𝜇 𝑥 𝑑𝑥

1
𝑥

   𝑒 𝜇 𝑥 𝑑𝑥

1
𝑥

   𝑒 𝜇 𝑥 𝑑𝑥

1
𝑥

   𝑒 𝜇 𝑥 𝑑𝑥
                                                            … 21  



IHSCICONF 2017                                                                                         Special Issue 
Ibn Al-Haitham Journal for Pure and Applied science       https://doi.org/ 10.30526/2017.IHSCICONF.1811
                                                                                                                    
 

 

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𝜕 ℓ 𝛽, 𝜆; 𝑥
𝜕𝛽𝜕𝜆

𝜆
𝑥

2
𝑥

 𝑙𝑛 𝑥   𝑒 𝜇 𝑥 𝑑𝑥

1
𝑥

   𝑒 𝜇 𝑥 𝑑𝑥
 

1
𝜆

𝑥
1

𝑥
 𝑙𝑛 𝑥  𝑒 𝜇 𝑥 𝑑𝑥 

1
𝑥

   𝑒 𝜇 𝑥 𝑑𝑥 

1
𝑥

   𝑒 𝜇 𝑥 𝑑𝑥
                 … 22  

 

3. Repeat step (2) until the convergence occurs. When the convergence occurs then the 

present β  and λ  be the MLEs of β and λ via NR algorithm which we referred to as 
β , λ . 

Now, depending on the invariant property of MLE, the MLE of the reliability function of 

inverse Weibull distribution via EM and NR algorithms, denoted by R t   and R t  
respectively, can be obtained by replacing β and λ in (3) by their MLE estimates as: 

𝑅 𝑡 1 𝑒    ;  𝑡 0                                                                                          … 23  

𝑅 𝑡 1 𝑒     ;  𝑡 0                                                                                          … 24  

Moment Estimations 
The method of moments is one the oldest method for deriving point estimators. The 
moment estimates for β and λ of inverse Weibull distribution can be found by the following 
two equations which are obtained by equating the first and the second population moments 
to the corresponding sample moments, that is: 

λ Г 1
1
β

1
n

E , X|x                                                                                                      …  25  

λ Г 1
2
β

1
n

E , X |x                                                                                                    …  26  

Note that, the direct form of the solutions to equations (25) and (26) could not be obtained. 
However, by using an iterative numerical process, we can obtain the parameter estimates as 
described below: 

Step (1) Set ℎ 0 and initial values of 𝛽 and λ, say 𝛽  and  λ  . 

Step (2)  At  h 1   iteration, using the expression (5) to compute the following 
conditional expectation, 



IHSCICONF 2017                                                                                         Special Issue 
Ibn Al-Haitham Journal for Pure and Applied science       https://doi.org/ 10.30526/2017.IHSCICONF.1811
                                                                                                                    
 

 

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E , X |x
x  e μ x  dx

x  e μ x  dx
 ;   r 1,2 

Step (3)  Solve the following equation for β, based on equations (25) and (26), to obtain the 
solution as β , 

Г 1
1
β

Г 1
2
β

∑ E , X|x

n ∑ E , X |x
 

Step (4)  Obtain the solution for λ, say λ , through the following equation, 

λ
∑ E , X|x

n Г 1
1

β

 

Step (5)  Setting h h 1, repeat step (2) to step (4) until convergence occurs. When the 
convergence occurs then the present β  and λ  be the moment estimates of β and λ 
which we referred to as β , λ . 

Now, depending on the moment estimates of the shape and scale parameters, the 
approximated moment estimate of the reliability function of inverse Weibull distribution at 

mission time t, denoted by R t , can be obtained by replacing  β and λ in equation (3) by 
their moment estimates as: 

R t 1 e    ;  t 0                                                                                  … 27  

Simulation Study 
In trying to illustrate and compare the algorithms as described above, a Monte-Carlo 
simulation study was perform to generate an independent identical distributed random 
samples, say x, according to inverse Weibull distribution through the adoption of inverse 
transformation method with size n = 20, 30 and 90 to take care of small, medium and large 
data sets. The number of sample replicated chosen to be (100). The shape parameter was 
chosen to be 3, 2.1, 1 and 0.5 and the scale parameter 0.5, 1, 3. Then, each observation of x 
was made fuzzy based on an appropriate selected membership function among the 
following eight membership functions in the FIS shown in figure (1). The simulation 
program has been written by using MATLAB (R2010b) program. The results of Monte-
Carlo simulation have been summarized in the tables (1)…(3). 



IHSCICONF 2017                                                                                         Special Issue 
Ibn Al-Haitham Journal for Pure and Applied science       https://doi.org/ 10.30526/2017.IHSCICONF.1811
                                                                                                                    
 

 

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Mathematics|385 
 

 

Figure (1): FIS used to Encode the Simulated Data [5 & 6] 

 

μ x

1  ;         x 0.05
0.25 x

0.2
;   0.05 x 0.25

0  ;        otherwise

 μ x

⎩
⎪
⎨

⎪
⎧

x 0.75
0.25

; 0.75 x 1

1.5 x
0.5

; 1 x 1.5  

0 ; otherwise    

 

μ x

⎩
⎪
⎨

⎪
⎧

x 0.05
0.2

;   0.05 x 0.25

0.5 x
0.25

;   0.25 x 0.5

0  ;            otherwise

 μ x

⎩
⎪
⎨

⎪
⎧

x 1
0.5

; 1 x 1.5

2 x
0.5

; 1.5 x 2  

0 ; otherwise    

 

μ x

⎩
⎪
⎨

⎪
⎧

x 0.25
0.25

;     0.25 x 0.5

0.75 x
0.25

;   0.5 x 0.75

0  ;            otherwise

 μ x

x 1.5
0.5

; 1.5 x 2    

3 x ; 2 x 3  
0 ; otherwise      

 

μ x

⎩
⎪
⎨

⎪
⎧

x 0.5
0.25

;   0.5 x 0.75

1 x
0.25

;    0.75 x 1

0    ;      otherwise  

 μ x
x 2 ; 2 x 3

1 ;      x 3      
0 ; otherwise        

 

The initial values required for proceeding with the Expectation-Maximization , Newton-
Raphson algorithms and moment method chosen to be the symmetrical rank regression 
estimators and the iterative process stops when the absolute difference between two 
successive iterations becomes less than ε 0.0001. The comparisons between parameter 
estimates were based on values from Mean Square Error (MSE) while it were based on 
values from Integrated Mean Square Error (IMSE) for the estimates of the reliability 
function [1], where: 

𝑀𝑆𝐸 𝛽

∑ 𝛽 𝛽𝐿

𝐿
                                                                                                          … 28  



IHSCICONF 2017                                                                                         Special Issue 
Ibn Al-Haitham Journal for Pure and Applied science       https://doi.org/ 10.30526/2017.IHSCICONF.1811
                                                                                                                    
 

 

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𝑀𝑆𝐸 𝜆

∑ 𝜆 𝜆𝐿

𝐿
                                                                                                           … 29  

𝐼𝑀𝑆𝐸 𝑅 𝑡
1
𝐿

1
𝑛

 𝑅 𝑡

𝑅 𝑡                                                                 … 30  

𝛽 , 𝜆  : is the estimate of 𝛽 and  𝜆 respectively at the j th replicate (run). 

𝐿: is the number of sample replicated. 

𝑛  : is the number of times chosen to be (4) where 𝑡 1,2,3,4 . 

𝑅 𝑡 : is the estimates of  𝑅 𝑡  at the 𝑗  replicate (run) and 𝑖  time. 

2. Conclusions and Recommendations 
Approximate maximum likelihood and moment estimations as non-Bayesian estimations 
have been used to estimate the parameters and reliability function of inverse Weibull 
distribution according to fuzzy data. The maximum likelihood estimators have been derived 
numerically based on two iterative techniques namely “Newton-Raphson” and the 
“Expectation-Maximization” techniques. In addition, we provide compared numerically 
through Monte-Carlo simulation study to obtained estimates of the parameters and 
reliability function in terms of their mean squared error (MSE) values and integrated mean 
squared error (IMSE) values respectively. 
The most important conclusions of Monte-Carlo simulation results are: 

Table (1): The maximum likelihood estimates based on Newton-Raphson algorithm 
introduced the best perform compared with other estimates with different values of the 
shape and scale parameters for all sample sizes except for small sample size with 
parameters, β 3, λ 0.5 and β 3, λ 3 . 

Table (2): The maximum likelihood estimates based on Newton-Raphson algorithm 
introduced the best perform compared with other estimates for all sample sizes except for 
large sample size with parameters β 2.1, λ 1, where Moment estimate is the best. 

Table (3): The maximum likelihood estimates based on NR introduced the best perform 
compared with other estimates with different values of the shape and scale parameters for 
all sample sizes with β 2.1, 3 and λ 0.5, 1 as well as for moderate sample size with 
β 2.1 and λ 3. On the other hand, the maximum likelihood estimates based on EM 



IHSCICONF 2017                                                                                         Special Issue 
Ibn Al-Haitham Journal for Pure and Applied science       https://doi.org/ 10.30526/2017.IHSCICONF.1811
                                                                                                                    
 

 

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Mathematics|387 
 

introduced the best perform compared with other non-Bayes estimates for all sample sizes 
with β=3 and λ=3 as well as for small and large sample sizes with β 2.1 and λ 3. 

Tables 1, 2 and 3 clearly show that as the sample size increases, the MSE and IMSE values 
of the estimates decrease. 
Based on this, we recommend, 
Using the maximum likelihood estimates based on Newton-Raphson algorithm for 
estimating the shape parameter of the inverse Weibull distribution especially with moderate 
and large sample sizes and we have to be careful in choosing the approximation techniques 
for estimating the shape parameter of this distribution when dealing with small sample size. 
Using the maximum likelihood estimates based on Newton-Raphson algorithm for 
estimating the scale parameter of the inverse Weibull distribution especially with small and 
moderate sizes and we have to be careful in choosing the approximation techniques for 
estimating the scale parameter of this distribution when dealing with large sample size. 
Using the maximum likelihood estimates based on Newton-Raphson algorithm for 
estimating the reliability function of the inverse Weibull distribution with β 2.1, 3 and 
λ 0.5, 1 and using the maximum likelihood estimates based on Expectation-
Maximization algorithm with β 3 and  λ 3  . 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 



IHSCICONF 2017                                                                                         Special Issue 
Ibn Al-Haitham Journal for Pure and Applied science       https://doi.org/ 10.30526/2017.IHSCICONF.1811
                                                                                                                    
 

 

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Mathematics|388 
 

Table (1): MSE Values for Non-Bayes Estimates of the Shape Parameter (β) of Inverse 
Weibull Distribution with Different Cases 

𝑛 MO ML Best 
Estimate EM NR 

β 2.1 , λ 0.5 
20 0.5925534 0.3276789 0.3210075 ML-NR 
30 0.5668387 0.1731912 0.1720976 ML-NR 
90 0.5026410 0.0807156 0.0802433 ML-NR 

β 3 , λ 0.5 
20 0.5945321 0.8529363 0.7876280 MO 
30 0.4313349 0.3780456 0.3622704 ML-NR 
90 0.4008172 0.0871851 0.0861431 ML-NR 

β 2.1 , λ 1 
20 1.0234089 0.4793374 0.4697234 ML-NR 
30 0.8987515 0.3765942 0.3700855 ML-NR 
90 0.8584988 0.1780456 0.1764240 ML-NR 

β 3 , λ 1 
20 0.8207644 0.7696865 0.7355563 ML-NR 
30 0.6615921 0.3721112 0.3652719 ML-NR 
90 0.5914806 0.1426038 0.1412609 ML-NR 

β 2.1 , λ 3 
20 2.4264343 1.4173415 1.3050008 ML-NR 
30 2.1658246 0.9206690 0.8674874 ML-NR 
90 2.1064719 0.5787366 0.5542025 ML-NR 

β 3 , λ 3 
20 1.2356461 0.8079456 0.8175703 ML-EM 
30 1.2346853 0.7552212 0.7170664 ML-NR 
90 1.1146329 0.2692730 0.2583540 ML-NR 

  
𝑛: Sample Size 
MO: Moment Estimate 
ML: Maximum Likelihood Estimate 
EM: Expectation-Maximization Algorithm 
NR: Newton-Raphson Algorithm 

 

 

 



IHSCICONF 2017                                                                                         Special Issue 
Ibn Al-Haitham Journal for Pure and Applied science       https://doi.org/ 10.30526/2017.IHSCICONF.1811
                                                                                                                    
 

 

For more information about the Conference please visit the websites:   
http://www.ihsciconf.org/conf/ 

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Mathematics|389 
 

Table (2): MSE Values for Non-Bayes Estimates of the Scale Parameter (λ) of Inverse 
Weibull Distribution with Different Cases 

n MO ML Best 
Estimate EM NR 

β 2.1 , λ 0.5 
20 0.0261692 0.0212376 0.0211103 ML-NR 
30 0.0185599 0.0146520 0.0145815 ML-NR 
90 0.0096941 0.0066130 0.0065777 ML-NR 

β 3 , λ 0.5 
20 0.0258200 0.0221413 0.0216447 ML-NR 
30 0.0187999 0.0153414 0.0151493 ML-NR 
90 0.0061082 0.0049155 0.0048935 ML-NR 

β 2.1 , λ 1 
20 0.1178850 0.0874576 0.0864178 ML-NR 
30 0.0539672 0.0495897 0.0493294 ML-NR 
90 0.0171838 0.0222780 0.0220901 MO 

β 3 , λ 1 
20 0.1266382 0.0949276 0.0932050 ML-NR 
30 0.0689319 0.0553591 0.0549835 ML-NR 
90 0.0431480 0.0176048 0.0172905 ML-NR 

β 2.1 , λ 3 
20 10.1555872 5.9423094 5.0189589 ML-NR 
30 6.6359697 3.0022532 2.5016068 ML-NR 
90 3.9878570 0.5756619 0.5583213 ML-NR 

β 3 , λ 3 
20 4.2045048 5.2462299 3.5490963 ML-NR 
30 3.8376384 5.1730105 3.0378761 ML-NR 
90 2.1648816 0.3457172 0.3411649 ML-NR 

 
𝑛: Sample Size 
MO: Moment Estimate 
ML: Maximum Likelihood Estimate 
EM: Expectation-Maximization Algorithm 
NR: Newton-Raphson Algorithm 
 

 



IHSCICONF 2017                                                                                         Special Issue 
Ibn Al-Haitham Journal for Pure and Applied science       https://doi.org/ 10.30526/2017.IHSCICONF.1811
                                                                                                                    
 

 

For more information about the Conference please visit the websites:   
http://www.ihsciconf.org/conf/ 

   www.ihsciconf.org  

Mathematics|390 
 

Table (3): IMSE Values for Non-Bayes Estimates of the Reliability Function of Inverse 
Weibull Distribution with Different Cases 

n MO ML Best 
Estimate EM NR 

β 2.1 , λ 0.5 

20 0.0038869 0.0030015 0.0029777 ML-NR 

30 0.0028681 0.0020454 0.0020271 ML-NR 

90 0.0019454 0.0010003 0.0009953 ML-NR 

β 3 , λ 0.5 

20 0.0027426 0.0025390 0.0024743 ML-NR 

30 0.0018299 0.0015330 0.0015252 ML-NR 

90 0.0007605 0.0005639 0.0005614 ML-NR 

β 2.1 , λ 1 

20 0.0068683 0.0050789 0.0050384 ML-NR 

30 0.0062469 0.0045062 0.0044766 ML-NR 

90 0.0041694 0.0024271 0.0024047 ML-NR 

β 3 , λ 1 

20 0.0037806 0.0033036 0.0032750 ML-NR 

30 0.0024769 0.0022311 0.0022310 ML-NR 

90 0.0016173 0.0008200 0.0008185 ML-NR 

β 2.1 , λ 3 

20 0.0196349 0.0138542 0.0140972 ML-EM 

30 0.0189882 0.0120847 0.0120259 ML-NR 

90 0.0182735 0.0106072 0.0106827 ML-EM 

β 3 , λ 3 

20 0.0040808 0.0031302 0.0032523 ML-EM 

30 0.0035055 0.0024299 0.0025396 ML-EM 

90 0.0032125 0.0015585 0.0015589 ML-EM 

 

𝑛: Sample Size 
MO: Moment Estimate 
ML: Maximum Likelihood Estimate 
EM: Expectation-Maximization Algorithm 
NR: Newton-Raphson Algorithm 
 

 



IHSCICONF 2017                                                                                         Special Issue 
Ibn Al-Haitham Journal for Pure and Applied science       https://doi.org/ 10.30526/2017.IHSCICONF.1811
                                                                                                                    
 

 

For more information about the Conference please visit the websites:   
http://www.ihsciconf.org/conf/ 

   www.ihsciconf.org  

Mathematics|391 
 

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