Comparing Three Different Estimators of Reliability Function of … · 2018. 9. 5. · effects, then a lognormal distribution must be the result. Many researchers study lognormal

Al-Mansour Journal/ Issue(30) 2018 ) 30مجلة المنصور/ العدد (

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Comparing Three Different Estimators of ReliabilityFunction of Lognormal Distribution

Rabha S. Kareem*,M.Sc.(Lecturer)

AbstractThe estimation of reliability function is important to indicate the ability of

machine and system to work without failure work for long time, this lead toincrease productivity, the research include estimating reliability function ofsome probability distribution (which is the lognormal) with two parameters( , ), where this distribution is necessary when the time failure ismeasured in hours, so it may be of large values, so transformation is takenon it and change values of ( ) into (log ). It is found that (log ) follownormal distribution ( , ), then estimating these parameters by maximumlikelihood and moments estimator, also introduce simple linear regressionin estimating ( , ) then [ ( )]. The comparison has been done throughsimulation using different sets of initial values for ( , ) and different setsof ( = 20,40,80), the results are compared using statistical measure meansquare error (MSE), and each experiment repeated ( = 1000 )

Keywords:Reliability Function,normal distribution, maximum likelihood,regression method estimation.

____________________*University of Information Technology & Communication

Rabha S. Kareem,M.Sc.(Lcturer)

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1. IntroductionThe normal distribution is one of important distribution that used for

estimation and test statistics in all applications, also in reliability estimationwhen the values that represent time to failure of some equipment aremeasured in hours, these values are distributed normally ( , )], but wecan use logarithmic transformation, i.e, to change the values of ( ) into(log ), this transformation is good for original data especially when thevalues of data are large, or represent multiplication, or geometric mean,also when the random variation represent the product of several randomeffects, then a lognormal distribution must be the result. Many researchersstudy lognormal distribution and have obtained a formula for meandifference of lognormal distribution. Also Jhon Norstad (2011) discussedimportant and basic properties of the normal and lognormal distributionswith its proof.

2. Definition of Lognormal Failure ModelThe probability density function ( . . ) of lognormal distribution with

two parameters ( , ) is given as;( ) = √ ( ) > 0 (1)

Also it can be rewritten by another formula ( . . ) depending on ( ),as given in equation (2), while the cumulative C.D.F is given as;( ) = Φ (2)

The p.d.f defined in equation (1) is obtained after applying logarithmtransformation on the variable [ ~ ( , )], so this transformation yield anew probability distribution called (Log. distribution), which is necessaryfor application in which the scales of units have large values.

The formula of (rth) moments origin is also derived by applying [ ′ =( )] on equation (1), this lead to;

′ = ( ) = (3)Equation below represent the C.D.F (in terms of median) from equation (2);( ) = Φ log = Φ(Z) (4)


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Where the standard values of (Z) are obtained by using;= log − log (5)

When the observations of failure time model have large values, then wecan take the logarithms, then the random variable ( = log ) have anormal distribution with mean ( ) and variance ( ) and its p.d.f is;( ; , ) = √ ( ) > 0 (6)

Depending on . . lognormal in equation (6) we can show that themaximum likelihood estimator of both parameters ( , ) are;̂ = = ∑

(7)

= ∑ ( ) (8)= (9)The variance;( ) = ( ) − ( ) = − = − 1 (10)

Also; = (11)The median time to failure;= (12)The mode of distribution;= (13)The reliability function for lognormal distribution is;( ) = 1 −Φ = 1 −Φ log (14)

Then;


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( )( ) = 1 −Φ (15)

Then the maximum likelihood estimator of (M) is the MLE for ( ̂);̂ = = ∑= ∑ ( − ) = ∑ log − ∑ (16)

Where, ( = log , = 1,2, … , ). This yield ( , , … , ) be also arandom sample taken from distribution [ ( , )], hence the maximumlikelihood estimator for ( & ) as shown in equation (19) and ( = ).

According to ( ̂ , );

( ) = 1 −Φ (17)

3. Estimation of two parametersFor lognormal failure model, we have ( ) observations represents a

random sample from [ ( , )], let ( = log , = 1,2, … , ), then the MLEof ( & ) as shown in equation (19), we can show that the estimators( ), then the maximum likelihood estimator of MTTF [ ( )], and( ) are;( ) = =( ) = − 1 (18)

Another required estimators using MLE as shown in equation (17), alsothe MLE for median of lognormal distribution as shown in equation (9).

4. Estimation by Regression LineWe have ( ) set observations of (log , ), where;= Φ [ ( )] = log − log (19)


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Here we have the observation ( ) and ( = log ), while the dependentvariable here is denoted by;= += ∑∑ ( ) (20)

= − = 1From value of ( , ), we can obtain ( ), where, ( ̂ = ).

5. SimulationTo generate random sample from logarithm normal ( , ), first of all we

generate random variable [ ~ (0.1)] by applying (Box – Muller) methodusing different initial values of ( & ), with different sample size, then;= ~ ( , )Where; == ( ) − 1Then choosing ( , ) as random variable from Uniform (0,1), then( & ) are exponential transformation of ( ) which are;= [ ( )]= [ ( )]Where; & = (0,1)


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Table (1): MSE for (MLE, MOM, REG) estimator parameters with ( = 1.5, = 1)


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Table (2): Values of [ ( )] estimated by (MLE, MOM, REG) when ( = 1.5, = 1)


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Table (3): MSE for (MLE, MOM, REG) estimator parameters with ( = 1, = 1.5)


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Table (4): Values of [ ( )] estimated by (MLE, MOM, REG) when ( = 1, = 1.5)


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Table (5): MSE for (MLE, MOM, REG) estimator parameters with ( = 2.5, = 2)


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Table (6): Values of [ ( )] estimated by (MLE, MOM, REG) when ( = 2.5, = 2)


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Conclusion After executing simulation experiment for all sets of initial values

of ( , ) and different set of (n), the best estimator is ( ), alsothe method of regression gives smallest (MSE) for estimation of( , ).

Also the estimation of two parameters ( , ) by ordinary leastsquare is necessary when using the model of regression model inforecasting.

The lognormal represent a good model to represent the dataespecially large value of ( ), where the logarithm transformationmake the data small and easy for evaluation.

We conclude that when the observations are negative we cannotapply lognormal distribution.


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References[1] A. Ramesh Kumar, V. Krishnan, (2017), “A Study on System Reliabilityin Weibull Distribution”, International Journal of Innovative Research inElectrical, Electronics, Instrumentation and Control Engineering.[2] Bahattchara, S. B. K. (1967), “Bayesian Approach to life testing andreliability estimation”, JASA, pp. (48 – 62).[3] Bassu A. P. (1964), “Estimates of Reliability for some Distributionuseful in life testing”, Technometrics, Vol.6, No. 2, pp. (215 – 219).[4] Edward Osakue, Lucky Anetor, (2016), “A Lognormal Reliability DesignModel”, International Journal of Research in Engineering and Technology.[5] Evan M., Hastings N. and Peacock P. (1993), “Statistical distributions”,second edition, Jhon Wiely and Sons, New York.[7] Fred Schenkelberg, (2012), “Use Lognormal Distribution”, MTBF : —ISSN 2168-4375.[8] Giovanni Girone, Fabio Manca, (2016), “The mean difference forlognormal distribution”, Applied Mathematics, 7, 824 – 828.[9] Jane Marshall, (2012), “An Introduction to Reliability and LifeDistributions”, Reliability and Life distributions.[10] K. S. Sultan, and A. S. Al-Moisheer, (2015), “Mixture of InverseWeibull and Lognormal Distributions: Properties, Estimation, andIllustration”, Mathematical Problems in Engineering Volume 2015 (2015),Article ID 526786, 8 pages.[11] Lu, H., He, Y. & Zhang, Y. (2014), Reliability-Based Robust Design ofMechanical Components with Correlated Failure Modes Based onMoment method, Advances in Mechanical Engineering.[12] Osakue, E. E., (2015), Lognormal Reliability-Based ComponentDesign, Technical Report, Dept. of Industrial Technologies, TexasSouthern University, Houston, Texas[13] Prasanta Basak, Indrani Basak, N. Balakrishnan, (2009), “Estimationfor the three-parameter lognormal distribution based on progressivelycensored data”, Computational Statistics & Data Analysis Volume 53 Issue10 .[14] Sinha S.K.(1985), “Bayes Estimation of the reliability function ofnormal distribution”, IEEE Vol.R-34, No.4, pp. (359 – 362).[15] S. K. Upadhyay, M. Peshwani, (2007), “Choice Between Weibull andLognormal Models: A Simulation Based Bayesian Study”,Communications in Statistics - Theory and Methods.


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مقارنة ثالث مقدرات للدالة المعولیة للتوزیع الطبیعي اللوغارتمي*رابحة سلیم كریم.م

المستخلصیتضمن ھذا البحث مقارنة ثالث طرائق لتقدیر كل من معلمة القیاس والشكل ودالة المعولة للتوزیع

ومقدرات االنحدار الخطي ,بعد تقدیر الطبیعي اللوغارتمي والطرائق ھي االمكان االعظم , العزوم ) n=20,40,80المعلمات قـُدرت ایضاً دالة المعولیة باستخدام المحاكاة ,حیث أًخذت ثالث حجوم عینات(

وتم تولید البیانات وقورنت نتائج المقدرات باستخدام المقیاس االحصائي (متوسط مربعات الخطأ ) ووجد فضل النھا تمتلك اصغر متوسط مربعات خطأ, وعند تنفیذ ان مقدرات االنحدار الخطي كانت ھي اال

) مرة وعرضت نتائج تقدیر المعلمات ومقدرات المعولیة L=1000تجارب المحاكاة تم تكرار كل تجربة (في جداول خاصة.

الدالة المعولیة ,التوزیع الطبیعي,االمكان االعظم,طریقة االنحدار.:الكلمات المفتاحیة

____________________جامعة تكنولوجیا المعلومات واألتصاالت*

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Comparing Three Different Estimators of Reliability Function of … · 2018. 9. 5. · effects, then a lognormal distribution must be the result. Many researchers study lognormal