Bootstrap Intervals of the Parameters of Lognormal Distribution Using Power Rule Model and Accelerated Life Tests
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1 Joural of Moder Applied Statistical Methods Volume 5 Issue Article --5 Bootstrap Itervals of the Parameters of Logormal Distributio Usig Power Rule Model ad Accelerated Life Tests Mohammed Al-Ha Ebrahem Yarmou Uiversity, Irbid, Jorda, m_hassab@hotmail.com Follow this ad additioal wors at: Part of the Applied Statistics Commos, Social ad Behavioral Scieces Commos, ad the Statistical Theory Commos Recommeded Citatio Ebrahem, Mohammed Al-Ha (5) "Bootstrap Itervals of the Parameters of Logormal Distributio Usig Power Rule Model ad Accelerated Life Tests," Joural of Moder Applied Statistical Methods: Vol. 5 : Iss., Article. DOI:.37/masm/63546 Available at: This Regular Article is brought to you for free ad ope access by the Ope Access Jourals at DigitalCommos@WayeState. It has bee accepted for iclusio i Joural of Moder Applied Statistical Methods by a authorized editor of DigitalCommos@WayeState.
2 Joural of Moder Applied Statistical Methods Copyright 6 JMASM, Ic. November, 6, Vol. 5, No., /5/$95. Bootstrap Itervals of the Parameters of Logormal Distributio Usig Power Rule Model ad Accelerated Life Tests Mohammed Al-Ha Ebrahem Departmet of Statistics Yarmou Uiversity Assumed that the distributio of the lifetime of ay uit follows a logormal distributio with parameters μ adσ. Also, assume that the relatioship betwee μ ad the stress level V is give by the power rule model. Several types of bootstrap itervals of the parameters were studied ad their performace was studied usig simulatios ad compared i term of attaimet of the omial cofidece level, symmetry of lower ad upper error rates ad the expected width. Coclusios ad recommedatios are give. Key words: Power rule model, logormal distributio, bootstrap itervals, accelerated life test. Itroductio The logormal distributio has may special features that allowed it to be used as a model i various real life applicatios. I particular, it is used i aalyzig biological data (Koch, 966), ad for aalyzig data i worplace exposure to cotamiats (Lyles & Kupper, 997). It is also of importace i modelig lifetimes of products ad idividuals (Lawless, 98). Various other motivatios ad applicatios of the logormal distributio may also be foud (see Johso et. al., 994, Scheider, 986). I a life testig experimet, the problem is that most uits have a very log life uder the ormal coditios. Therefore, by the time the experimet is completed ad a estimate of the reliability is obtaied, the results will be outdated. To overcome this delay, accelerated life testig was itroduced (Ma. et. al., 974). I a accelerated life testig experimet a certai umber of uits are subected to a stress that is higher tha the ormal stress. The Mohammed Al-Ha Ebrahem is Assistat Professor i the Departmet of Statistics at Yarmou Uiversity, Irbid-Jorda. His research iterest is i reliability, accelerated life test ad o-parametric regressio models. m_hassab@hotmail.com. experimet is repeated uder differet values of stress. I order to do so, some relatioship betwee the parameters of the time to failure distributio of the uit ad the correspodig stress level must be postulated. It is assumed that desity fuctio of the time to failure of a uit depeds o oe parameter sayθ, ad the eviromet depeds o oe stress V ad that the relatioship betwee C θ ad V is give by θ = where C ad P P V are positive costats. This relatioship is ow as the power rule model. Cosider the iterval estimatio for the parameters of the logormal distributio after reparametrizig the locatio parameter μ as a fuctio of the stress V usig power rule model. The performace of the bootstrap ad Jacife itervals (Efro & Tibshirai, 993) i term of attaimet of the omial cofidece level, symmetry of lower ad upper error rates ad the expected width of the itervals will be compared. The Model ad The Maximum lielihood Estimatio It is assumed that the lifetime (T) of ay uit follows a logormal distributio with locatio parameter μ ad scale parameter σ. The probability desity fuctio of T is give by (Lawless, 98): 38
3 38 PARAMETERS OF LOGNORMAL DISTRIBUTION f () t = exp tσ π ( lt μ) σ, < t <. () The locatio parameter μ was reparameterized as a fuctio of the stress V usig the power rule C model μ =, therefore c ad σ are the ew V P parameters of the model. The uow parameters c ad σ were estimated usig complete samples. The -th sample is obtaied by usig uits ad the value V for the stress, =,,.,. The lielihood fuctio of the complete samples is give by: ad σˆ Cˆ = i= = = l t i / v = / v p p cˆ l ti p = i= v = (4) (5) L( μ, σ ) = e = σ σ = i= (π ) (l t = i μ ) Π Π t = i= i () C Usig the power rule model μ = =,, P V.,, the lielihood fuctio is give by: L( C, σ ) = e = σ σ = i= = (π ) c l ti p v o Π Πt = i= i (3) It is easy to show that the Maximum lielihood estimators of C ad σ are give by: It is obvious that Ĉ is a ubiased estimator of C while σˆ is a biased estimator of σ. The Percetile Iterval The methods of derivig cofidece itervals preseted i this sectio ad sectio 4 are based o the parametric bootstrap approach (Efro & Tibshirai, 993); they are costructed by resamplig from the estimated parametric distributio. To costruct the percetile iterval, a simulatio of the bootstrap distributio of Ĉ ad σˆ is doe by resamplig from the parametric model of the origial data. That is, a B bootstrap sample is geerated ad for each sample Ĉ ad σˆ are calculated usig equatio (4) ad (5) respectively. The calculated values * * are deoted by Ĉ ad ˆ σ. Let Ĝ deotes the cumulative * distributio of Ĉ, the ( α ) % percetile iterval of C is ˆ α ˆ α G, G, similarly let Ĝ deotes the cumulative * distributio of ˆ σ, the ( α ) % percetile iterval of σ is ˆ α ˆ α G, G.
4 MOHAMMED AL-HAJ EBARAHEM 383 The Bias Corrected ad Accelerated Iterval (BCa Iterval) The bias corrected ad accelerated iterval is costructed by calculatig two umbers â ad ẑ called the accelerated ad the bias correctio factor respectively, they are calculated usig the followig formulas ( Cˆ(.) Cˆ( i) ) i= a ˆ = (6) 3 / 6 ( Cˆ(.) Cˆ( i) ) i= where C ˆ( i ) is the maximum lielihood estimator of C usig the origial data excludig the i-th Cˆ( i) i= observatio ad C ˆ (.) =, = = The value of ẑ is give by 3 Cˆ * #( < Cˆ) zˆ = Φ (7) B where Φ(.) is the stadard ormal cumulative distributio fuctio. The ( α ) % BCa ˆ ˆ G α, G α where ( ) iterval of C is ( ) ( ) ad α = Φ z ˆ zˆ + zα / + aˆ( zˆ + z α / ) zˆ + z α / α = Φ + z ˆ (8) aˆ( zˆ + z α / ) where z α is the α quatile of the stadard ormal distributio. I the same way, the ( α ) % BCa iterval of σ ca be costructed. Jacife Iterval A ( α ) % Jacife iterval of C ( Efro ad Tibshirai, 993) is costructed as follows:. Cˆ(.) ± Z ˆ, where (α / ) S Jac ( Cˆ(.) Cˆ( i) ) Sˆ Jac =, Ĉ (.), C ˆ( i ) i= ad were defied i sectio 4. Similarly, the ( α ) % Jacife iterval of σ by replacig C by σ i the above iterval. Simulatio Study A simulatio study is coducted to ivestigate the performace of the itervals discussed i sectios 3, 4 ad 5 above. The idices of the simulatio study are: : The umber of logormal populatios, i this study =. : Sample size from the first logormal populatio, i this study = 5,, 3. : Sample size from the secod logormal populatio, i this study = 5,, 3. C : Parameter of the power rule model, i this study C = 3. P : I this study P =.3. V : The value of stress for the first logormal populatio, i this study V =. V : The value of stress for the secod logormal populatio, i this study V =. σ : I this study σ =. B: The umber of bootstrap samples, i this study B =. For each combiatio of ad samples are geerated ad a ( α ) % Percetile iterval is costructed, BCa iterval ad Jacife iterval for C ad σ. Two values are cosidered for α,.5 ad.. The followig were obtaied for each iterval: - The expected width (IW): the average of widths of the itervals. - Lower error rate (LER): the fractio of itervals that fall etirely above the true parameter.
5 384 PARAMETERS OF LOGNORMAL DISTRIBUTION 3- Upper error rate (UER): the fractio of itervals that fall etirely below the true parameter. 4- Total error rate (TER): the fractio of itervals that did ot cotai the true parameter value. Results ad Coclusios The results are give i tables. Table has simulatio results of the percetile iterval of the parameter C usig α =. 5. Table has simulatio results of the BCa iterval of the parameter C usig α =. 5. Table 3 has simulatio results of the Jacife iterval of the parameter C usig α =. 5. Table 4 has simulatio results of the percetile iterval of the parameter C usig α =.. Table 5 has simulatio results of the BCa iterval of the parameter C usigα =.. Table 6 has simulatio results of the Jacife iterval of the parameter C usig α =.. Table 7 has simulatio results of the percetile iterval of the parameter σ usig α =. 5. Table 8 has simulatio results of the BCa iterval of the parameter σ usig α =.5. Table 9 has simulatio results of the Jacife iterval of the parameter σ usig α =.5. Table has simulatio results of the percetile iterval of the parameter σ usig α =.. Table has simulatio results of the BCa iterval of the parameter σ usig α =.. Table has simulatio results of the Jacife iterval of the parameter σ usig α =.. From these results the followig ca be cocluded: For the parameter C, the three itervals have almost the same expected width, ad the expected width decreases as the sample sizes icreases. I term of attaimet of coverage probability ad symmetry of lower ad upper rates, the three itervals behave i the same way. It is recommeded that the Jacife iterval be used because its calculatio is simpler tha the BCa ad the percetile itervals. For the parameter σ, the expected width for the percetile iterval is early smaller tha the other two itervals. O the other had, i term of attaimet of coverage probability ad symmetry of lower ad upper rates, the BCa iterval behaves the best. It is therefore recommeded that the BCa iterval be used i this case.
6 MOHAMMED AL-HAJ EBARAHEM 385 Table. Percetile Iterval of the parameter C with α =. 5 IW LER UER TER Table. BCa Iterval of the parameter C with α =. 5 IW LER UER TER Table 3. Jacife Iterval of the parameter C with α =. 5 IW LER UER TER
7 386 PARAMETERS OF LOGNORMAL DISTRIBUTION Table 4. Percetile Iterval of the parameter C with α =. IW LER UER TER Table 5. BCa Iterval of the parameter C with α =. IW LER UER TER Table 6. Jacife Iterval of the parameter C with α =. IW LER UER TER
8 MOHAMMED AL-HAJ EBARAHEM 387 Table 7. Percetile Iterval of the parameter σ with α =. 5 IW LER UER TER Table 8. BCa Iterval of the parameter σ with α =. 5 IW LER UER TER Table 9. Jacife Iterval of the parameter σ with α =. 5 IW LER UER TER
9 388 PARAMETERS OF LOGNORMAL DISTRIBUTION Table. Percetile Iterval of the parameter σ with α =. IW LER UER TER Table. BCa Iterval of the parameter σ with α =. IW LER UER TER Table. Jacife Iterval of the parameter σ with α =. IW LER UER TER
10 MOHAMMED AL-HAJ EBARAHEM 389 Refereces Efro, B. & Tibshirai, R. (993). A itroductio to the bootstrap. New Yor: Chapma ad Hall. Johso, N. L, Kotz, S., & Balarisha. (994). Cotiuous uivariate distributios: vol. New Yor: Wiley Koch, A.L. (966). The logarithm i biology. Joural of Theoretical Biology,, Lawless, J.F. (98). Statistical models ad methods for lifetime data. New Yor: Wiley. Lyles, R. H., Kupper, L. L., & Rappaport, S. M. (997). Assessig regulatory compliace of occupatioal exposures via the balaced oe-way radom effects ANOVA model. Joural of Agricultural, Biological, ad Evirometal Statististics,, Ma, N. R., Schafer, R. E., & Sigpurwalla, N. D. (974). Methods for statistical aalysis of reliability ad life data. New Yor: Joh Wiley ad Sos Ic. Scheider, H. (986). Trucated ad cesored samples from ormal populatios. New Yor: Marcel Deer. Efro, B. & Tibshirai, R. (993). A itroductio to the bootstrap. New Yor, N.Y.: Chapma ad Hall. Jeigs, D. (987). How do we udge cofidece itervals adequacy? The America Statisticia, 4(4), Kulldorff, G. (96). Estimatio from grouped ad partially grouped samples. New Yor, N.Y.: Wiley. Meeer, Jr, W. (986). Plaig Life tests i which uits are ispected for failure. IEEE Tras. o Reliability R-35, Pettitt, A. N., & Stephes, M. A. (977). The Kolmogrov-Smirov goodess-of-fit statistic with discrete ad grouped data. Techometrics, 9, 5.
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