RELIABILITY ESTIMATION FOR TWO PARAMETERS LOG-LOGISTIC DISTRIBUTION

Makki  A. Mohammed  Salih, Ahmed  J. Kamees

Abstract


In this paper, we study two parameters Log-Logistic distribution, applied two estimation methods
(Moment Method (MOM) and Modification Moment Method (MM)) to estimate  parameters and reliability
function for this distribution.
We used the simulation to generate random data follow the distribution on three models of the real values
of the  parameters:  )4,5(:),5.3,5.1(:),3(: 321   AAA with  sample
size (n =10,25,50,75,100) and replicate sample (N=1000), and taking five different failure times for each case to
estimate reliability function.
Comparisons have been made between the obtained results from the estimators using the Mean Square
Errors (MSE), the results indicate that:-Two parameters estimators of (MM)are better than estimators of (MOM), and the reliability estimators
of (MOM)are better than estimators of (MM).
Keywords: Log–Logistic distribution, Moment Method, Modification Moment Method


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References


In this paper, we study two parameters Log-Logistic distribution, applied two estimation methods

(Moment Method (MOM) and Modification Moment Method (MM)) to estimate parameters and reliability

function for this distribution.

We used the simulation to generate random data follow the distribution on three models of the real values

of the parameters: )4,5(:),5.3,5.1(:),3(: 321   AAA with sample

size (n =10,25,50,75,100) and replicate sample (N=1000), and taking five different failure times for each case to

estimate reliability function.

Comparisons have been made between the obtained results from the estimators using the Mean Square

Errors (MSE), the results indicate that:-Two parameters estimators of (MM)are better than estimators of (MOM), and the reliability estimators

of (MOM)are better than estimators of (MM).

Keywords: Log–Logistic distribution, Moment Method, Modification Moment Method

INTRODUCTION

In 2003 M. O. Ojo and A. K. Olapade, studied

the generalized Logistic and the generalized Log-Logistic distributions are considered. Some theorems

that characterize the generalized Logistic distribution

are proved[7].In 2006 Coºkun Kuº and Mehmet Fedai

Kaya, studied, maximum likelihood estimates for the

parameters of the Log-Logistic distribution are

obtained using the EM algorithm based on a

progressive Type-II right censored sample. An

illustrative example is also given[5]. In 2006 R.R.L.

Kantam, G.Srinivasa Rao & B. Sriram , sampling plans

in which items that are put to test, to collect the life of

the items in order to decide upon accepting or rejecting

a submitted lot, are called reliability test plans. The

basic probability model of the life of the product is

specified as the well-known Log-Logistic distribution

with a known shape parameter. For a given producer’s

risk, sample size, termination number, and waiting time

to terminate the test plan are computed. The prefer

ability of the test plan over similar plans existing in

the literature is established with respect to cost and

time of the experiment[6]. In 2007 Yan Yan Zhou, Jie

Mi · Shengru Guo, studied demonstrates the existence

and uniqueness of the MLEs of the parameters of

the Logistic distribution under mild conditions with

grouped data. The times with the maximum failure

rate and the mode of the p.d.f. of the Log-Logistic

distribution are also estimated based on the MLEs.

The methodology is further studied with simulations

and exemplified with a data set with artificially

introduced grouping from a locomotive life test

study[9]. In 2009 Rao, Srinivasa, et al. studied reliability

function of the well –known Log-Logistic distribution

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Reliability Estimation For Two Parameters Log-Logistic Distribution

model is considered and viewed as parametric function.

for certain known combinations of its shape parameter,

its scale parameter is estimated by different methods

of estimation from right censored samples. The

methods of estimation are show to be asymptotically

equivalent to MLE method of estimation[8].

The Log-Logistic distribution economics is a

continuous probability distribution for a non-negative random variable. It is used in survival

analysis as a parame-tric for events whose rate

increases initially and decreases later, for example

mortality rate from cancer following diagnosis or

treatment. It has also been used in hydrology to model

stream flow and precipitation and in economics as a

simple model of the distribution of wealth or in come.

The probability density function and cumulative

distribution function for two parameters Log– Logistic

distribution was suggested by Balakrishnan (1987) and

given by[3]:

...(1)

...(2)

Where 0;1;0  x , is scale

parameter and is shape parameter

Since

   xRxXPxF r  1)(

So, the reliability function )(xR is the

conditional probability of failure in the time interval

, x) given that the system has survived to time x.

...(3)

)(





x

x

xf



x

xF

(



x

xR

(

ESTIMATION METHODS

In this section we discuss two methods to

estimate two unknown parameters for Log-Logistic

distribution in complete data.

Moment Method (MOM )

This method is one of the simplest techniques

commonly used in the field of parameter estimation[4]

.In a wide variety of problems the parameter to be

estimated is some known function of given finite

number of moments about zero.

The moment method leads to get consistent and

unbiased estimators, in this case the estimators are

asymptotically normally distribution[1] .

The rth moments of sample and population for

two – parameters Log-Logistic distribution are given,

respectively:

...(4)

...(5)

From equation (4) and (5) the first moment,

where (r=1), are given from follows:

… (6)

The second moments, where (r=2) in equation

(4) and (5) are given from follows:

…(7)

  n

x

xE

n

i

r

ir

r

 

 1 )(





r

r r

r

sin

'



 



sin

)( 1 )(

n

x

xEx

n

i i

 

 



sin

2

)(

n

xn

i i

 

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Reliability Estimation For Two Parameters Log-Logistic Distribution

Now, solving equations (6) and (7) by fixed

point iteration method, the first technique form (6),

we get:

Then:

… (8)

And, from equation (7), we get:

sin

ˆ

x



 ˆsin



x



sin

ˆ 1

x

MOM

sinˆ

ˆ2

x

 

sin

ˆ2ˆ 1



x

MOM

MOM



 2

ˆ22sinˆ 

x

… (9)

And, the second technique from equation (6), we get:

… (10)

And from equation (7), we get:



 0

sinˆ 

x



sin

ˆ 2



x

MOM

… (11)

So, to estimate the reliability approximation

)(ˆ xR substitute equations (8) and (9) in equation (3)

,we get

… (12)

And substituted equations (10) and (11) in equation

(3),we get:

… (13)

Modification Moment Method (MM)

In year (1982) the two researchers Whitten

and Cohen suggest new modify by used equation:

… (14)

i represents the views rank after arranged in

ascending order, is estimated unbiased for

function distribution and by replacement

by plotting position formula[2].

Then

… (15)

From two formulas(2) and (15), we get:

ˆ

sinˆ

ˆ 2

 

MOM

MOM

MOM

x



ˆ

sinˆ

ˆ2 2

x



 22

ˆ2ˆ

sinˆ 

x

ˆ

ˆ

ˆ

 MOM

MOM

MOM

x

R 

 

ˆ

ˆ

ˆ

 MOM

MOM

MOM

x

R 

  )()(ˆ )()( ii XFXFE 

( )1(

n

XF

ni

n

iXF i ,...,2,1,

)( 

)1(

 nx

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Reliability Estimation For Two Parameters Log-Logistic Distribution

… (16)

By taking the natural logarithm for equation

(16), we get:

… (17)

… (18)

And form equation (17), we get:

… (19)

Now, the variance of sample its estimate of

variance of statistical population and variance defined

by:

… (20)

Since

… (21)

Then

… (22)

n

x





)1(

)1(



n

x



)1(

)(ˆ 1

x

Ln

nLn

MM

)()1(

nLn

x

Ln 

)()1(

nLn

x

Ln 

  1

)1(ˆ 

 

MM

MM nx

 

)1( )()( nxLnLn

)(1)()( )1( nLnxLnLn

 

)(1)()( )1( nLnLnxLn

 

)(1)1(

nLn

x

Ln





 22

)()()( xExExVar 

)( SxVar 

 

 n

i i xx

n

S 1

)(

 

n

i i xx

n 1

)(

sin

sin

And substituted the first and second moment

of population for Log-Logistic distribution in

equation(22), we get:

… (23)

… (24)

Now, from equation (23)

… (25)

So, to estimate the reliability approximation

)(ˆ xR substitute equations (18) and (19) in equation

(3),we get

 

n

i i xx

n 1

)(

sin

sin





 

sin

sin

1(

)(ˆ 2

n

xxn

i i

MM

  

 n

i i xxn 1

)(

sin

sin

1( 



  

 n

i i xx

n MM

MM

)(

sin

sin

ˆ)1(

ˆ





  

n

i i xx

n 1

)(

sin

sin







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Reliability Estimation For Two Parameters Log-Logistic Distribution

… (26)

And substituted equations (24) and (25) in

equation (3), we get:

… (27)

SIMULATION STUDY

We obtained, in the above sections, moment

and modify moment estimates for two parameters

Log-Logistic distribution. We adopted the Mean

Squared Errors (MSE), in order to assess the statistical

performances of these estimates, a simulation study

is conducted, and using generated random samples

for different sizes are to compute each estimator Log-Logistic distribution[2].stages of generation and

estimation parameters and reliability are as follows:

a. Specify default values for ( then choosing

the default values for real parameters in three

cases are arranged the following table

Table 1. Default values for the distribution

parameters

ˆ

ˆ

ˆ

 MM

MM

MM

x

R 

ˆ

ˆ

ˆ

 MM

MM

MM

x

R 

Parameters

A

 

3 1A

5 1.5 2A

5 3A

b. Choosing five different sizes for sample (n=10, 25,

, 75,100) and generated data which is follows

Log-Logistic distribution by using the following:

From (2) we obtained:

If U = F(x) where U is continuous random

on (0, 1) then we obtained a random sample from

the following:

c. We computed estimators MOMˆ from equations

(8) and (11), estimators MMˆ from equations (20)

and (24), MOMˆ from equations (9) and (10),

estimators MMˆ from equations (18) and (25),

estimators )(ˆ xRMOM obtained from equations

(12) and (13) and estimators )(ˆ xRMM

obtained

from equations (26) and (27). The real time to

estimate reliability is given in table 2:

Table 2. Real times of reliability for three cases

U

Ux

 x

xF

)(

 x

xF

)(



 xxF

xF

)(1

)(

ti 1A 2A 3A

t1 0.2 0.1 .75

t2 4.2 1.66 4.81

t3 8.2 3.22 8.87

t4 12.2 4.78 12.93

t5 16.2 6.34 16.99

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Reliability Estimation For Two Parameters Log-Logistic Distribution

d. The above steps are replicated (1000) times and

the Mean Square Errors (MSE) are computed for

different sample sizes.

, Where is

any parameters and N is replicated of samples.

THE RESULTS

After applying both methods on the data

generated to estimate the parameters, the results of

(MSE) of these estimates, as shown in the following

tables.

Results of

Table 3. MSE(ˆ ) for AA1

Table 4. MSE(ˆ ) for AA2

n

Methods

MOM-1 MOM -2 MM -1 MM -2 BEST

.64880 .44654 .14467 35.81379 MM-1

.48325 .22813 .05522 13.12261 MM-1

.46317 .13850 .02401 9.84696 MM-1

.34554 .08866 .01591 5.66612 MM-1

.25993 .06359 .01209 3.40908 MM-1

n

Methods

MOM-1 MOM -2 MM -1 MM -2 BEST

.78405 .51660 .22350 65.85587 MM-1

.51265 .25865 .08569 19.75606 MM-1

.37610 .14465 .03749 11.86174 MM-1

.25944 .09336 .02479 6.97863 MM-1

.18096 .06873 .01885 4.74853 MM-1

n

Methods

MOM-1 MOM -2 MM -1 MM -2 BEST

.10479 .68801 .02638 7.00668 MM-1

.07197 .03148 .01010 2.29844 MM-1

.00610 .01998 .00440 1.50479 MM-1

.04379 .01294 .00291 .88207 MM-1

.03065 .00946 .00221 .60251 MM-1

Table 5. MSE(ˆ ) for AA3

Results of

Table 6. MSE(ˆ ) for AA1

Table 7. MSE(ˆ ) for AA2

Table 8. MSE(ˆ ) for AA3

n

Methods

MOM-1 MOM -2 MM -1 MM -2 BEST

32.46197 39.81207 9.02067 42.37429 MM-1

5.36669 6.73557 2.43589 7.14195 MM-1

2.85661 3.95788 1.33692 3.69841 MM-1

1.94869 2.70742 .92555 2.41931 MM-1

1.62628 1.72375 .76343 1.95635 MM-1

N

Methods

MOM-1 MOM -2 MM -1 MM -2 BEST

62.25411 65.57915 16.06369 74.91019 MM-1

10.49572 10.20079 4.33485 12.70474 MM-1

5.56356 5.14508 2.37691 6.57743 MM-1

3.74637 3.41192 1.64543 4.30159 MM-1

3.09684 2.80642 1.35723 3.47638 MM-1

n

Methods

MOM-1 MOM -2 MM -1 MM -2 BEST

46.08802 92.66202 12.28988 57.3071 MM-1

7.72091 25.8101 3.31743 9.71962 MM-1

4.10219 3.76685 1.81977 5.03567 MM-1

2.77639 2.53460 1.25978 3.29341 MM-1

2.30462 2.25949 1.03912 2.66158 MM-1

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Reliability Estimation For Two Parameters Log-Logistic Distribution

Results of )(ˆ xR

MSE( )(ˆ xR ) for

Table 9. Results for (n=10)

Table 10. Results for (n=25)

Table 11. Results for (n=50)

n=10

ti

Methods

MOM-1 MOM -2 MM -1 MM -2 BEST

2 3.01065E-07 4.27122E-07 1.12231E-04 1.89205E-03 MOM-1

2 3.64155E-02 3.21540E-02 1.61863E-02 2.04398E-01 MM-1

2 3.01152E-03 2.71569E-03 4.26335E-03 1.17318E-01 MOM-2

2 3.78605E-04 5.66211E-04 1.81279E-03 5.32035E-02 MOM-1

2 6.43011E-05 2.04840E-04 9.84784E-04 2.42468E-02 MOM-1

n=50

ti

Methods

MOM-1 MOM -2 MM -1 MM -2 BEST

2 8.23517E-08 2.01502E-03 1.45524E-05 2.97303E-04 MOM-1

2 1.61102E-02 1.58981E-02 5.00941E-03 1.07893E-01 MM-1

2 2.37506E-03 2.61829E-02 1.82902E-03 3.45369E-02 MM-1

2 5.29207E-04 7.10809E-04 6.25720E-04 1.24886E-02 MOM-1

2 1.54470E-04 4.14123E-04 2.80969E-04 4.89831E-03 MOM-1

n=25

ti

Methods

MOM-1 MOM -2 MM -1 MM -2 BEST

2 1.41151E-07 1.61485E-03 2.90861E-05 2.88315E-04 MOM-1

2 2.28632E-02 1.56588E-02 8.43082E-03 1.41424E-01 MM-1

2 2.15227E-03 2.36228E-02 2.50957E-03 5.63125E-02 MOM-1

2 2.43905E-04 6.26188E-04 8.97646E-04 2.08199E-02 MOM-1

2 4.88993E-05 3.40367E-04 4.21278E-04 1.13742E-02 MOM-1

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Reliability Estimation For Two Parameters Log-Logistic Distribution

Table 12. Results for (n=75)

Table 13. Results for (n=100)

MSE( )(ˆ xR ) for

Table 14. Results for (n=10)

n=10

ti

Methods

MOM-1 MOM -2 MM -1 MM -2 BEST

1 1.71079E-08 4.08047E-07 4.96417E-05 1.21590E-03 MOM-1

66 5.16385E-02 2.52760E-02 1.68170E-02 1.88036E-01 MM-1

22 4.81584E-03 1.42709E-02 5.46026E-03 1.68921E-01 MOM-1

78 4.60020E-04 5.44287E-04 2.01751E-03 8.23447E-02 MOM-2

34 7.20303E-05 1.94775E-04 9.92671E-04 4.19260E-02 MOM-1

n=75

ti

Methods

MOM-1 MOM -2 MM -1 MM -2 BEST

2 9.23429E-08 2.28998E-03 1.36257E-05 1.81215E-04 MOM-1

2 1.31994E-02 1.56810E-02 3.82575E-03 8.50550E-02 MM-1

2 1.92239E-03 2.78151E-02 1.54611E-03 2.36043E-02 MM-1

2 3.06988E-04 7.17508E-04 5.22778E-04 8.73974E-03 MOM-1

2 7.12579E-05 4.38829E-04 2.33830E-04 3.75763E-03 MOM-1

n=100

ti

Methods

MOM-1 MOM -2 MM -1 MM -2 BEST

2 6.93945E-08 1.96825E-03 9.65933E-06 6.40485E-05 MOM-1

2 1.02605E-02 1.54849E-02 3.27120E-03 7.03139E-02 MM-1

2 1.61192E-03 2.74099E-00 1.38469E-03 1.34596E-02 MM-1

2 2.79934E-04 4.88916E-04 4.50460E-04 3.90565E-03 MOM-1

2 5.94841E-05 2.68710E-04 1.93633E-04 1.90526E-03 MOM-1

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Reliability Estimation For Two Parameters Log-Logistic Distribution

Table 15 . Results for (n=25)

Table 16 . Results for (n=50)

Table 17. Results for (n=75)

n=25

ti

Methods

MOM-1 MOM -2 MM -1 MM -2 BEST

1 2.11225E-08 5.53073E-06 9.80228E-06 1.81141E-03 MOM-1

66 2.77319E-02 1.86036E-02 6.20132E-03 1.42497E-01 MM-1

22 3.22715E-03 2.74445E-03 3.32797E-03 8.60677E-02 MOM-2

78 2.95514E-04 4.46994E-04 1.02551E-03 2.92788E-02 MM-1

34 4.79710E-05 1.78742E-04 4.26407E-04 1.45766E-02 MOM-1

n=50

ti

Methods

MOM-1 MOM -2 MM -1 MM -2 BEST

1 1.16781E-08 3.47453E-06 4.20534E-06 1.20372E-05 MOM-1

66 1.79551E-02 9.72114E-03 2.73938E-03 1.16712E-01 MM-1

22 3.15263E-03 2.30052E-03 2.43195E-03 5.04259E-02 MOM-2

78 5.06418E-04 4.28100E-04 7.20501E-04 1.70211E-02 MOM-2

34 9.10715E-05 1.70544E-04 2.84776E-04 5.79297E-03 MOM-1

n=75

ti

Methods

MOM-1 MOM -2 MM -1 MM -2 BEST

1 6.66916E-09 2.10394E-06 4.43883E-06 1.38372E-05 MOM-1

66 1.33432E-02 6.64329E-03 1.85490E-03 9.91020E-02 MM-1

22 2.39988E-03 1.98326E-03 2.05434E-03 3.32522E-02 MOM-2

78 2.76783E-04 3.73064E-04 6.02972E-04 1.05372E-02 MOM-1

34 4.73482E-05 1.42827E-04 2.37004E-04 3.87255E-03 MOM-1

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Reliability Estimation For Two Parameters Log-Logistic Distribution

Table 18 . Results for (n=100)

MSE( )(ˆ xR ) for

Table 19. Results for (n=10)

Table 20 . Results for (n=25)

n=100

ti

Methods

MOM-1 MOM -2 MM -1 MM -2 BEST

1 6.07956E-09 8.84345E-07 2.94460E-06 5.71245E-05 MOM-1

66 9.92017E-03 5.23719E-03 1.41449E-03 8.50606E-02 MM-1

22 1.99741E-03 1.72661E-03 1.84944E-03 2.00657E-02 MOM-2

78 2.13790E-04 2.72661E-04 5.22816E-04 4.41834E-03 MOM-1

34 3.24408E-05 8.43410E-05 196401E-04 1.90728E-03 MOM-1

n=10

ti

Methods

MOM-1 MOM -2 MM -1 MM -2 BEST

.75 8.12570E-07 3.05404E-06 1.57740E-04 1.83601E-03 MOM-1

81 5.39816E-02 4.72926E-02 1.48031E-02 1.56776E-01 MM-1

87 7.59291E-03 5.99706E-03 7.21620E-03 2.18521E-01 MOM-2

93 6.43634E-04 6.27993E-04 2.42026E-03 1.21022E-01 MOM-2

99 8.47751E-05 1.78254E-04 1.10521E-03 6.60352E-02 MOM-1

n=25

ti

Methods

MOM-1 MOM -2 MM -1 MM -2 BEST

.75 6.12071E--07 4.29296E-06 4.54094E-05 1.26251E-03 MOM-1

81 2.99058E-02 2.11165E-02 5.08422E-02 1.25828E-01 MOM-2

87 5.11034E-03 4.19521E-03 4.49213E-03 1.21214E-01 MOM-2

93 4.55697E-04 4.55367E-04 1.28217E-03 4.75194E-02 MOM-2

99 7.16722E-05 1.17567E-04 4.88296E-04 1.92595E-02 MOM-1

Magistra No. 101 Th. XXIX Desember 2017

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Reliability Estimation For Two Parameters Log-Logistic Distribution

Table 21 . Results for (n=50)

Table 22 . Results for (n=75)

Table 23 . Results for (n=100)

n=50

ti

Methods

MOM-1 MOM -2 MM -1 MM -2 BEST

.75 3.47820E-07 2.69700E-06 2.40037E-05 1.04448E-04 MOM-1

81 1.96311E-02 1.12446E-02 2.20979E-03 1.06429E-01 MM-1

87 4.25879E-03 3.26002E-03 3.25387E-03 7.37866E-02 MM-1

93 4.96378E-04 3.85315E-04 9.11723E-04 2.44709E-02 MOM-2

99 6.23107E-05 8.99472E-05 3.28941E-04 8.71389E-03 MOM-1

n=75

ti

Methods

MOM-1 MOM -2 MM -1 MM -2 BEST

.75 2.79400E-07 7.67238E-07 2.17070E-05 9.41733E-05 MOM-2

81 1.40278E-02 6.98868E-03 1.36816E-03 9.57195E-02 MM-1

87 3.15673E-03 2.72151E-03 2.73126E-03 4.89905E-02 MOM-2

93 3.07882E-04 3.19221E-04 7.65307E-04 1.42645E-02 MOM-1

99 4.21477E-05 6.39635E-05 2.73741E-04 4.87993E-03 MOM-1

n=100

ti

Methods

MOM-1 MOM -2 MM -1 MM -2 BEST

.75 2.67606E-07 7.46499E-07 1.57863E-05 9.94518E-05 MOM-1

81 1.04431E-02 5.14501E-03 9.94920E-04 8.30561E-02 MM-1

87 2.73788E-03 2.48491E-03 2.46198E-03 3.16266E-02 MM-1

93 2.56242E-04 2.71702E-04 6.70096E-04 6.01749E-03 MOM-1

99 3.48102E-05 4.66089E-05 2.28574E-04 2.15103E-03 MOM-1

Magistra No. 101 Th. XXIX Desember 2017

ISSN 0215-9511

Reliability Estimation For Two Parameters Log-Logistic Distribution

CONCLUSION

analyzed the results of the scale parameter, values

of (MSE), in tables(3) ,(4) and (5) shows that the

(MM-1) is better than the others estimators in all

cases the sample and the values of the parameters.

analyzed the results of the shape parameter, values

of (MSE), in tables(6), (7) and (8) shows that the

(MM-1) is better than the others estimators in all

cases the sample and the values of the parameters.

The Reliability approximation, through the study

and analyzed of the results obtained from the tables

(9,10,11,12,13) in the first case, and tables

(14,15,16,17,18) in the second case, and tables

(19,20,21,22,23) in the third case, we found that

the order of performance estimation methods for

most of the cases are as follows:

Table 24 . The order of preference for the reliability

estimators

Order 1 2 3 4

Method MOM-1 MM-1 MOM-2 MM-2

Magistra No. 101 Th. XXIX Desember 2017

ISSN 0215-9511

Reliability Estimation For Two Parameters Log-Logistic Distribution

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