A DATA DRIVEN PARAMETER ESTIMATION FOR THE THREE- PARAMETER WEIBULL POPULATION FROM CENSORED SAMPLES

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1 Mathematcal ad Computatoal Applcatos, Vol. 3, No., pp Assocato fo Scetfc Reseach A DATA DRIVEN PARAMETER ESTIMATION FOR THE THREE- PARAMETER WEIBULL POPULATION FROM CENSORED SAMPLES Ahmed M. M. Sulta Egypt A Foce, Cao, Egypt amsulta_@yahoo.com Abstact- A method s descbed fo the calculato of the thee-paamete Webull dstbuto fucto fom cesoed samples. The method toduces a data dve techque based o a adapted Gaussa lke keel to match the cesog scheme. The method mmzes the Came vo Mses dstace fom a o-paametc desty estmate ad the paametc estmate at the ode statstcs. The mamum lkelhood estmatos ae foud ad a compaso s made wth the ew estmato. A Mote Calo epemet of sze 000 s coducted to test the pefomace of the ew paamete estmato techque. The mea tegated squae eo s take as a measue of the closeess of the estmated desty ad the tue desty. Key Wods- No-paametc desty, Webull cesoed samples, Gaussa keel, type II cesog, hybd methods, Came vo Mses statstc. INTRODUCTION The method of momet, the method of mamum lkelhood, ad othe methods have cosdeed the estmato of the paametes of Webull populato based o a cesoed sample. I ths pape, a appoach usg a adapted o-paametc desty estmato s toduced as a methodology fo the paamete estmato. Secto dscusses the soluto of the log lkelhood equatos fo the cesoed sample. The method of soluto s a modfcato of the classcal Newto-Raphso teatve scheme. The method s based o the umecal soluto of the log lkelhood equato usg a quas Newto method ad a actve set stategy to mamze the log lkelhood fucto subject to smple bouds o the dstbuto paametes. The method s suveyed ad a stoppg ule s stated. I secto 3 the applcato of a o-paametc desty estmato to obta estmates of the paametes of the thee-paamete Webull dstbuto fom a cesoed sample s dscussed. A adapted keel s used whch s a Gaussa lke keel wth a fte ght tal. A Mote Calo compaso of the mamum lkelhood estmatos ad the mmum dstace estmatos s gve usg the tegated squaed eo (ISE) betwee the tue desty ad the estmated tue model. Samples of sze 0, 0, ad 30 cesoed at the 7th, th, ad 0th ode statstc espectvely ae used. The epemet s doe fo thousad Mote Calo epettos. A compaso s made betwee the mamum lkelhood estmatos ad the ew estmatos fo locato paamete 0, wth scale paamete ad 0 ad fo shape paamete 3, 4,, ad 6 tables ad fgues. The esults ae show secto 4. These esults dcate a mpovemet of the ew method ove the classcal mamum lkelhood method.

2 30 A. M. M. Sulta. MAXIMUM LIKELIHOOD PARAMETER ESTIMATION The mamum lkelhood estmato fo the paametes of the Webull dstbuto has bee studed etesvely fo complete ad cesoed samples. The studes clude those by Hate ad Mooe (96, 967) whee they studed the mamum lkelhood estmato of the gamma ad Webull populato, fom cesoed samples. They also studed the asymptotc vaaces ad covaaces of mamum lkelhood estmatos fom cesoed samples fom Webull ad gamma populatos. Cohe (96) studed the mamum lkelhood estmato the Webull dstbuto based o complete ad cesoed samples. He also studed (97) the mult-cesoed samplg case of the thee-paamete Webull dstbuto. Some esults o complete ad cesoed samplg fom the thee paamete Webull dstbuto wee show by Wycoff et al.(980). Cohe et. al. (984) toduced modfed estmatos fo the paametes of the thee-paamete Webull wth smalle bases ad smalle vaaces. The pobablty desty fucto of the thee-paamete Webull deoted by W(,, )wth locato, scale, ad shape s gve by: f ( ;,, ) ep whee < <, > 0, > 0. The coespodg cumulatve dstbuto fucto s gve by: F( ;,, ) ep th Now, cosde that a sample of sze has bee cesoed at the ode statstc usg a type II cesog mechasm. The esultg desty fo the fst ode statstcs wll be gve as:! f(,..., ) f( ) [ F( )] ( )!! ( ) ep ep! Takg the logathm fo the above desty gves the followg log-lkelhood fucto: [ ] ( ) *! L log log log log! The patal devatves fo the log-lkelhood fucto wth espect to the thee ukow paametes (,, ) ae:

3 A Data Dve Paamete Estmato 3 L L L log log log * * * Equatg the patal devatves to zeo ad solvg the system of olea equatos smultaeously gves a estmato ˆ, ˆ, ˆ, ˆ Θ l that mamzes the loglkelhood fucto ad mamzes the lkelhood fucto as well. The system of the 3-o lea equatos fo the mamum lkelhood,, Θ s solved usg a umecal techque. The method s kow as the hybd method. Ths method s bascally a teatve method based o Newto-Raphso method. Such methods eed to compute 3 compoets of L ad 9 etes of L. Seveal othe modfcatos ae toduced by Powell (970) to eleve such a poblem by computg the dffeece appomatos stead of the dect computato of L. I Powell s teatve scheme the devatve s ot just scaled by a small facto but by toducg a egatve multple of the gadet of Θ L such that the decto fo the coecto the dffeet teatos wll be sesble whe the Jacoba becomes almost sgula. Fo detals about cases whe method ca, ad dffeet factos that affect the ug tme of the method, see Powell (970). A accuacy of.0 was used fo the absolute dffeece betwee two successve Θ s whle the Eucldea om accuacy was elaed sce the mea tegated squae eo (MISE) ctea s to be used latte fo the compaso ad the teest was the covegece of the Θ paamete maly. The algothm dd ot covege a few cases (umbe bold tables -3 the fst colum) whch wee ecluded fom the Mote Calo esults. Ths happeed because the method was seachg fo a zeo of the system of olea equatos Θ L 0 by mmzg the quadatc fom Θ Θ L L T o the sum of squaes of the

4 3 A. M. M. Sulta mamum lkelhood equatos. I ths case, the mmum would ot gve a zeo of the system. The same tal guess s chose fo all the dffeet Mote Calo samples of sze 000. The esults fom the pevous fo sample szes 0, 0, ad 30 cesoed at the 7th, th, ad 0th espectvely ae show. The paametes used fo the Mote Calo epemetato ae 3, 4,, ad 6 fo the shape paamete, ad 0 fo the scale paamete, ad 0 fo the locato paamete. 3. MINIMUM DISTANCE ESTIMATION I ths secto of the pape, we fd estmatos of the paametes,,. These estmatos ae the mmum dstace estmatos that mmze a goodess of ft statstc. Ths goodess of ft statstc s take as the Came vo Mses statstc W whch measues the tegal of the squaed dffeece betwee the desty ad the sample empcal dstbuto fucto. Ths W s defed as: W [ Fo F ] dfo ˆ whee F ˆ ( ) s the sample empcal dstbuto fucto ad F 0 ( ) s a completely specfed dstbuto fucto. The coespodg computatoal fom s: 0. W F0 ( ( ) ) 0. Ths computatoal fom uses the step fucto as a estmato fo F ˆ ( ). The basc oto ths secto of the pape s to mplemet the cocept of opaametc desty estmato to eplace the step fucto epesetg the sample empcal dstbuto fucto. Of couse to do that t s eeded to defe a keel ad the paamete to be used wth that keel. I ou case a adapted Gaussa keel togethe wth a heustc o empcal choce fo the wdow wdth ae toduced. Fst, the defto of the ew adapted Gaussa keel was dve by how to beeft fom the fact that the sample s ght cesoed sample. Also, the defto takes cae of that the sample s a odeed sample. Ths adapted keel takes the fom: e < < τ K( ) πφ( τ) 0 τ whee τ detemes a theshold fom the ght that gves a zeo weght to the -values beyod that τ ad φ(τ ) s the C.D.F of the stadad omal dstbuto. Ths τ value wll be used to compesate the odeg of the sample whch case t wll cosde the fomato that X X fo all wth as the ght cesog lmt. Ths ca smply be show fom the way the keel o the bump s placed ove each obsevato. Ths keel s placed ove each obsevato such that a zeo weght ( mass ) s gve fo obsevato X at ad beyod X fo all obsevatos othe tha X.

5 A Data Dve Paamete Estmato 33 Whle fo X, the theshold s abtay chose to be at multples of X ( take at multples of X ou case ). Thus, the keel at ode statstc X wll be : fo -, ad K h π φ 0 ( X ) ( ) e X h < < X X X( R) h K < < e X h π φ( X ) 0 X fo. Secod, the optmal value of the wdow wdth h ( the MISE sese) depeds o the choce of the keel K, the udelyg ukow desty f() ad the sample sze.e. f K. f f f ( ) h opt. 3 wth eplct epesso fo h opt gve as: / / / / h m { K ( t) dt} { f d} opt whee m deotes the keel secod momet. A easoable appomato fo ths optmal value fo bascally a omal sample was suggested to be h k whee k s a eal costat. Although ths appomato smplfes the optmal epesso fo the wdow wdth ad woks fe wth the omal dstbuto t s ot as good fo othe dstbutos. A alteatve fo computg the wdow wdth that s moe effcet computatoally ad gves a good mpovemet ths applcato s to choose a empcal h whch equals c s whee s epesets the data dve paamete fom the cesoed sample ad epesets the cesoed sample sze. Ths s s equal to g g Γ g Γ.The ( g, g ) ae tal guess fo both the scale ad shape g g paametes of the Webull desty. These ae chose as scaled sample stadad devato of the cesoed sample wth scale 4.0 ad 3.0 fo both values espectvely. Suggested h togethe wth the adapted keel showed a mpovemet MISE besdes beg smple, wthout a eed fo etesve computatos. The followg fgue (Fg..) shows a eample fo the use of ths ew o-paametc desty wth the toduced keel ad the chose wdow wdth. ( ) ( )

6 34 A. M. M. Sulta The sample used the eample s of sze 0 cesoed at the th odeed obsevato ad s fom a Webull dstbuto wth locato paamete, scale paamete, ad shape paamete 3. The odeed statstcs ae , , , , , 8.377, , 8.499, , , , , , , ad The data dve wdow wdth ths case s h METHODOLGY Both esults fom MLE computatos ad the ew techque ae show the followg tables (Table, Table, ad Table 3). Table. Results fom M.C sze 000 fo type II Rght Cesoed sample of sze 7 out of 0 Webull (loc., sca., sha.) MISE CvM MISE MLE W(0,,3) 3 W(0,,4) 8 W(0,,) 6 W(0,,6) w(0,0,3) W(0,0,4) W(0,0,) W(0,0,6) ( ) ( ) ( ) ( ) ( ) (0.0873) ( ) ( ) (0.3) ( ) ( ) ( ) ( ) ( ) ( ) ( )

7 A Data Dve Paamete Estmato 3 Table. Results fom M.C sze 000 fo type II Rght Cesoed sample of sze out of 0 Webull (loc., sca., sha.) MISE CvM MISE MLE W(0,,3) ( ) W(0,,4) ( ) W(0,,) ( ) W(0,,6) ( ) w(0,0,3) ( ) W(0,0,4) ( ) W(0,0,) ( ) W(0,0,6) ( ) ( ) ( ) ( ) ( ) ( ) (0.067) ( ) ( ) Table 3. Results fom M.C sze 000 fo type II Rght Cesoed sample of sze 0 out of 30 Webull (loc., sca., sha.) MISE CvM MISE MLE W(0,,3) ( ) W(0,,4) ( ) W(0,,) ( ) W(0,,6) ( ) w(0,0,3) ( ) W(0,0,4) ( ) W(0,0,) ( ) W(0,0,6) ( ) (0.088) (0.0330) ( ) ( ) ( ) ( ) ( ) ( ) The tables show the esultg MISE togethe wth ts stadad devato betwee backets fo samples of sze 0, 0, ad 30 cesoed at the 7th, th, ad 0th ode statstc fo dffeet paamete values fo both the ew poposed estmato cocuetly wth the modfed olea method fo solvg the mamum lkelhood equatos. I addto, the tables show that the ew techque has a sgfcat mpovemet ove the MLE method fo shape paametes 3, 4,, ad 6. A quck look at the esults fom table, fo eample, wthout ovegeealzg coclusos depcts a bette MISE ad smalle stadad devato fo the poposed method.

8 36 A. M. M. Sulta Thus, the ew techque shows a sgfcat mpovemet ove the MLE method. The mpovemet MISE ages fom close but yet smalle value of MISE to almost 3. tmes smalle case of locato 0, scale 0, ad shape 6. The vaatos h togethe wth the coespodg vaatos MISE dcate that the method s a adaptve oe the sese that the choce of the paamete h that s data depedet vaes wth the vaato of the dstbuto paametes ad the sample sze. The fal cocluso s that the pevously descbed method s ecommeded fo use as a alteatve to the MLE method fo estmatg the paametes of the Webull dstbuto based o ght cesoed samples fo up to sample szes 30. REFERENCES. Cohe, A. C. ad Whtte, B. J., Estmato the thee paamete Webull dstbuto Techometcs, 7, 347-3, Cohe, A. C. ad Dg, Y., Modfed momet estmato fo the thee paamete Webull dstbuto J. Qual. Tech., 6, 9-67, Cohe, A. C., Mamum lkelhood estmato the Webull dstbuto based o complete ad cesoed samples. Techometcs, 7, 79-88, Cohe, A. C., Multcesoed samplg the thee -paamete Webull dstbuto. Techometcs, 7, 347-3, 97.. Hate, H. L. ad Mooe, A. H., Mamum lkelhood estmato of the paametes of gamma ad Webull populato fom complete ad cesoed samples. Techometcs, 7, , Hate, H. L. ad Mooe, A. H., Asymptotc vaaces ad covaaces of mamum lkelhood estmatos fom cesoed samples of the paametes of Webull ad gamma dstbuto. A. Math. Stat., 38, 7-70, Otega, J. M.; Rheboldt, W. C., 970. Iteatve Soluto of Nolea equatos Seveal Vaables. Academc Pess. New Yok ad Lodo. 8. Powell, M. J. D., 970. "A hybd method fo olea equatos," Numecal methods fo No lea Algebac equatos, P. Rabwtz edto. 8. Wycoff, J., Ba, L., ad Eglehadt, M., Some complete cesoed samplg esults fo the thee paamete Webull dstbuto J. Statst. Comp. Smul., II, 39-, 980.