Int. J. Anal. Appl. (2023), 21:80 

 

 

Received: Jun. 5, 2023. 

2020 Mathematics Subject Classification. 60E05. 

Key words and phrases. failure rate; age distribution; randomly right censored data. 

 

https://doi.org/10.28924/2291-8639-21-2023-80 © 2023 the author(s) 

ISSN: 2291-8639  

1 

 

A Class of Tests for Testing Better Failure Rate at Specific Age Distribution With 

Randomly Right Censored Data  

Gamal R. Elkahlout* 

Faculty of Business School, Arab Open University, Riyadh, Saudi Arabia 

*Corresponding author: g.elkahlout@arabou.edu.sa 

ABSTRACT: A device has a better failure rate at specific age 𝑡0 property, denoted by 𝐵𝐹𝑅 − 𝑡0 if its failure rate r(t) increases 

for 𝑡 ≤ 𝑡0 and for 𝑡 > 𝑡0, r(t) is not less than its value at 𝑡0. A test statistic is proposed to test exponentiality versus 

𝐵𝐹𝑅 − 𝑡0  based on a randomly right censored sample of size n. Kaplan-Meier estimator is used to estimate the empirical 

life distribution. Properties of the test are measured by power estimates, estimated risks, and test of normality. The 

efficiency loss due to censoring is investigated by using tests for censored sample data. 

 

1. INTRODUCTION 

The concept of ageing for engineering devices, biological organs or their corresponding 

systems has been characterized by various life distribution classes. The increasing failure rate (IFR) 

class of life distributions is the most used and have all other notions of ageing in reliability literature. 

Among these notions are the increasing failure rate average (IFRA), new better than used 

(NBU), new better than used failure rate (NBUFR), new better than used in average failure rate 

(NBAFR), decreasing mean remaining life (DMRL), new better than used in expectation (NBUE) 

and harmonic new better than used in expectation (HNBUE). See ([14], [23], [21], [6]) for definitions, 

properties and interrelationships of these classes of life distributions. 

In many practical situations, it is familiar that properties of life distributions may not be 

completely observed after a specific time. This arises in data collection in companies for their 

commodities with guarantee for some fixed time 𝑡0, say. 

https://doi.org/10.28924/2291-8639-21-2023-80


2 Int. J. Anal. Appl. (2023), 21:80 

 

 

In study of [15] for the new better than used at specific age 𝑡0 (𝑁𝐵𝑈 − 𝑡0) have considered 

the hypothetical cancer patients problem. It is interesting to investigate the problem of testing 

whether new diagnosed cancer patient has smaller chance of survival than a patient with similar initial 

diagnosis and survived, on treatment for a certain year. The decreasing mean remaining life at a 

specific age 𝑡0 (𝐷𝑀𝑅𝐿 − 𝑡0) is defined in similar line by [10]. 

In a study by [2] introduced the class of better failure rate at a specific age 𝑡0 (𝐵𝐹𝑅 − 𝑡0) 

and its dual class (𝑊𝐹𝑅 − 𝑡0). Its closure properties under some reliability operations are studied.  

Test statistic for testing exponentiality against 𝐵𝐹𝑅 − 𝑡0 or its dual class  𝑊𝐹𝑅 − 𝑡0 is also proposed 

for the complete sample. 

Different classes of life distributions based on a random censored samples, have been studied 

in references such as ([26], [27]) for testing NBU and IFR classes of life distributions, [20] for testing 

IFRA class of life distributions and ([3], [4])  for testing NBRU and NBRUE classes of life distributions 

and their dual classes. 

In research for [12] defined classes of life distributions IFRA-to and NBU-t0. The properties 

of these classes are studied, and a nonparametric test is proposed which is designed to test the 

hypothesis whether NBU-t0 element is strictly new better than used after time 𝑡0. 

A paper for [19] about NBU class made by [15] to investigate the testing of new better than 

used at specified age (NBU-t0) based on a U-statistic whose kernel depends on sub-sample minima. 

A member of the class of tests proposed by [17] for this problem belongs to the class of tests are 

covered and distribution properties of the class of test statistics are studied. The performances of a 

few members of the proposed class of tests are studied in terms of Pitman asymptotic relative 

efficiency.  

In a paper for [9] he introduced some properties of the new better than used in convex 

ordering at age t0 (NBUC− t0) and new better than used of second order (2) at age t0 (NBU (2)− 

t0) classes of life distributions, where the survival probability at age 0 is greater than or equal to the  

conditional survival probability at specified age t0> 0. Preservation properties of the two classes under 

various reliability operations and shock model are arriving according to homogeneous Poisson process 

are established. 

Researchers [13] have defined two classes of life distributions, namely new better than used 

in expectation at specific age  𝑡0 (𝑁𝐵𝑈𝐸 − 𝑡0) and harmonic new better than used in expectation at 

specific age t0 (𝐻𝑁𝐵𝑈𝐸 − 𝑡0) and their dual classes (𝑁𝑊𝑈𝐸 − 𝑡0) and (𝐻𝑁𝑊𝑈𝐸 − 𝑡0). The closure 



3 Int. J. Anal. Appl. (2023), 21:80 

 

 

properties under various reliability operations such as convolution, mixture, mixing and the 

homogeneous Poisson shock model of these classes are studied. Nonparametric tests are proposed 

to test exponentiality versus the NBUE-t0 and HNBUE-t0 classes. The critical values and the powers 

of these tests are calculated to assess the performance of the tests. They show that the proposed 

tests have high efficiencies for some commonly used distributions in reliability.  

A test statistic has been built by [7] for two classes of life distributions defined earlier by [1] 

namely new better than used renewal failure rate (NBURFR) and new better than average renewal 

failure rate (NBARFR). These two classes include many other classes of life distributions. Test 

statistics for testing of exponentiality as a null hypothesis against these two renewal ageing criteria, 

and their duals are derived in the case of randomly censored samples. Percentiles tables, power 

estimates, estimated risks are calculated. The normality of their test statistics is also studied.   

 

2. THE 𝑩𝑭𝑹 − 𝒕𝟎 AND  𝑾𝑭𝑹 − 𝒕𝟎 CLASSES 

Let 𝑇 be a life length of a device with continuous distribution F, survival function �̄�(𝑡) = 1 −

𝐹(𝑡) and failure rate 𝑟(𝑡), then it is called better failure rate at time 𝑡0 𝐵𝐹𝑅 − 𝑡0 (𝑊𝐹𝑅 − 𝑡0) if  

          𝑟(𝑠) ≤ (≥)𝑟(𝑡)        ∀      𝑠 < 𝑡 < 𝑡0         &      𝑡 ∈ [0, 𝑡0],       (2.1)  

and 

𝑟(𝑡0) ≤ (≥)𝑟(𝑡)   ∀   𝑡 ≥ 𝑡0              (2.2)  

This means that any device of age 𝑡0 or less has smaller failure rate than an older device, whereas 

a device of age 𝑡0 or more cannot have a failure rate less than 𝑟(𝑡0). 

 

3. TESTING EXPONENTIALITY VERSUS 𝑩𝑭𝑹 − 𝒕𝟎  AND 𝑾𝑭𝑹 − 𝒕𝟎 CLASSES 

In this section we consider the problem of testing: 

𝐻0: 𝐹(𝑡) = 1 − 𝑒
−𝜆𝑡0  ,  i.e. 𝑟(𝑡) = 𝑟(𝑡 + 𝑥)∀𝑥 ≥ 0, 0 ≤ 𝑡 ≤ 𝑡0, versus 

𝐻1: 𝐹is 𝐵𝐹𝑅 − 𝑡0  i.e. 𝑟(𝑡)  is increasing for 𝑡 ≤ 𝑡0 and 𝑟(𝑡) ≥ 𝑟(𝑡0) for 𝑡 ≥ 𝑡0. 

This test is based on randomly right-censored data by using Kaplan-Meier estimator [11] for the 

empirical survival function ( , ),( ) ( )Z i ni i    1  . 

Let  {𝑇𝑖 }𝑖 = 1,2, . . . , 𝑛 be independent and identically distributed non-negative continuous random 

variables having a common distribution 𝐹  and survival function �̄�(𝑡) = 1 − 𝐹(𝑡). 

Let {𝑌𝑖 }𝑖 = 1,2, . . . , 𝑛 be independent and identically distributed random variables according to a 

continuous censoring distribution 𝐻. {𝑇𝑖 } and {𝑌𝑖 } are independent of each other. The censoring 



4 Int. J. Anal. Appl. (2023), 21:80 

 

 

distribution 𝐻 is usually, but not necessary, unknown and is considered as nuisance parameter. The 

pairs (𝑇1, 𝑌1), . . . . . , (𝑇𝑛 , 𝑌𝑛 ) are defined on a common probability space. 

In the censored situations of sample size n, the 𝑇1, . . . . . . . . . 𝑇𝑛 are not completely observed but the 

pairs (𝑍1, 𝛿1), . . . . . . , (𝑍𝑛 , 𝛿𝑛 ) are observed data, where 

𝑍𝑖 = 𝑚𝑖𝑛(𝑇𝑖 , 𝑌𝑖 ) , ∀𝑖 = 1,2, . . . . , 𝑛      and      𝛿𝑖 = {
1  if  𝑇𝑖 ≤ 𝑌𝑖
0  if  𝑇𝑖 > 𝑌𝑖

 

 

4. TEST STATISTICS FOR 𝑩𝑭𝑹 − 𝒕𝟎 

 In the 𝐵𝐹𝑅 − 𝑡0 the required test can be based on the estimation of the parameter 

 M(F) = ∬ F̄(s)F̄(t){r(s)-r(t)}dF(s)
0<z<t≤t0

dF(t) + ∫ F̄(t0)F̄(t){r(t0)-r(t)}dF(t)
∞

t0
        (4.1) 

The statistic 𝑀(𝐹) = 0 under 𝐻0 whereas 𝑀(𝐹) < 0 under 𝐻1 where strict inequality is due to the 

continuity of the underlying distribution F. Also the statistic 𝑀(𝐹) gives a measure of deviation from 

the exponentiality towards the 𝐵𝐹𝑅 − 𝑡0 property. 

For the present of randomly censored data, we base our test on the Kaplan-Meier estimator of �̄�𝑛 , 

see [11], which is given by 

)(
0

1
)(

)(

)(
:

nn
zt

in

in
tF

i

i
tzi










+−

−
= 







 



             (4.2) 

Where  0 = 𝑧(0) ≤ 𝑧(1) ≤. . . . . . . ≤ 𝑧(𝑛) ,  denote ordered sample of  𝑧1, 𝑧2, . . . . . . , 𝑧𝑛 and  

𝛿(1), 𝛿(2), . . . . . , 𝛿(𝑛)  are the corresponding 𝛿𝑖   for  the ordered 𝑧(𝑖)  for 𝑖 = 1,2, . . . . , 𝑛 . 

Note that 𝛿(𝑛)  is taken as one, that is  𝑧(𝑛) is treated as uncensored observation whether it is the 

case or not. The corresponding density function and failure rate of  �̄̂�𝑛 (𝑡) are given in the following. 

𝑓𝑛 (𝑡) =
�̄̂�𝑖−1,𝑛(𝑡)−�̄̂�𝑖,𝑛(𝑡)

𝑧(𝑖)−𝑧(𝑖−1)
                             (4.3) 

where  �̄̂�𝑛 (𝑡) = 1 − �̂�𝑛 (𝑡),  

Hence 

𝑓𝑛 (𝑡) = (𝑧(𝑖) − 𝑧(𝑖−1))
−1

[∏ {
𝑛−𝑖

𝑛−𝑖+1
}{𝑖−1:𝑧(𝑖)≤𝑡}

𝛿(𝑖)
− ∏ {

𝑛−𝑖

𝑛−𝑖+1
}{𝑖:𝑧(𝑖)≤𝑡}

𝛿(𝑖)
]  

= (𝑧(𝑖) − 𝑧(𝑖−1))
−1

[∏ {
𝑛−𝑗

𝑛−𝑗+1
}𝑖−1𝑗=1

𝑍(𝑖)≤𝑡

𝛿(𝑗)
− ∏ {

𝑛−𝑗

𝑛−𝑗+1
}𝑖𝑗=1

𝛿(𝑗)
]  

 = (𝑧(𝑖) − 𝑧(𝑖−1))
−1

∏ {
𝑛−𝑗

𝑛−𝑗+1
}𝑖−1𝑗=1

𝛿(𝑗)
[1 − {

𝑛−𝑖

𝑛−𝑖+1
}

𝛿(𝑖)
]  

ni ,....,2,1=



5 Int. J. Anal. Appl. (2023), 21:80 

 

 

= (𝑧(𝑖) − 𝑧(𝑖−1))
−1

[[1 − {
𝑛−𝑖

𝑛−𝑖+1
}

𝛿(𝑖)
] ∏ {

𝑛−𝑗

𝑛−𝑗+1
}𝑖−1𝑗=1

𝛿(𝑗)
]                          (4.4) 

It is reasonable to base a test of exponentiality versus 𝐵𝐹𝑅 − 𝑡0  property on a consistent 

estimator𝑀(𝐹), given by (3) and then exploit 𝑀(�̄̂�𝑛 ),w here �̄̂�𝑛 the Kaplan-Meier estimator. In fact 

the Kaplan-Meier estimator has been shown to be weakly convergent by ([5], [16], [22])  and strongly 

consistent by ([24], [18]). 

The test statistic in (3) can by simplified in the following. For easy calculation we write  𝑀(𝐹) =

𝐴 + 𝐵, where 

            𝐴 = ∬ �̄�(𝑠)�̄�(𝑡){𝑟(𝑠) − 𝑟(𝑡)}𝑑𝐹(𝑠)𝑑𝐹(𝑡)
0<𝑠<𝑡<𝑡0

                      (4.5) 

and    𝐵 = ∫ �̄�(𝑡0)�̄�(𝑡){𝑟((𝑡0) − 𝑟(𝑡))}𝑑𝐹(𝑡)
∞

𝑡0
                       (4.6) 

Now 

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( )
( )
( )

( ) ( ) ( ) ( )
( )
( )

( ) ( )

( )  ( ) ( ) ( ) ( )  ( ) 

( )  ( ) ( )
( )( )

( ) 

( )  ( ) ( ) ( )   ( )  dttftFtdFtFdssf

dttf
sF

tdFtFdssf

dttfsdFsFtdFtFdssf

dtsdFtf
tF

tf
tFsFtdFdssf

sF

sf
tFsF

tdFsdFtrtFsFtdFsdFsrtFsFA

tt t

t
t

t t

tt t

t tt t

t tt t

2

0

2

0 0

2

2

0

2

0

0 0

2

2

00 0

2

0 00 0

0 00 0

00

00

00

0

00

1
2

1

2

 










 

 





−−







=

−







=

−







=

−=

−=

           (4.7) 

and 

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( )
( )
( )

( ) ( ) ( )
( )
( )

( )

( ) ( ) ( ) ( ) ( ) 

( ) ( )( )  ( ) ( ) 

( ) ( )( ) ( ) ( ) 




















−−=

−=

−=

−=

−=

0

0
0

00

00

00

2

0

2

00

2

00

2

00

0

0

0
0

000

2
1

2
1 2

t

t
t

tt

tt

tt

dttftFtFtf

dttftFtFtf

dttftFtdFtFtf

tdF
tF

tf
tFtFtdF

tF

tf
tFtF

tdFtrtFtFtdFtrtFtFB

               (4.8) 



6 Int. J. Anal. Appl. (2023), 21:80 

 

 

Hence 

𝑀(𝐹) = ∫ [∫ {𝑓(𝑠)}2𝑑𝑠
𝑡

0

]
𝑡0

0

�̄�(𝑡)𝑑𝐹(𝑡) −
1

2
∫ [{�̄�(𝑡)}2 − 1]

𝑡0

0

{𝑓(𝑡)}2𝑑𝑡 

−
1

2
𝑓(𝑡0)(�̄�(𝑡0))

2
− �̄�(𝑡0) ∫ {𝑓(𝑡)}

2𝑑𝑡
∞

𝑡0
                  (4.9) 

or 

𝑀(𝐹) = ∫ [∫ {𝑓(𝑠)}2𝑑𝑠
𝑡

0

]
𝑡0

0

�̄�(𝑡)𝑓(𝑡)𝑑𝑡 −
1

2
∫ {�̄�(𝑡)}2

𝑡0

0

{𝑓(𝑡)}2𝑑𝑡 +
1

2
∫ {𝑓(𝑡)}2𝑑𝑡

𝑡0

0

 

−�̄�(𝑡0) ∫ {𝑓(𝑡)}
2𝑑𝑡

∞

𝑡0
−

1

2
𝑓(𝑡0)(�̄�(𝑡0))

2
                                          (4.10) 

Now substituting the empirical values of �̄̂�(𝑡) and 𝑓𝑛 (𝑡)in (12), then the statistics 𝑀𝑛(𝐹) in the case 

of randomly censored data defined earlier will be  

𝑀𝑛 (𝐹) = ∑ [∑{𝑓𝑛(𝑍(𝑗))}
2

(𝑍(𝑗) − 𝑍(𝑗−1))

𝑖

𝑗=1

]

𝑚

𝑖=1

�̄�𝑛 (𝑍(𝑖))𝑓𝑛 (𝑍(𝑖))(𝑍(𝑖) − 𝑍(𝑖−1)) 

+
1

2
∑{𝑓𝑛 (𝑍(𝑖))}

𝑚

𝑖=1

2

(𝑍(𝑖) − 𝑍(𝑖−1)) −
1

2
∑{�̄�𝑛 (𝑍(𝑖))}

2
𝑚

𝑖=1

{𝑓𝑛 (𝑍(𝑖))}
2

(𝑍(𝑖) − 𝑍(𝑖−1)) 

                   −�⃑�𝑛 (𝑍(𝑚)) ∑ {𝑓𝑛(𝑍(𝑖))}
𝑛
𝑖=𝑚

2
(𝑍(𝑖) − 𝑍(𝑖−1)) −

1

2
𝑓𝑛 (𝑍(𝑚0)){�̄�𝑛(𝑍(𝑚0))}

2
            (4.11) 

F Zn i( )( )  is the Kaplan-Meier, product limit estimator of Fn  and given by [11] and defined as 

  F Z F Zn j n j
k

j

n k
n k

k

( ) ( )
( ) ( )

( )

= − =
−
− +

=

1 1
1



                           (4.12) 

Also, Z i( )  and ( )i  are defined earlier, and 


( ) ( )d Z Zi i i= − −1 , 

Here 𝑍(𝑛) is considered as an actual observation, whether or not it is censored, and in this 

case 𝜕𝑛 is taken as one to avoid appearance of indefinite values in the empirical calculation of the 

statistic 𝑀𝑛 (𝐹). 

For a specific significance level  = 0  and a sample size n n= 0 , we reject 𝐻0 in favor of 

𝐻1, that F has the 𝐵𝐹𝑅 − 𝑡0 property, for small negative values of  𝑀𝑛 (𝐹). In otherwards, we reject 

𝐻0 in favor of 𝐻1 if the observed or calculated values of the statistic is less than or equal the tabulated 

value of the statistic for the same value of 𝛼0 and 𝑛0. 

For testing  𝐻0 against 𝐻1 that F has the 𝑊𝐹𝑅 − 𝑡0 property, we reject 𝐻0 for small negative 

values of −𝑀𝑛(𝐹), or for large positive values of  𝑀𝑛 (𝐹). 



7 Int. J. Anal. Appl. (2023), 21:80 

 

 

The asymptotic normality of the sequence, such as √𝑛[𝑀𝑛(𝐹) − 𝑀(𝐹)] , has been dealt with 

in the literature under the following crucial assumption. 

      ) Sup G t H t t( ) ( ) , ,1 1 0− −       for some  0 1              (4.13) 

where G and H are the distribution functions of the actual values of X and the censored values Y 

respectively. Here G and H are assumed to have support on  )0, . 

The above sequence, or stochastic process, converges weakly as n → to a Gaussian process 

with zero mean, for this result and the details of the corresponding variance function see [22]. 

A simulation of a small sample is calculated. In practice, simulated percentiles for small 

samples are commonly used by applied statisticians and reliability analysts. 

Based on test statistics in (4.1), empirical evaluation in (4.11) using Kaplan-Meier, product 

limit estimator in (4.12), the lower percentiles in the  0.01, 0.05, and 0.10 regions and upper 

percentiles in the 0.90, 0.95, and 0.99 regions for the sample sizes  n=10(2),30(5) and 50 are 

presented in table 1 for the 𝐵𝐹𝑅 − 𝑡0 and  𝑊𝐹𝑅 − 𝑡0  test statistics.   

 

5. PROPERTIES OF THE TEST   

In the following, we will cover three properties of the test, namely, power estimates, estimated 

risks, and test of normality. 

5.1 Power Estimates 

 The power estimates of the test statistic 𝑀𝑛 (𝐹) are considered for the significance level 

=0.05 and for commonly used distributions in reliability modeling. These distributions are Gamma, 

Weibull, Pareto, and Rayleigh with the following survival functions and failure rates. 

Gamma distribution: 

    F t x e dx t
t

x
( )

( )
, , ,=  

− −



1

0 0
1

 


                    (5.1) 

  ( ) r t ut e
u

du
−

= +
−

−


1
1

1

0

( )
( )

  

Weibull distribution: 

   F t t t( ) exp( ) , ,= −  


 0 0                    (5.2) 

    r t t( ) =
−


 1

. 

Pareto distribution: 



8 Int. J. Anal. Appl. (2023), 21:80 

 

 

  ( )F t t t( ) , , ,= +  
−

1 0 0
1

                      (5.3) 

   ( )r t t t( ) , ,= +  
−

1 0 0
1

  . 

Rayleigh distribution: 

  F t t t( ) exp( ) ,= − 
1
2

2
0                    (5.4) 

   r t t( ) =                 t  0 

For further references about these distributions, one can refer to [25]. Table 2 contains the 

power estimates of 𝑀𝑛 (𝐹)  test statistic with respect to these distributions. The estimates are based 

on the average of five runs each with 1000 simulated samples of sizes n=10,20,30,40 and 50 with 

significance level =0.05. 

The power estimates of 𝐵𝐹𝑅 − 𝑡0 statistic for Gamma distribution increases more rapidly as 

 increases than the other distributions, namely, Weibull, Pareto, and Rayleigh. This indicates that 

the power estimate increases as the class tends to be more 𝐵𝐹𝑅 − 𝑡0.  The same pattern for the 

power estimates can be noted when the sample size increases. 

This table shows that the percentage of  = 1 ( i.e. the percentage of the uncensored random 

variables in the sample) is approximately fixed when the sample size increases for the same parameter 

and decreases as the parameter of the distribution increases for all the distributions, namely, Gamma, 

Weibull, Pareto and Rayleigh. This study shows that the percentage of censored values is dependent 

on the parameter values of the distribution more than the sample size. The resulting estimates 

indicate that the proposed statistic is suitable for different small sample sizes in reliability applications. 

 5.2 Estimated Risks  

 The estimated risks (ER) of the statistic 𝑀𝑛(𝐹) are given by ; 

 𝐸𝑅(𝑀𝑛(𝐹)) =
1

𝑚
∑ (𝑋𝑘 − �̅�𝑀(𝐹))

2𝑚
𝑘=1                                  (5.5) 

where X
DA
~  is the mean value, Xi  is the statistic value corresponding to the distribution under study, 

BFR-t0, and m is the sample size. Estimated risks are summarized in Table 3. 

 It is noted that as the sample size increases, the estimated risks decrease, and the mean value is 

increasing. More precisely, ER/n decreases as n increases. For a large sample size ER/n approaches 

to zero. This indicates that our test is a powerful one. 

 5.3 A Test of Normality 

The Kolmogorov-Smirnov (KS) test is applied to check how well the underlying statistic 

𝑀𝑛 (𝐹)  tends to normality with unspecified mean and variance. For testing normality let S be the 



9 Int. J. Anal. Appl. (2023), 21:80 

 

 

empirical distribution function based on the random sample X X X
n1 2

, ,....., . The test statistics D is 

defined as the greatest vertical distance between standardized version of 𝑀𝑛 (𝐹) , denoted by , and 

S. Symbolically,  

   D S= −sup                                     (5.6) 

Here we utilize the modified Kolmogorov-Smirnov test of normality proposed by [8] which 

accommodates the sample estimated normal parameters. 

The D values are given in Table 4. By comparing the calculated KS value with the tabulated 

one, one accepts the hypothesis approach those to normality for the considered sample sizes. 

 

   Table 1: Critical values for 𝑀𝑛 (𝐹) statistic for testing 𝐵𝐹𝑅 − 𝑡0  and 𝑊𝐹𝑅 − 𝑡0   ageing  properties 

n 0.01 0.05 0.1 0.9 0.95 0.99 

10 -3.6457 -2.431 -1.9567 0.2251 0.6764 1.7745 

12 -3.2025 -2.1203 -1.7422 0.398 0.8059 1.8555 

14 -2.9597 -1.9717 -1.617 0.4558 0.9362 2.177 

16 -2.5969 -1.8157 -1.4816 0.601 1.1027 2.3588 

18 -2.5062 -1.7031 -1.3924 0.6615 1.1451 2.4567 

20 -2.2232 -1.6249 -1.3085 0.7522 1.1715 2.328 

22 -2.1796 -1.5047 -1.242 0.7844 1.3103 2.4974 

24 -1.9614 -1.4286 -1.1464 0.8456 1.3761 2.773 

26 -1.8858 -1.317 -1.0815 0.963 1.4602 3.0735 

28 -1.7508 -1.261 -1.0356 1.0441 1.6226 2.8913 

30 -1.6563 -1.197 -0.9633 1.007 1.4842 2.8249 

35 -1.5217 -1.0807 -0.8538 1.1309 1.6884 3.1401 

40 -1.3643 -0.926 -0.7355 1.2301 1.7746 3.0854 

45 -1.2075 -0.8482 -0.6432 1.307 1.8649 3.0055 

50 -1.1091 -0.7636 -0.5834 1.373 1.8875 3.2362 

 

  



10 Int. J. Anal. Appl. (2023), 21:80 

 

 

Table 2 : Power estimates for 𝑀𝑛 (𝐹)   statistic for testing 𝐵𝐹𝑅 − 𝑡0  and 𝑊𝐹𝑅 − 𝑡0  ageing 

properties. 

Distribution 
Par. 

Sample size 

10 20 30 40 50 

F (Gamma) 2 0.009 0.012 0.014 0.016 0.021 

percentage of Delt=1   62 62 63 62 62 

F (Gamma) 3 0.162 0.223 0.271 0.272 0.347 

percentage of Delt=1   50 50 50 50 50 

F (Gamma) 4 0.562 0.613 0.652 0.665 0.691 

percentage of Delt=1   40 41 40 40 41 

F (Gamma) 5 0.834 0.855 0.862 0.954 1 

percentage of Delt=1   32 32 32 32 32 

F (Gamma) 6 0.999 1 1 1 1 

percentage of Delt=1   26 26 26 26 26 

F (Weibull)  0.25 0.08 0.12 0.18 0.3 0.46 

percentage of Delt=1   32 35 38 36 37 

F (Weibull)  0.5 0 0.02 0.08 0.1 0.18 

percentage of Delt=1   40 39 38 38 40 

F (Weibull)   2 0.1 0.14 0.2 0.32 0.48 

percentage of Delt=1   49 49 51 52 51 

F (Weibull)   3 0.11 0.12 0.13 0.14 0.16 

percentage of Delt=1   38 42 41 41 40 

F (Weibull)  4 0.15 0.19 0.25 0.28 0.35 

percentage of Delt=1   35 34 35 35 35 

F (Weibull)   5 0.17 0.23 0.47 0.53 0.6 

percentage of Delt=1   30 29 29 30 29 

F (Weibull)  6 0.26 0.47 0.64 0.72 0.82 

percentage of Delt=1   23 23 25 25 25 

F (Pareto)  0.25 0.093 0.118 0.094 0.082 0.087 

percentage of Delt=1   83 83 83 83 83 

F (Pareto) 0.5 0.217 0.238 0.25 0.216 0.228 

percentage of Delt=1   81 80 80 80 80 

F (Pareto)   2 0.385 0.505 0.569 0.582 0.618 

percentage of Delt=1   63 63 63 63 63 

F (Pareto)  3 0.325 0.436 0.459 0.506 0.53 

percentage of Delt=1   54 55 55 55 54 

F (Pareto)   4 0.275 0.321 0.303 0.341 0.371 



11 Int. J. Anal. Appl. (2023), 21:80 

 

 

percentage of Delt=1   48 48 48 48 48 

F (Pareto)  5 0.229 0.219 0.227 0.2 0.2 

percentage of Delt=1   43 43 43 43 43 

F (Pareto)  6 0.156 0.123 0.116 0.107 0.127 

percentage of Delt=1   39 38 38 38 39 

F (Rayleigh)   0.013 0.121 0.147 0.153 0.16 

percentage of Delt=1   58 58 58 58 58 

 

Table 3 : Mean and estimated risks  for  𝑀𝑛(𝐹)  statistic  for testing 𝐵𝐹𝑅 − 𝑡0  and 𝑊𝐹𝑅 −

𝑡0  ageing properties. 

n Mean ER ER/n n Mean ER ER/n 

10 -0.3321 0.9927 0.0993 31 0.5667 0.8982 0.0290 

11 -0.2782 0.9567 0.0870 32 0.5036 0.7378 0.0231 

12 -0.1888 0.9000 0.0750 33 0.5329 0.7943 0.0241 

13 -0.2058 0.7048 0.0542 34 0.5706 0.8068 0.0237 

14 -0.0598 0.9599 0.0686 35 0.6364 0.8101 0.0231 

15 -0.0485 0.7991 0.0533 36 0.6030 0.7587 0.0211 

16 0.0075 0.9237 0.0577 37 0.6182 0.7333 0.0198 

17 0.0579 0.8699 0.0512 38 0.6362 0.7163 0.0188 

18 0.0768 0.7815 0.0434 39 0.6821 0.7759 0.0199 

19 0.1144 0.8101 0.0426 40 0.6886 0.9857 0.0246 

20 0.1872 0.9038 0.0452 41 0.6787 0.7548 0.0184 

21 0.2494 0.8292 0.0395 42 0.7043 0.8144 0.0194 

22 0.2207 0.7853 0.0357 43 0.7457 0.7930 0.0184 

23 0.2370 0.7266 0.0316 44 0.7248 0.7257 0.0165 

24 0.3127 0.8435 0.0351 45 0.7847 0.8133 0.0181 

25 0.3467 0.8647 0.0346 46 0.7854 0.7610 0.0165 

26 0.3367 0.7591 0.0292 47 0.8035 0.8464 0.0180 

27 0.3529 0.7320 0.0271 48 0.8510 0.8627 0.0180 

28 0.4842 0.9700 0.0346 49 0.8423 0.8315 0.0170 

29 0.4765 0.9224 0.0318 50 0.8303 0.8485 0.0170 

30 0.4503 0.8198 0.0273 
    

 



12 Int. J. Anal. Appl. (2023), 21:80 

 

 

Table 4 : Test of normality  for𝑀𝑛 (𝐹)  statistic  for testing 𝐵𝐹𝑅 − 𝑡0  and 𝑊𝐹𝑅 − 𝑡0  ageing 

properties 

n D n D 

10 0.073 31 0.093 

11 0.0835 32 0.1153 

12 0.0853 33 0.1037 

13 0.0874 34 0.1158 

14 0.0794 35 0.0976 

15 0.0882 36 0.1144 

16 0.0892 37 0.1209 

17 0.0894 38 0.1044 

18 0.0803 39 0.0915 

19 0.0907 40 0.1147 

20 0.0774 41 0.0883 

21 0.0913 42 0.0891 

22 0.0925 43 0.0925 

23 0.0917 44 0.113 

24 0.099 45 0.1095 

25 0.0951 46 0.1096 

26 0.0806 47 0.1116 

27 0.1002 48 0.0987 

28 0.1153 49 0.1232 

29 0.0963 50 0.0809 

30 0.0784     

 

 

Acknowledgement: The author extends his appreciation to the Arab Open University for funding 

this work through AOU research fund No. (AOURG-2023-016). 

 

Conflicts of Interest: The author declares that there are no conflicts of interest regarding the 

publication of this paper. 

 



13 Int. J. Anal. Appl. (2023), 21:80 

 

 

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