DISCRIMINATION BETWEEN WEIBULL AND LOGNORMAL DISTRIBUTIONS FOR LIFETIME DATA

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1 Joural of Research (Scece), Bahaudd Zakarya Uversty, Multa, Paksta. Vol. 7, No., Aprl 006, pp ISSN 0-0 DISCRIMINATION BETWEEN WEIBULL AND LOGNORMAL DISTRIBUTIONS FOR LIFETIME DATA G. R. Pasha Departmet of Statstcs Bahaudd Zakarya Uversty, Multa emal: M. Shuab Kha Departmet of Statstcs Govt Zamdar College, Gujrat emal: Ahmed Hesham Pasha Departmet of Electrcal Egeerg, Bahaudd Zakarya Uversty, Multa emal: Abstract Relablty ad falure data, both from lfe testg ad from -servce records, are ofte modeled by the Webull or Logormal dstrbutos so as to be able to terpolate ad/or extrapolate results. The am of ths research s to dscrmate betwee the Webull ad Logormal dstrbutos for complete samples. Both relablty models Webull ad Logormal are the llustrated. Meda rak regresso (MRR) ad maxmum lkelhood estmato (MLE) data- fttg methods are descrbed ad goodess-of-ft usg maxmum lkelhood rato () ad most powerful varat (MPI) tests. We fd the Webull dstrbuto better for fttg to lfetme data whe comparg wth the Logormal dstrbuto. Keywords: Maxmum lkelhood rato test, meda rak regresso, most powerful varat test, Webull ad Logormal dstrbutos. INTRODUCTION A lfe tme dstrbuto model ca be ay probablty desty fucto f (t) defed over the rage of tme (0, ). The correspodg cumulatve dstrbuto F(t) s very useful fucto as t gves probablty that a radomly selected ut wll fal by tme t. Cohe [95], Dubey [966], Schltzer [966], Lock [973], Gross ad Lure [977], Ba [978], Gbbos ad Vace [98], Lawless [98] ad Aberethy [994] amog may others fd that there are a umber of methods for fttg lfe data pots to a dstrbuto. We are gog to dscrmate two most frequetly used lfe tme dstrbutos, Webull ad Logormal dstrbutos. Ths dscrmato s based o goodess of ft tests after estmatg the both lfetme models by two popular methods, amely, meda rak regresso (MRR) ad maxmum lkelhood estmator (MLE). The goodess-of-ft test ca test whether the complete lfe data 03 J. res. Sc., 006, 7(), 03-4

2 04 G. R. Pasha, M. Shuab Kha, Ahmed Hesham Pasha are from the uderlyg dstrbuto or ot. It s ofte foud that the lfe data, whch has bee plotted o the relevat probablty paper, ft both the Webull ad Logormal les very well. How do we objectvely judge whch model s the better choce? Aberethy [996] ad Fulto [995] developed a graphcal goodess-offt test ad the p-value model. Frstly, the lfe tme dstrbuto models: Webull dstrbuto ad Logormal dstrbuto have bee descrbed. Secodly, the data descrpto s gve. The estmato method MRR, ad MPI tests are also explaed ths secto. Thrdly, the data aalyss ad dscusso of the results s preseted whle last secto cocludes the research work. THE LIFETIME DISTRIBUTION MODELS I ths secto Webull ad Logormal dstrbutos have bee descrbed. WEIBULL DISTRIBUTION The Webull probablty dstrbuto has three parameters η, β ad. It ca be used to represet the falure probablty desty fucto (PDF) wth tme, so that: t t0 β β t t ( ) 0 β η fw () t = ( ) e η > 0, β > 0, t 0 > 0, < t0 < t () η η ; where β s the shape parameter (determg what the Webull PDF looks lke) ad s postve ad η s a scale parameter ad s also postve, s a locato or shft or threshold parameter (sometmes called a guaratee tme, falure-free tme or mmum lfe), t0 be ay real umber, If t = 0 0 the the Webull dstrbuto s sad to be two-parameter. LOGNORMAL DISTRIBUTION The probablty desty fucto (PDF) for 3-parameter logormal dstrbuto s: ρ t t0 l θ > 0, ρ > 0, t > 0, < t θ 0 < t, () ρ fl ( t ) = exp, π( t t ) 0 where ρ s the shape parameter,θ s the scale parameter ad s the locato parameter. The uts of ρ, θ ad are the same as the Webull case. The Logormal s sad to be a two-parameter dstrbuto whe = 0. The restrctos o the values of t, Eq. (). 0 θ, ρ for the Logormal dstrbuto are as stated

3 DISCRIMINATION BETWEEN WEIBULL AND LOGNORMAL DISTRIBUTIONS 05 METHODOLOGY We take the lfetme data of data 30 electrc tubes lghts used by Kale ad Sha [97]. The data are gve as Table A Appedx. The X values the data lst represet actual electrc tubes lfe data hours. The data was aalyzed by fttg Webull ad Logormal models. The Meda Rak Regresso (MRR) ad the Maxmum Lkelhood Estmato (MLE) were appled to estmate the ukow parameters. For the dscrmato betwee the Webull ad the Logormal, the ad MPI tests were used. MEDIAN RANK REGRESSION (MRR) FOR COMPLETE SAMPLES The lfetme data, whe plotted o probablty paper are approxmately lear, so the parameters ca be estmated by usual least squares (LS) method. Regressg Y o X mmzes the sum of squares of resdual varato the Y drecto, whereas regressg X o Y mmzes t the X drecto. Berkso [950] studed these two regressos ad suggested that the scale wth larger error should be regarded as the depedet varable. For lfe data aalyss, the tme-to falure, X always shows much more error tha the meda raks especally for -servce falure data. So, Aberethy [994] cocluded that the better method for probablty paper plot s to regress X o Y. He also showed that regressg X o Y has a better accuracy tha regressg Y o X. So our study cosders both regressos; X o Y ad Y o X. The Webull cumulatve dstrbuto fucto (CDF), deoted by F(t), s: ( t t0 β η F ( ) ) w t = e (3) The lear form of the resultg Webull CDF ca be represeted by a rearraged verso of Eq. (3): l t = ll + lη β () FW t Comparg ths equato wth the lear form y = Bx + A, leads to y = l t ad x = l l (). FW t By mmzg β ad η usg the LS method oe obtas: ad x x ˆ = = β =, (5) x y x y exp = = ˆ = = η = y x ˆ β, = (4) (6)

4 06 G. R. Pasha, M. Shuab Kha, Ahmed Hesham Pasha where s the sample sze ad ˆ dcates a estmate. The mathematcal expresso for x ad y are: ( ) x = l l ad y FW t = lt. 0.3 F ( t ) ca be estmated by usg Beard s formula,, whch s a good approxmato to the meda rak estmator [Tobas ad Trdade 986, Aberethy 994]. The Beard s meda rak was used because t showed the best F. The procedure performace ad s the most wdely used rak to estmate ( ) for rakg complete data s as follows:. Lst the tme to falure data from small to large.. Use Beard s formula to assg meda raks to each falure. 3. Estmate the β ad η by Eqs. (5) ad (6). F( t ) â ˆ the βˆ ad ηˆ ca easly be obtaed. The estmator of ρ s the sample correlato coeffcet, ρˆ gve by: The s estmated from the meda raks. Oce ad b are obtaed, ( x x )( y y ) = ˆρ = (7) ( x x ) ( y y ) = = The Logormal CDF s the tegral of the PDF from 0 to tme- to-falure t. It ca be wrtte terms of the stadard Normal CDF as: ρ t t 0 F L ( t ) = Φ l (8) θ The Logormal CDF, whe plotted agast approprate probablty axes, appears lear ad so ca be represeted by a rearraged verso of Eq. (8) as: Z p t (9) l = + l θ ρ Comparg ths wth the lear form y = B x + A, leads to y = ad x = Z p. The same LS procedure, as appled for the Webull dstrbuto, yelds. t lt ˆρ = = x = x y x = = = x y (0)

5 ad DISCRIMINATION BETWEEN WEIBULL AND LOGNORMAL DISTRIBUTIONS 07 ˆ = = θ = exp x = where z p, ad y y x ˆ ρ () = l t z p = φ ( z p ) s the percetle of the stadard t = z p ca be estmated usg Berard s formula. The same rakg procedure was used as appled for the Webull dstrbuto. The rak table for Webull s the same as that of Logormal as show Table A. Normal CDF, whch s wdely tabulated. Aga F( ) φ( ) THE MAXIMUM LIKELIHOOD RATIO () TEST For, the hypothess settg s as follows: H : uderlyg dstrbuto s the Webull dstrbuto. 0 H : uderlyg dstrbuto s the Logormal dstrbuto. Level of sgfcace s set at α = 0.0,0.05,0.0,0. 0 The test statstcs (TS) s: TS = ( πσˆ e ) = t f w ( t ) Dumoceaux et al. [973] also proposed the reverse hypothess, as follows: H 0 : Uderlyg dstrbuto s the Logormal dstrbuto. H : Uderlyg dstrbuto s the Webull dstrbuto. The, the test statstcs (TS) of becomes: TS = () π e t f w ( t ) (3) = σˆ The ull hypothess was rejected the favor of alteratve hypothess wheever TS TS (tabulated). THE MOST POWERFUL INVARIANT (MPI) TEST Ket [979] defed the most powerful varat (MPI) test statstcs for selecto betwee the Webull ad Logormal dstrbutos gve by: ˆ β ˆ β Γ( ) t ( ˆ σ π ) e TS = MPI = (4) t ˆ β = where s sample sze, βˆ ad σˆ ca be determed by the method of MLE.

6 08 G. R. Pasha, M. Shuab Kha, Ahmed Hesham Pasha If TS >, the the data could come from the Webull dstrbuto alterately, If TS <, the the data could come from the Logormal dstrbuto. So Eq. () ca be rearraged as: = ˆ β t e t ˆ β = > < ˆ β After takg logarthms of both sdes: ˆ β = lt + l = t ˆ β > < Γ( )( σˆ π ) (5) l( ) ( )l(ˆ βσ ˆ π ) l[ Γ( )] (6) Let the left-had sde of the expresso equate to A ad the rght-had sde of the expresso equate to B, to smplfy the equalty. Hece, f A > B, the ths dcates that the samples could come from the Webull dstrbuto. Coversely, f B>A, the t dcates that the samples could come from the Logormal dstrbuto. RESULTS AND DISCUSSION The rak table s show Table A. The Webull probablty plot s obtaed usg Webull Y-Bath TM as show Fg.. From Fg., estmates of ηˆ = ad βˆ =.0435 were obtaed. By plottg the data (x o y) o probablty paper the tred may appear curved, ether cocave dow or up. These curvatures may dcate that the org of the data s ot the same as the zero from whch the lfe data has bee measured. I such cases a locato parameter may be eeded to make the orgs cocde. Cocave dowward mples that a value > 0 s eeded to covert the data. Cocave upward mples that a value < 0 s eeded. I practce cocave dowward stuatos are more ofte see tha cocave upward [Aberethy 994]. The case of >0 dcates that falures caot occur utl after a certa perod of tme has elapsed. Durg ths perod the uts caot fal. Ths s why sometmes, called guaratee parameter. The case of < 0 dcates that some duty may have occurred. Fg. llustrates how to deal wth the 3-parameter Webull dstrbuto. Usg Webull Y-Bath TM ηˆ = , βˆ =.555 ad ˆ = ca be readly obtaed. Fg. shows the tube lght falures usg MRR. Note that the fal result of η must be adjusted for t0 to retur to the orgal lfe scale, so that ηˆ = = The result s ˆ = whe correlato coeffcet CC = reaches the maxmum usg a graphcal method. s,

7 DISCRIMINATION BETWEEN WEIBULL AND LOGNORMAL DISTRIBUTIONS 09 Fg. : -P Webull Aalyss for Tube Lght Data. Fg. : 3-P Webull Aalyss for Tube Lght Data.

8 0 G. R. Pasha, M. Shuab Kha, Ahmed Hesham Pasha Whe the data (y o x) was plotted o probablty paper the tred appeared curved. Here aga the tube lght data [Kale ad Sha 97] s take to llustrate how to deal wth the -parameter Webull dstrbuto. Usg Webull Y - Bath TM ηˆ = , βˆ =.0080 ad smlarly for 3-parameter Webull dstrbuto TM usg Webull Y-Bath ηˆ = , βˆ =.0, ˆ = -3.6 ca be readly obtaed. Results show the tube lght falures usg MRR. Note that the fal result of η must be adjusted for to retur to the orgal lfe scale, so that ηˆ = = The result s ˆ = -3.6 whe CC = reaches the maxmum usg a graphcal method. W SMTTH TM, as show Fg. 3, ths termology stads for at-log of the logvalue mea. The sdf s a multpler / dvsor ad represets the at-log of lfe log-value stadard devato [Fulto 995]. The estmated values obtaed were mual = = θˆ ad sdf = The value of sdf = 3.76 leads to a result for ρˆ = Fg. 3. -P Logormal Aalyss for Tube Lght Data. The 3-parameter Logormal aalyss for the tube lght data are smlar to the Webull case as show Fg. 4. The estmated values obtaed were mual = 56 = θˆ, ˆ = -73. ad sdf =.4. The value of sdf =.4 leads to a result for ρˆ = The fal result of θ must be adjusted for to retur to the orgal lfe scale. So θˆ = = 53.8.

9 DISCRIMINATION BETWEEN WEIBULL AND LOGNORMAL DISTRIBUTIONS Fg. 4: 3-P Logormal Aalyss for Tube Lght Data. The results of the MRR method are summarzed as preseted Table. Table : Summary of Results for Tube Lght Data Usg MRR Method ( = 30). Dstrbutos Webull Logormal Method Parameters -P 3-P Parameters -P 3-P η θ MRR β ρ t t CC CC GOODNESS-OF-FIT There are may methods of measurg goodess-of-ft such as Ch-square test ad test etc. Dumoceaux et al. [973], Lawless [98], O Coor [99] ad Dodso [994] used other methods such as MPI tests to evaluate the dfferece betwee the Webull ad Logormal dstrbutos. The ad MPI tests were cosdered for the preset study. Usg estmated values of ukow parameters Eq. (), the value of statstcs was foud to bets = Whe ths value was compared wth the crtcal values gve Table for dfferet α, the ull hypothess was accepted all the cases, whch led to the cocluso that the Webull s a statstcally sgfcatly better ft tha the Logormal. H 0

10 G. R. Pasha, M. Shuab Kha, Ahmed Hesham Pasha By cosderg reverse hypothess proposed by Dumoceaux et el. [973], oe has TS =.444. The same cocluso ca aga be draw for the better ft of Webull dstrbuto over that of Logormal. Fally for MPI test, A = > B = The results of the MPI tests were also foud to be cosstet wth tests. Therefore, It ca be deduced that the data could have come from the Webull dstrbuto. Table : Crtcal Values of Test for Dscrmatg betwee the Webull ad Logormal Dstrbutos. α = 0. α = 0. α = α = 0. 0 TSW TSL TSW TSL TSW TSL TSW TSL CONCLUSIONS The Webull dstrbuto ad Logormal dstrbutos are extesvely used relablty ad lfe testg. We have cocluded that whle comparg these two dstrbutos, Webull dstrbuto provdes better ft to lfetme data. We also ote that ad MPI tests are cosstet wth ths dscrmato betwee the Webull wth the Logormal dstrbuto. Refereces Aberathy, R.B. (994) The New Webull Hadbook, SAE Professoal Developmet. Aberathy, R.B. (996) The New Webull Hadbook, d ed., Dept. AT Housto, Texas, USA.. Aberathy, R.B. ad Fulto, W. (995) Webull NEWS. Fall. Ba, L..J. (978) Statstcal Aalyss of Relablty ad Lfe-Testg Models: Theory ad Methods, New York, USA.. Berkso, J. (950) Two Regressos, J. Amer. Stat. Assoc., 45, Cohe, A.C. (95) Estmatg Parameters of Logarthmc-Normal Dstrbutos by Maxmum Lkelhood, J. Amer. Stat. Assoc., 46, 06-. Dodso, B. (994) Webull Aalyss, ASQC Qualty Press, Mlwaukee, Wscos, USA.. Dubey, S.D. (966) Comparatve Performace of Several Estmators of the Webull Parameters, Proceedgs of the 0 th Techcal Coferece of the Amerca Socety for Qualty Cotrol, Dumoceaux, R., Atle, C.E. ad Haas, A. (973) Lkelhood Rato Test for Dscrmato betwee Two Models wth Ukow Locato ad Sale Parameters, Techometrcs, 5, 9-7.

11 DISCRIMINATION BETWEEN WEIBULL AND LOGNORMAL DISTRIBUTIONS 3 Fulto, W. (995) Fulto Fdgs, Webull SMITH, Mote Carlo SMITH & Vsual SMITH software, 5 W. Sepulveda Blvd. #800, Torrace, CA 9050, USA. Gbbos, D.I. sd Vace, L.C. (98) A Smulato Study of Estmators for the - Parameter Webull Dstrbuto, IEEE Trasactos o Relablty, R-30(), Gross, A.J. ad Lure, D. (977) Mote Carlo Comparsos of Parameter Estmators of the -parameter Webull Dstrbuto, IEEE Trasactos o relablty, R-6(.5), Kale, B.K. ad Sha, S.K. (97) Estmato of Expected Lfe the Electrc Compoets, Techometrcs, 4, Ket, J.E.A. (979) A Effcet Procedure for Selectg amog Fve Relablty Models, North Carola State Uversty, Ralegh, USA. Lawless, J.F. (98) Statstcal Models ad Methods of Lfetme Data, Joh Wley, New York, USA. Lock, M.O. (973) Relablty, Mataablty, ad Avalablty Assessmet, Hayde Book Co. New Jersey, USA. O Coor, P.D.T. (99) Practcal Relablty Egeerg, 3 rd ed., Joh Wley, New York, USA. Schltzer, L.D. (966) Mote Carlo Evaluato ad Developmet of Webull Aalyss Techques for Small Sample Szes, May -5. Tobas, P.A. ad Trdade, D.C. (986) Appled Relablty, Va Nostrad Rehold Co. New York, USA.

12 4 G. R. Pasha, M. Shuab Kha, Ahmed Hesham Pasha Appedx Table A: Webull Aalyss for Electrc Tubes Lght data. Rak Table Set Pot x-value Quatty Beard s Rak E E E