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Parametric Models in Survival Analysis for Lung Cancer Patients  

 

Layla A. Ahmed 

layla.aziz@garmian.edu.krd 

Department of Mathematics, College of Education, University of Garmian, Kurdistan Region, Iraq 

 

 

 

 

 

Abstract 

    The aim of this study is to estimate the survival function for the data of lung cancer patients, 

using parametric methods (Weibull, Gumbel, exponential and log-logistic). 

Comparisons between the proposed estimation method have been performed using statistical 

indicator Akaike information Criterion, Akaike information criterion corrected and Bayesian 

information Criterion, concluding that the survival function for the lung cancer by using 

Gumbel distribution model is the best. The expected values of the survival function of all 

estimation methods that are proposed in this study have been decreasing gradually with 

increasing failure times for lung cancer patients, which means that there is an opposite 

relationship failure times and survival function. 

Keywords: Survival analysis, Weibull distribution, Gumbel distribution, Exponential 

distribution, Log-logistic distribution. 

1.Introduction 

     Survival analysis is a branch of statistics which deals with the analysis of time to events, 

such as death in biological organisms and failure in mechanical systems. The topic of survival 

analysis is called reliability theory or reliability analysis in engineering, and duration analysis 

or duration modeling in economics or event history analysis in sociology [1]. In many applied 

sciences such as medicine, engineering and finance, amongst others, modeling and analyzing 

lifetime data are crucial.  

Several lifetime distributions have been used to model such type of data. The quality of the 

executions and in a statistical analysis depends heavily on the presupposed probability 

distributions [2].  

Parametric methods which involve the exponential, Weibull, lognormal, gamma and extreme 

value distribution have been widely used for fitting survival data [3]. 

Ibn Al Haitham Journal for Pure and Applied Science 

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

 

Doi: 10.30526/34.2.2617 

Article history: Received, 7,January,2020, Accepted 20 ,Februry,2020, Published in April 2021 

 

mailto:layla.aziz@garmian.edu.krd


  

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Gumbel showed that the Weibull distribution and the type 𝐼𝐼𝐼 smallest extreme value 

distribution are the same [4]. 

Log- logistic distribution is a very important reliability model as it fits well in many applied 

situations of reliability data analysis. Another advantage with the log- logistic distribution lies 

in its closed form expression for survival and failure rate functions that makes it important 

over log- normal distribution [5]. 

Salman and Farhan [6] estimated the survival function for the patients of lung cancer; they 

used several nonparametric estimation methods, and concluded that the shrinkage was the 

best method. 

      The main objective of this research is to estimate the survival function for the data of lung 

cancer patients, by using parametric methods and determine the best and most efficient 

distribution. 

 

2. Theoretical Part  

2.1 Survival Analysis  

       Survival function is the probability that a system or component will survive without 

failure during a specified time interval [0, 𝑡] under given operating conditionals, denoted by 𝑆, 

which is defined as [7]: 

 𝑆(𝑡) = 𝑃(𝑇 > 𝑡) = ∫ 𝑓(𝑢)𝑑𝑢   , 𝑡 ≥ 0
∞

𝑡
                                                              (1) 

Where, 𝑇 is a random variable, 𝑡 is the time of death. 

The survival function 𝑆(𝑡) is the probability that the patient will survival till time 𝑡. Survival 

probability is usually assumed to approach zero as age increases, with: 

 𝑆(0) = 1 

 lim
𝑡→∞

𝑆(𝑡) = 0 

 𝑆(𝑡) is decreasing and continuous from right side. It is linked with the failure distribution 

function 𝐹(𝑡) and in fact, it is the complement of it, i.e.: 

 𝑆(𝑡) = 1 − 𝐹(𝑡)                                                                                             (2) 

 
𝑑

𝑑𝑡
𝑆(𝑡) = −𝑓(𝑡)                                                                                                     (3) 

2.2 Life Time Distribution Function   

    The life time distribution function, is defined as the complement of the survival function 

[1], 

𝐹(𝑡) = 𝑃(𝑇 ≤ 𝑡) = 1 − 𝑆(𝑡)                                                                                 (4) 

If 𝐹(𝑡) is differentiable then the derivative, which is the density function of the lifetime 

distribution is, 

 𝑓(𝑡) =
𝑑

𝑑𝑡
𝐹(𝑡)                                                                                                        (5) 



  

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The function 𝑓(𝑡) is sometimes called the event density; it is the rate of death or failure events 

per unit time. 

2.3 Hazard Function  

    Hazard function is also known as the immediate failure rate [8]. This is the limit of the 

conditional probability that an item will fail in the time interval [𝑡, 𝑡 + ∆𝑡]  when we know 

that the item is functioning at time t is 

 𝑃(𝑡 < 𝑇 ≤ 𝑡 + ∆𝑡 𝑇⁄ > 𝑡) =
𝑃(𝑡<𝑇≤𝑡+∆𝑡)

𝑃(𝑇>𝑡)
 

                                           =
𝐹(𝑡+∆𝑡)−𝐹(𝑡))

𝑆(𝑡)
                                                                (6) 

By dividing this probability by the length of the time interval, ∆𝑡, and letting ∆𝑡 → 0, we get 

the rate function ( ℎ(𝑡)) , and  it is defined as: 

 ℎ(𝑡) = lim
∆𝑡→0

𝐹(𝑡+∆𝑡)−𝐹(𝑡)

𝑆(𝑡)
      

 ℎ(𝑡) =
𝑑𝐹(𝑡)

𝑑𝑡

1

𝑆(𝑡)
 

 ℎ(𝑡) =
𝑓(𝑡)

𝑆(𝑡)
                                                                                                               (7) 

Where 𝑓(𝑡) 𝑖𝑠 the failure density functions, and 𝑆(𝑡) is the survival function. 

Then, 

 𝐻(𝑡) =
−𝑑𝑙𝑛𝑆(𝑡)

𝑑𝑡
                                                                                                       (8) 

 𝑆(𝑡) = exp (− ∫ ℎ(𝑠)𝑑𝑠)
𝑡

0
                                                 

So hazard is instantaneous mortality rate conditional on previous survival, and the integrated 

form of cumulative hazard 

 𝐻(𝑡) = ∫ ℎ(𝑠)𝑑𝑠
𝑡

0
= − ln 𝑆(𝑡)                                                                               (9) 

3. Parametric Methods  

3.1. Weibull Distribution   

      The Weibull distribution is continuous distribution. It is one of the most widely applied 

life distributions in reliability analysis [9] and [10].  

The probability density function is: 

 𝑓(𝑡) = {
𝛼

𝜆
(

𝑡

𝜆
)𝛼−1𝑒

−(
𝑡

𝜆
)𝛼

𝑡 > 0

0 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
                                                                  (10) 



  

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Where ( 𝛼 > 0) is shape parameter and ( 𝜆 > 0) is the scale parameter of the distribution. The 

mean and variance of Weibull distribution are respectively: 

 𝐸(𝑡) = 𝜆Γ(1 +
1

𝛼
)                                                                                                 (11)                                        

 𝑉(𝑡) = 𝜆2[Γ (1 +
1

𝛼
) − Γ2(1 +

1

𝛼
)                                                                       (12) 

The cumulative distribution function is defined as: 

 𝐹(𝑡) = 1 − 𝑒
−(

𝑡

𝜆
)𝛼

, 𝑡 > 0                                                                                      (13) 

The survival function is defined as: 

 𝑆(𝑡) = 𝑒
−(

𝑡

𝜆
)𝛼

                                                                                                        (14) 

The failure rate (or hazard rate) function is given by: 

 ℎ(𝑡) =
𝛼

𝜆
(

𝑡

𝜆
)𝛼−1 = 𝛼𝜆−𝛼 𝑡𝛼−1                                                                                (15) 

3.2. Gumbel Distribution  

    Gumbel (1958) denotes this distribution of the type I distribution of the smallest extreme, 

called the Gumbel distribution of the smallest extreme [8]. The Gumbel distribution is a very 

common distribution due to its global applicability in several fields and its wide applications 

[11] and [13]. The probability density function is: 

 𝑓(𝑡) = {
1

𝛽
exp [−(𝑧 + exp(−𝑧))] −∞ < 𝑡 < ∞

0 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
                                                 (16) 

where 𝑧 =
𝑡−𝜇

𝛽
,  𝜇  is the location parameter, and 𝛽  is the scale parameter of the distribution.  

The mean and variance of 𝑊𝑒𝑖𝑏𝑢𝑙𝑙 distribution are respectively: 

 𝐸(𝑡) = 𝜇 + 𝛽𝛾 ,                                                                                                      (17) 

  𝛾  is Euler's constant (0.577215) 

 𝑉(𝑡) = (𝛽𝜋)2/6                                                                                                      (18) 

The cumulative distribution function is defined as: 

 𝐹(𝑡) = 1 − exp[− exp(−𝑧)] , −∞ < 𝑡 < ∞                                                          (19) 

The survival function is defined as: 

 𝑆(𝑡) = exp [− exp(−𝑧)]                                                                                          (20) 

The failure rate (or hazard rate) function is given by: 

 ℎ(𝑡) =

1

𝛽
exp [−(𝑧+exp(−𝑧))]

exp [− exp(−𝑧)]
                                                                                         (21) 



  

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3.3.Exponential Distribution  

     The exponential distribution is a special case of two- parameter 𝑊𝑒𝑖𝑏𝑢𝑙𝑙 distribution (or 

gamma distribution) when the shape parameter is (𝛼 = 1) in equation 10, then the probability 

density function is [11]: 

 𝑓(𝑡) = {
1

𝜆
 𝑒

−
𝑡

𝜆 𝑡 > 0

0 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
                                                                                    (22) 

where 𝜆 > 0 is the scale parameter of the distribution, and the mean and variance of 

exponential distribution are respectively: 

 𝐸(𝑡) =  𝜆                                                                                                                  (23) 

 𝑉(𝑡) = 𝜆2                                                                                                                 (24) 

The cumulative distribution function is defined as: 

 𝐹(𝑡) = 1 − 𝑒
−

𝑡

𝜆   , 𝑡 > 0                                                                                            (25) 

The survival function is defined as: 

 𝑆(𝑡) = 𝑒
−

𝑡

𝜆                                                                                                                (26) 

The failure rate (or hazard rate) is given by: 

 ℎ(𝑡) =
1

𝜆
                                                                                                                    (27) 

3.4. Log- Logistic Distribution  

     Log- logistic distribution is widely used in survival analysis when the failure rate function 

presents an unmoral shape [5]: 

𝑓(𝑡) = {
𝛽

𝛼
(

𝑡

𝛼
)𝛽−1[1 + (

𝑡

𝛼
)𝛽]−2 𝑡 > 0

0 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
                                                            (28) 

where ( 𝛼 > 0) is scale parameter and ( 𝛽 > 0) is the shape parameter of the distribution. 

The mean and variance are respectively [12], 

𝐸(𝑡) =
𝜋𝛼𝛽−1

sin(𝜋𝛽−1)
     , 𝛽 > 0                                                                                         (29) 

And 

𝑉(𝑡) =
2𝜋𝛼2𝛽−1

sin(2𝜋𝛽−1)
− [

𝜋𝛼𝛽−1

sin(𝜋𝛽−1)
]2  , 𝛽 > 2                                                                    (30) 

The cumulative distribution function is defined as: 

 𝐹(𝑡) = 1 − [1 + (
𝑡

𝛼
)𝛽]−1                                                                                      (31) 

Also the survival function is defined as: 



  

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 𝑆(𝑡) = [1 + (
𝑡

𝛼
)𝛽]−1                                                                                                  (32) 

The hazard function is given by: 

ℎ(𝑡) =
𝛽

𝛼
(

𝑡

𝛼
)𝛽−1[1 + (

𝑡

𝛼
)𝛽 ]−1                                                                                     (33) 

4. Goodness of Fit Test 
         In order to compare the distributions, we consider some other criterion like Akaike 

information Criterion (𝐴𝐼𝐶), Akaike information criterion corrected (𝐴𝐼𝐶𝐶) and Bayesian 

information Criterion (𝐵𝐼𝐶) for the real data set [2]. The best distribution corresponds to 

lower 𝐴𝐼𝐶, 𝐴𝐼𝐶𝐶, and 𝐵𝐼𝐶 values [7] and [13]: 

 𝐴𝐼𝐶 = −2 log 𝐿 + 2𝑘                                                                                               (34) 

 𝐴𝐼𝐶𝐶 = 𝐴𝐼𝐶 +
2𝑘(𝑘+1)

𝑛−𝑘−1
                                                                                              (35) 

 𝐵𝐼𝐶 = −2 log 𝐿 + 𝑘 log 𝑛                                                                                         (36) 

Where k is the number of parameters in the statistical model, 𝑛 the sample size and 𝐿 is the 

maximized value of the likelihood function for the estimated model. 

5. Data Analysis and Results  
     The dataset used in this study consists of a sample of (118) lung cancer patients obtained 

from Salman and Farhan [6] and given in Table 2. 

      We can find the estimated value of the parameters and its confidence intervals for the 

distributions by using maximum likelihood estimation method as follows: 

Table1:  Maximum likelihood estimates parameters of the distributions 

𝑀𝑜𝑑𝑒𝑙 𝐸𝑠𝑡𝑖𝑚𝑎𝑡𝑒𝑠 95% 𝐶. 𝐼 

 𝑊𝑒𝑖𝑏𝑢𝑙𝑙 𝛼 = 3.05 
𝜆 = 373.58 

[2.62 −  3.54] 
      [351.30 −  397.28] 

 
𝐺𝑢𝑚𝑏𝑒𝑙 

 

 

𝐸𝑥𝑝𝑜𝑛𝑒𝑛𝑡𝑖𝑎𝑙 
 

 

𝜇 = 394.02 
𝛽 = 102.78 

 

𝜆 = 337.15 
 

 

[374.42 −  413.61] 
  [89.521 − 118.01] 

 

[281.49 −  403. 82] 

𝐿𝑜𝑔 −  𝑙𝑜𝑔𝑖𝑠𝑡𝑖𝑐 𝛽 = 5.80 
𝛼 = 0.26 

[5.72 −  5.88] 
[0.22 –  0.30] 

 

         The survival function estimations of time are obtained by substituting these estimated 

values of the parameters in equations (14), (20), (26), and (32) as shown in table 2. And the 

survival plots of the W.D, G.D, E.D and L.L.D are shown in figures 1, 2, 3, and 4 

respectively. 

 



  

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Figure 1: The curve of Weibull distribution for survival function 

 

 

 

 

 

Figure 2: The curve of Gumbel distribution for survival function  

 

 

Figure 3: The curve of exponential distribution for survival function  

 

600500400300200100

100

80

60

40

20

0

Shape 3.04788

Scale 373.580

Mean 333.835

StDev 119.621

Median 331.253

IQR 167.610

Failure 118

Censor 0

Table of Statistics

T ime

P
e

rc
e

n
t

6005004003002001000-100

100

80

60

40

20

0

Loc 394.015

Scale 102.781

Mean 334.688

StDev 131.821

Median 356.344

IQR 161.626

Failure 118

Censor 0

Table of Statistics

T ime

P
e

rc
e

n
t

16001400120010008006004002000

100

80

60

40

20

0

Mean 337.153

StDev 337.153

Median 233.696

IQR 370.400

Failure 118

Censor 0

Table of Statistics

T ime

P
e

rc
e

n
t



  

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Figure 4: The curve of log logistic distribution for survival function  

 

 

 

Table 2: Estimated values of the survival function [6] 

120010008006004002000

100

80

60

40

20

0

Loc 5.80194

Scale 0.256581

Mean 369.695

StDev 200.104

Median 330.940

IQR 189.054

Failure 118

Censor 0

Table of Statistics

T ime

P
e

rc
e

n
t

No. Time

/Day 
�̂�(𝑾. 𝑫) �̂�(𝑮. 𝑫) �̂�(𝑬. 𝑫) �̂�(𝑳. 𝑫) No. Time

/Day 
�̂�(𝑾. 𝑫) �̂�(𝑮. 𝑫) �̂�(𝑬. 𝑫) �̂�(𝑳. 𝑫) 

1 
3 1.000 

0.978 
0.991 1.000 

60 
341 0.469 

0.550 
0.364 0.471 

2 
37 0.999 

0.969 
0.896 0.999 

61 
342 0.466 

0.547 
0.363 0.468 

3 
72 0.993 

0.957 
0.808 0.997 

62 
345 0.456 

0.538 
0.359 0.460 

4 
75 0.993 

0.956 
0.801 0.997 

63 
349 0.444 

0.524 
0.355 0.448 

5 
91 0.987 

0.949 
0.763 0.994 

64 
354 0.428 

0.508 
0.349 0.435 

6 
100 0.982 

0.944 
0.743 0.991 

65 
357 0.419 

0.498 
0.347 0.427 

7 
103 0.981 

0.943 
0.737 0.989 

66 
363 0.400 

0.477 
0.341 0.411 

8 
121 0.968 

0.932 
0.699 0.981 

67 
364 0.397 

0.474 
0.339 0.408 

9 
127 0.964 

0.928 
0.686 0.977 

68 
364 0.397 

0.474 
0.339 0.408 

10 
140 0.951 

0.919 
0.660 0.966 

69 
366 0.391 

0.467 
0.338 0.403 

11 
154 0.935 

0.908 
0.633 0.952 

70 
367 0.388 

0.464 
0.337 0.401 

12 
156 0.933 

0.906 
0.629 0.949 

71 
368 0.385 

0.460 
0.336 0.398 

13 
164 0.922 

0.899 
0.615 0.939 

72 
371 0.376 

0.449 
0.333 0.391 

14 
186 0.888 

0.876 
0.576 0.904 

73 
373 0.369 

0.443 
0.331 0.386 

15 
211 0.839 

0.845 
0.535 0.853 

74 
380 0.349 

0.419 
0.324 0.369 

16 
212 0.837 

0.844 
0.533 0.850 

75 
387 0.328 

0.393 
0.317 0.352 

17 
213 0.835 

0.842 
0.532 0.848 

76 
387 0.328 

0.393 
0.317 0.352 

18 
217 0.826 

0.836 
0.525 0.838 

77 
392 0.314 

0.375 
0.313 0.341 

19 
218 0.824 

0.835 
0.524 0.836 

78 
393 0.311 

0.372 
0.312 0.339 

20 
221 0.817 

0.831 
0.519 0.828 

79 
397 0.300 

0.357 
0.308 0.330 

21 
221 0.817 

0.831 
0.519 0.828 

80 
399 0.295 

0.350 
0.306 0.325 

22 
233 0.789 

0.812 
0.501 0.797 

81 
400 0.292 

0.346 
0.305 0.323 



  

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23 
240 0.772 

0.789 
0.491 0.778 

82 
400 0.292 

0.346 
0.305 0.323 

24 
241 0.769 

0.798 
0.489 0.775 

83 
401 0.289 

0.343 
0.304 0.321 

25 
243 0.764 

0.794 
0.486 0.769 

84 
402 0.286 

0.339 
0.304 0.319 

26 
249 0.748 

0.784 
0.478 0.752 

85 
407 0.273 

0.322 
0.299 0.309 

27 
254 0.735 

0.774 
0.471 0.737 

86 
409 0.268 

0.314 
0.297 0.305 

28 
266 0.701 

0.749 
0.454 0.701 

87 
419 0.242 

0.279 
0.289 0.285 

29 
273 0.681 

0.735 
0.445 0.679 

88 
421 0.237 

0.272 
0.287 0.281 

30 
276 0.672 

0.728 
0.441 0.669 

89 
421 0.237 

0.272 
0.287 0.281 

31 
277 0.669 

0.726 
0.440 0.667 

90 
422 0.235 

0.269 
0.286 0.279 

32 
278 0.666 

0.724 
0.438 0.664 

91 
422 0.235 

0.269 
0.286 0.279 

33 
281 0.657 

0.717 
0.435 0.654 

92 
423 0.232 

0.266 
0.285 0.278 

34 
290 0.630 

0.695 
0.423 0.626 

93 
427 0.222 

0.252 
0.282 0.270 

35 
301 0.596 

0.667 
0.410 0.591 

94 
428 0.220 

0.247 
0.281 0.269 

36 
301 0.596 

0.667 
0.410 0.591 

95 
430 0.215 

0.242 
0.279 0.265 

37 
301 0.596 

0.667 
0.410 0.591 

96 
446 0.179 

0.191 
0.266 0.238 

38 
302 0.593 

0.665 
0.408 0.588 

97 
450 0.171 

0.178 
0.263 0.232 

39 
304 0.587 

0.659 
0.406 0.582 

98 
454 0.163 

0.167 
0.260 0.226 

40 
304 0.587 

0.659 
0.406 0.582 

99 
416 0.249 

0.289 
0.291 0.291 

41 
306 0.581 

0.654 
0.404 0.576 

100 
463 0.146 

0.141 
0.253 0.213 

42 
307 0.577 

0.651 
0.402 0.573 

101 
470 0.133 

0.123 
0.248 0.203 

43 
307 0.577 

0.651 
0.402 0.573 

102 
477 0.122 

0.106 
0.243 0.194 

44 
308 0.574 

0.649 
0.401 0.569 

103 
481 0.115 

0.097 
0.240 0.189 

45 
313 0.558 

0.635 
0.395 0.554 

104 
481 0.115 

0.097 
0.240 0.189 

46 
313 0.558 

0.635 
0.395 0.554 

105 
483 0.112 

0.093 
0.239 0.186 

47 
314 0.555 

0.632 
0.394 0.551 

106 
483 0.112 

0.093 
0.239 0.186 

48 
318 0.542 

0.621 
0.389 0.539 

107 
497 0.092 

0.066 
0.229 0.170 

49 
330 0.504 

0.585 
0.376 0.503 

108 
511 0.074 

0.044 
0.220 0.155 

50 
331 0.501 

0.582 
0.375 0.499 

109 
512 0.073 

0.043 
0.219 0.154 

51 
332 0.498 

0.579 
0.374 0.497 

110 
512 0.073 

0.043 
0.219 0.154 

52 
332 0.498 

0.579 
0.374 0.497 

111 
516 0.069 

0.038 
0.216 0.150 

53 
334 0.491 

0.573 
0.371 0.491 

112 
517 0.068 

0.037 
0.216 0.149 

54 
334 0.491 

0.573 
0.371 0.491 

113 
519 0.066 

0.034 
0.215 0.148 

55 
335 0.488 

0.569 
0.370 0.488 

114 
533 0.052 

0.021 
0.206 0.135 

56 
335 0.488 

0.569 
0.370 0.488 

115 
534 0.051 

0.021 
0.205 0.134 

57 
335 0.488 

0.569 
0.370 0.488 

116 
535 0.050 

0.19 
0.205 0.133 

58 
335 0.488 

0.569 
0.370 0.488 

117 
540 0.046 

0.016 
0.202 0.129 

59 
338 0.479 

0.560 
0.367 0.479 

118 
550 0.039 

0.010 
0.196 0.121 



  

117 

 

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (2) 2021 

 

Table 3. Criteria for comparison 

𝑀𝑜𝑑𝑒𝑙 −𝐿𝐿 𝐴𝐼𝐶 𝐴𝐼𝐶𝐶 𝐵𝐼𝐶 

 𝑊𝑒𝑖𝑏𝑢𝑙𝑙 737.89 1479.78 1479.88 1479.92 

𝐺𝑢𝑚𝑏𝑒𝑙 
𝐸𝑥𝑝𝑜𝑛𝑒𝑛𝑡𝑖𝑎𝑙 

729.93 
804.82 

1463.86 
1611.65 

1463.96 
1611.68 

1464.00 
1611.71 

𝐿𝑜𝑔 − 𝑙𝑜𝑔𝑖𝑠𝑡𝑖𝑐 757.03 1518.06 1518.16 1518.20 

 

     The results in Table 3 indicate that the 𝐺𝑢𝑚𝑏𝑒𝑙 distribution has the lowest 𝐴𝐼𝐶, 𝐴𝐼𝐶𝐶 and 

𝐵𝐼𝐶 values than the Weibull, log- logistic, and exponential. Hence Gumbel distribution leads 

to a better fit than the other three distributions. 

6. Conclusions  
      From the practical work, it is concluded that the Gumbel distribution has the lowest 𝐴𝐼𝐶, 

𝐴𝐼𝐶𝐶 and 𝐵𝐼𝐶 values than the Weibull, exponential, and log- logistic distributions. We 

conclude that the survival function for the lung cancer by using Gumbel distribution model is 

the best. And the expected values of the survival function of all estimation methods which are 

proposed in this article has been decreasing progressively with increasing failure times for 

lung cancer patients: this means that there is an opposite relationship failure times and 

survival function. 

 

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