Comparison of Methods for Estimation of Sample Sizes under the Weibull Distribution

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1 Iteratioal Joural of Applied Egieerig Research ISSN Volume 12, Number 24 (2017) pp Research Idia Publicatios. Compariso of Methods for Estimatio of Sample Sizes uder the Weibull Distributio Ju Hyoug Lim Departmet of Idustrial ad Maagemet Egieerig, Kyoggi Uiversity Graduate School, Gyeoggi16227, Republic of Korea. Yog Soo Kim* Departmet of Idustrial ad Maagemet Egieerig, Kyoggi Uiversity, Gyeoggi 16227, Republic of Korea. Abstract Recet techological developmets have led to icreases i the umber ad complexity of idividual parts of multifuctioal systems. Therefore, it is difficult to collect failure data due to time ad cost limitatios i idustrial sites, ad oly cesored failure data ca be collected. It is ecessary to esure accuracy of reliability data for small sample sizes to reflect realistic coditios. I this study, we compared the accuracy of reliability measures calculated by covetioal statistical ad Bayesia methods cosiderig the failure mode, sample size, ad cesored ratio accordig to a six-step procedure; a compariso of accuracy of prior iformatio based o the Bayesia method was also performed. Based o the results, guidelies for estimatio methods for sample size are provided. Keywords: Bayesia iferece, least squares estimatio, maximum likelihood estimatio, MCMC, Weibull distributio INTRODUCTION The umber ad complexity of idividual parts of multifuctioal systems have icreased with rapid techological developmet, leadig to ew failure modes. I particular, the reliability of systems ad equipmet must be accurately assessed i key idustries (e.g., defese, shipbuildig, etc.) to avoid damage ad disasters caused by simple faults or failures [1, 2]. To determie the reliability of the system quatitatively, reliability aalysis is widely used to idetify the failure distributio by aalyzig the failure data of the system. Thus, large amouts of failure data are required; however, it is geerally difficult to collect failure data due to time ad cost limitatios. Usually oly cesored failure data that caot precisely idetify the failure time are available [3]. To reflect realistic coditios, it is therefore importat to esure accurate reliability for small sample sizes. Covetioal statistical methods, such as maximum likelihood estimatio (MLE) ad least squares estimatio (LSE), are widely used to estimate parameters of the failure distributio. I geeral, MLE has high accuracy for parameter estimatio whe the sample size is large. However, for small sample sizes, the Bayesia approach is kow to be more accurate [4]. The Bayesia approach estimates the parameters through the posterior distributio usig give iformatio (the prior distributio) for the subject, such as the system or parts. The Bayesia estimator, i.e., a posterior distributio, is geerally difficult to calculate. It uses various algorithms for this purpose; the Markov chai Mote Carlo (MCMC) method is widely used [5, 6]. Ahmed et al. (2010) ad Ibrahim et al. (2012) compared the estimatio performace betwee MLE ad the Bayesia method usig Jeffrey s distributio i various sample sizes [7, 8]. Kudu ad Raqab (2012) used the Bayesia method to calculate the posterior distributio for the shape parameter of the Weibull distributio accordig to prior distributios that are iformative ad o-iformative [9]. I this research, we assumed a Weibull distributio to represet the iitial, accidetal, ad wear failure. The accuracy of the reliability measure B 10 calculated by covetioal statistical methods ad the Bayesia method cosiderig the failure mode, sample size, ad cesored ratio were compared to justify the use of the Bayesia approach. I additio, the Bayesia method was assessed for accuracy of prior iformatio. Based o our results, we provide estimatio method guidelies for the sample size. PROCEDURES AND METHODS Study procedure The study procedure (Fig. 1) cosists of six steps for compariso of the estimatio accuracy betwee covetioal statistical methods ad Bayesia methods. The robustess of the Bayesia method for effectiveess of prior iformatio was also evaluated

Iteratioal Joural of Applied Egieerig Research ISSN 0973-4562 Volume 12, Number 24 (2017) pp. 14273-14278 Research Idia Publicatios. http://www.ripublicatio.

2 Iteratioal Joural of Applied Egieerig Research ISSN Volume 12, Number 24 (2017) pp Research Idia Publicatios. MSE = m (B 10 B 10i ) (2) m Next, aalysis of variace (ANOVA) was performed for experimetal poits of the calculated D ad MSE. The four factors tested are give below: 1. Sample size: 10, 20, 30, 40, Shape parameter: 0.5, 1.0, 2.0, Proportio of observed failure data (POFD): 20%, 40%, 60%, 80%, 100% 4. Estimatio method: LSE, MLE, NB, IB ( 80%), IB ( 40%), IB (0%), IB (+40%), IB (+80%) where NB idicates o-iformative Bayesia method, ad IB idicates iformative Bayesia method. Fially, followig the procedure, we outlie guidelies for the estimatio method with respect to sample size, based o the ANOVA results. The estimatio methods are discussed i the sectios that follow. Figure 1: Study procedure for compariso of estimatio accuracy. We first set the umber of factors, such as the values of parameters ad sample size. Scale parameters were fixed at 10 ad shape parameters were set at 0.5, 1.0, 2.0, ad 3.0 to compare the chages with respect to the failure mode. The sample size was divided ito 10, 20, 30, 40, ad 50. Fially, the proportios of observed failure data (POFD) were 20%, 40% 60%, 80%, ad 100%, where POFD of 100% idicates complete failure. Secod, to determie the effectiveess of prior iformatio, deviatios of 80%, 40%, +40%, ad +80% of the shape parameters were used for the mea value of the prior distributio; for example, give a shape parameter of 1.0, deviatio values were set at 0.2, 0.6, 1.4, ad 1.8, respectively. These values were ot aalyzed by covetioal statistical methods. Third, 1000 samples were geerated for all test poits ad the parameters were estimated by LSE, MLE, ad the Bayesia method. The reliability measure B 10 was calculated usig the estimated parameters. The fourth step ivolves calculatio ad aalysis of the accuracy of B 10 with respect to the mea ad variace. Here, D, is defied as the absolute value of the deviatio of the mea of estimated B 10 with respect to the true value, ad variace is the mea squared error (MSE) o B 10, as give below: D = B 10 B 10, where B 10 = m B 10i m (1) Covetioal statistical methods We used LSE ad MLE as covetioal statistical methods. Give a set of radom lifetimes t 1,, t, the probability desity fuctio of the Weibull distributio is give below i terms of the scale parameter α ad the shape parameter β: f(t) = β α (t α ) β 1 exp [ ( t α ) β]. LSE estimates ukow parameters by miimizig the sum of squared errors usig a liear regressio model. To estimate ukow parameters through LSE, the cumulative distributio fuctio (cdf) of the Weibull distributio uses a log trasformatio: l[ l(1 F(t i ))] = βl(t i ) βl(α). (4) Variable trasformatio is performed as y i = l[ l(1 F(t i ))] ad x i = l(t i ). Usig the least squares method, the sum of the squared error (SSE) model, A, is give by A = [y i (βx i βl(α))]. (5) To miimize the SSE, Eq. (5) is partially differetiated ito α ad β. The values of α ad β were both calculated to be zero. The calculated value is a parameter of the Weibull distributio. The MLE is a method for fidig the parameter that maximizes the likelihood fuctio, which idicates a fuctio of the parameters of a statistical model for the give data. The likelihood fuctio of the Weibull distributio for a give data set is (3) 14274

3 Iteratioal Joural of Applied Egieerig Research ISSN Volume 12, Number 24 (2017) pp Research Idia Publicatios. L(t i α, β) = f(t i ). (6) For complete failure data, the likelihood fuctio of the Weibull distributio is as follows: L(t i α, β) = ( β α ) β 1 t i exp [ ( t i α )β ]. (7) The log likelihood fuctio for Eq. (7) is give below: l(l(t i α, β)) = lβ + βl ( 1 α ) + (β 1) (lt i) β ( 1 α ) t i. To miimize the log likelihood fuctio, Eq. (8) is partially differetiated ito ad, respectively, to fid a parameter value that becomes zero. The partial differetial equatios for log likelihood fuctio are described by Eqs. (9) ad (10). As it caot be solved by a simple operatio, Eq. (10) is calculated usig umerical aalysis methods, such as the Newto Rapso method: ll α ll β = βα + βα (β+1) β t i = β + l (1 α ) + lt i ( t β i α ) Bayesia method (8) = 0, (9) l ( t i α ) = 0. (10) The Bayesia method based o Bayes theorem estimates the parameters usig the observed data ad iformatio about the data. The data represet the probability of the parameter through a likelihood fuctio ad the iformatio about the data is represeted by a prior distributio, which is a probability distributio. The posterior distributio ad the prior distributio are as follows: p(θ t i ) = p(t i θ)π(θ) p(t i ) p(t i θ)π(θ), (11) where θ is a parameter of the probability distributio, p(θ t i ) is the posterior distributio, π(θ) is the prior distributio, ad p(t i θ) is the likelihood fuctio. The posterior distributio for the Weibull distributio calculated usig Eq. (11) is as follows: p(α, β t i ) L(t i α, β)π(α, β). (12) It is difficult to calculate the posterior distributio by a simple operatio as it is usually provided i the form of a double itegral. I this study, the Metropolis Hastigs algorithm of MCMC was used to calculate the posterior distributio for the Weibull distributio. The prior distributio of the Bayesia method was divided ito iformative prior distributio ad o-iformative prior distributio, accordig to the presece or absece of iformatio, respectively. I the absece of prior iformatio about the parameters or past experiece, a o-iformative prior distributio that performs a miimal role for the prior distributio ca be applied. Zaidi et al. (2012) used Jeffrey s distributio as a o-iformative prior distributio uder the Weibull distributio [10]. Jeffrey s distributio ca be expressed as a Fisher iformatio matrix. The Fisher iformatio matrix o the Weibull distributio is as follows: π(α, β) I(α, β), (13) 2 lf(t α, β) α I(α, β) = E 2 2 lf(t α, β) α β 2 lf(t α, β) 2 α β 1 2 lf(t α, β) = ( αβ ). (14) β 2 The o-iformative prior distributio o the Weibull distributio is determied accordig to Eq. (13). The joit posterior distributio usig the likelihood fuctio ad Jeffrey s distributio is as follows: π(α, β) 1 αβ, (15) (1/α) β+1 β 1 t β 1 π(α, β t i ) = i exp[ (t i /α) β ] (1/α) β+1 β 1 t β 1. (16) i exp[ (t i /α) β ] dαdβ 0 0 Whe there is prior iformatio or theoretical kowledge about the parameter to be estimated, the posterior distributio ca be determied usig the prior distributio. I this study, we assumed that the prior distributio of the shape parameter ad the scale parameter o the Weibull distributio is a gamma distributio [11]: α~gamma(a, b), (17) β~gamma(c, d), (18) where a is a scale parameter o α, b is a shape parameter o α, c is a scale parameter o β, ad d is a shape parameter o β. Hyperparameters a, b, c, ad d form a set, such that the mea of each gamma distributio is the true value of ad ad the variace of each gamma distributio is 1. We assumed that we have appropriate iformatio if the mea of the prior distributio is the true value of the parameter. Thus, whe comparig the performace of the Bayesia method for the accuracy of prior iformatio, we ca compare the results accordig to the mea of the prior distributio by fixig the variace to a value of 1. RESULTS AND DISCUSSION ANOVA was performed for each of MSE ad D for a total of 800 experimetal poits, ad the iteractio was cosidered as the secod iteractio. The ANOVA tables for this are show i Tables 1 ad 2. The MSE ca be affected by all of the factors, 14275

4 Iteratioal Joural of Applied Egieerig Research ISSN Volume 12, Number 24 (2017) pp Research Idia Publicatios. ad the ifluece of the sample size ad the shape parameter is sigificat based o the F value. I the mai effect, the MSE decreased as the sample size ad POFD icreased, ad the shape parameter was small. The MSEs of Bayesia methods were geerally smaller tha the others, ad the egative deviatio of iformative Bayesia methods had a lower MSE tha the positive deviatio. The iteractio of the sample size ad the shape parameter was the greatest effect i the iteractios. The mai effect of D was similar to the results of the MSE, ad all factors were sigificat. I additio, the sample size ad shape parameters had the strogest ifluece, ad the iteractio betwee the shape parameter ad estimatio method was the greatest amog the iteractios. This was because as the shape parameter icreased, the D for the iformative Bayesia method ( 80%) was about twice those of the other methods. The o-iformative Bayesia method is preferred whe the POFD exceeds 40%. Table 1: ANOVA results for the MSE Factor Df Adj SS Adj MS F P Sample size Shape parameter POFD Method Sample size*shape parameter Sample size*cesor ratio Sample size*method Shape parameter*cesor ratio Shape parameter*method Cesor ratio*method Error Total Table 2: ANOVA results for D Factor Df Adj SS Adj MS F P Sample size Shape parameter POFD Method Sample size*shape parameter Sample size*cesor ratio Sample size*method Shape parameter *Cesor ratio Shape parameter *Method Cesor ratio*method Error Total

5 Iteratioal Joural of Applied Egieerig Research ISSN Volume 12, Number 24 (2017) pp Research Idia Publicatios. The MSE ad D accordig to the estimatio method icludig all sample sizes are show i Fig. 2. The solid lie represets the mea value ad the dashed lie represets the error rage of 20% o the mea value. As show i Fig. 2, the iformative Bayesia method showed better performace with more accurate prior iformatio. However, it is recommeded to uderestimate the value compared to the true parameter whe prior iformatio is ucertai. The LSE, MLE, ad oiformative Bayesia methods did ot show better performace tha the iformative Bayesia method. Fially, ANOVA was performed agai for the MSE ad D usig the shape parameter, estimatio method, ad POFD, for the sample size that was the most sigificat. For a quatitative idex, the rak of the guidelies is based o the mea value of the MSE ad D give by the ANOVA results (the give mea). The MSE ad D were reaalyzed for sample sizes of 10, 20, ad 30, as show i Fig. 3. Figure 2: MSE ad D for the estimatio methods. Figure 3: MSE ad D for sample sizes of 10, 20, ad

6 Iteratioal Joural of Applied Egieerig Research ISSN Volume 12, Number 24 (2017) pp Research Idia Publicatios. Table 3: Guidelies for the estimatio methods Sample size Classical methods Bayesia methods LSE MLE NB IB ( 80%) IB ( 40%) IB (0%) IB (+40%) IB (+80%) CONCLUSION I this paper, we compared the accuracy of estimatio methods usig various factors, icludig sample size, shape parameter, ad POFD, from the viewpoit of both the mea ad variace. It was cofirmed that the accuracy of the estimatio icreased as the sample size icreased. For a sample size greater tha 30, the performace of the estimatio methods was similar. Thus, guidelies for the estimatio methods were established based o sample sizes of 10, 20, ad 30, as show i Table 3. For a sample size of 10, the results showed that it is better to use the iformative Bayesia method. If there is o iformatio, the the LSE is recommeded. For a sample size of 20, the iformative Bayesia method showed the best performace. If o iformatio is available, oe of LSE, MLE, or NB ca be used. For a sample size of 30, the iformative Bayesia method was the best ad MLE was the secod best. Whe the iformative Bayesia method is used, the accuracy of the estimatio improves with greater accuracy of the prior iformatio. If the prior iformatio is ucertai, it is better to uderestimate the value tha to use the true value of the parameter. ACKNOWLEDGEMENT This work was supported by Kyoggi Uiversity Research Grat REFERENCES [1] Hauske, Kjell. "Strategic defese ad attack for reliability systems." Reliability Egieerig & System Safety Vol 93 (11), pp , [2] Eiseberg, D., et al. "A methodology to evaluate water ad wastewater treatmet plat reliability." Water sciece ad techology Vol 43 (10), pp 91-99, [3] Lord, Domiique, ad Luis F. Mirada-Moreo. "Effects of low sample mea values ad small sample size o the estimatio of the fixed dispersio parameter of Poisso-gamma models for modelig motor vehicle crashes: a Bayesia perspective." Safety Sciece Vol 46 (5), pp , [4] Berger, James O. Statistical decisio theory ad Bayesia aalysis. Spriger Sciece & Busiess Media, [5] Walters, Carl, ad Doald Ludwig. "Calculatio of Bayes posterior probability distributios for key populatio parameters." Caadia Joural of Fisheries ad Aquatic Scieces Vol 51 (3), pp , [6] Li, Jig. "A itegrated procedure for bayesia reliability iferece usig MCMC." Joural of Quality ad Reliability Egieerig 2014, [7] Ahmed, A. O. M., H. S. Al-Kutubi, ad N. A. Ibrahim. "Compariso of the Bayesia ad maximum likelihood estimatio for Weibull distributio." joural of mathematics ad statistics Vol 6 (2), pp , [8] Mohammed A, Al Omari, et al. "Bayesia survival ad hazard estimate for Weibull cesored time distributio." Joural of Applied Scieces Vol 12, pp , [9] Ahmed, Al O. "Compariso of the Bayesia Methods o Iterval-Cesored Data for Weibull Distributio.", [10] Zaidi, Abdelaziz, Belkacem Ould Bouamama, ad Mocef Tagia. "Bayesia reliability models of Weibull systems: State of the art." Iteratioal Joural of Applied Mathematics ad Computer Sciece Vol 22 (3), pp , [11] Hamada, Michael S., et al. Bayesia reliability. Spriger Sciece & Busiess Media,

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