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1 Iteratoal Joural of Advacemets Research & Techology, Volume 3, Issue 10, October Usg webull dstrbuto the forecastg by applyg o real data of the umber of traffc accdets sulama durg the perod ( ) Samra Muhammad Salh Uversty of Sulama - College of Admstrato ad Ecoomc / Statstc Departmet. Sulama / Iraq Samra19690@yahoo.com Abstract: I ths study we tred the applcato of oe cotuous probablty dstrbutos,especally Webull probablty dstrbuto, whch s used the study of the relablty ad qualty cotrol the forecastg.real data of the umber of traffc accdets for the cty of Sulama cty durg ( ) has bee appled: 1) Covert(trasferece) Webull dstrbuto to lear regresso to estmate the parameters. ) How to use the geeralzed Webull dstrbuto fdg probablty of the occurrece of traffc accdets. Keywords: Webull dstrbuto, relablty, regresso, ordary least square ad lear estmato techques. 1. Itroducto: The problem of estmatg the ukow parameters of the probablty dstrbutos s mportat ssues that have receved ad terest to researchers ad others that terested mathematcal statstcs, by gvg the developmet of estmato methods ad cotrast, whch recall that for accuracy estmato we ca fd the best estmator for these parameters. I ths paper we focused o the Essece of the dstrbuto of Webull, I order to estmate the parameter of ths dstrbuto.i 1990 (AL-Badha) reached to a alteratve formula for the dstrbuto of Webull amed "geeralzed Webull dstrbuto ", to avod the problems that facg researchers estmatg the parameters of the dstrbuto, especally whe hs parameters exceeds the value of (3.6), so suggest a ew formula several ways, t was the most mportat method of least squares. Research Problem: I ths study we focus o the basc research to the possblty of usg Webull probablty dstrbuto to forecast ad determe the modaltes ad methods used to estmate ukows parameters.the am of the research o the theoretcal kowledge ad appled to the estmato methods for the parameters of the Webull dstrbuto of Webull ad hs methods ad focus o the possblty to use t forecastg. That appled o real data of the umber of traffc accdets Sulama durg ( ).. Methodology ad research hypotheses: ths research, we reled o the descrptve aalytcal approach the characterzato of the dstrbuto of Webull, ad how to estmate ts features. The data of traffc accdets to the cty of Sulama derved from the Drectorate of Traffc Sulama durg the exteded perod of tme betwee ( ).I ths research, we focused o a basc premse of the approprateess of the data dstrbuto ad geeralzed Webull Copyrght 014 ScResPub.

2 weather, geeral survval. the Webull T Tt T relato the Ta T practcal wd T Iteratoal Joural of Advacemets Research & Techology, Volume 3, Issue 10, October dstrbuto wth two parameters. Also how smlar the real values through probablstc dstrbuto ad Webull. 3. The theoretcal sde: 3.1Essece Webull Dstrbuto: Here we wll dscuss the theoretcal framework for the dstrbuto ad ad stuatos through the relablty ad hazard fucto ad geeral lear methods to estmate parameters of ths dstrbuto. Suppose that (X) have cotuous radom varable wth Webull three parameters dstrbuto whch are ( θλκ,, ), (k) represeted as shape parameter, ( λ )scale parameter, ad (θ )Locato parameter, ad as log as: ( λ ) ( k ) 0, 0,() θ real value, ad Tgves 0, for X 0 f x for X e λ λ Tdesty k ( ) K 1 x θ k X θ, 0 (1) λ Tcotued Tas follows T: Let κ α λ, ( ) β κ ad X θ X λ ca we wrte the equato 1 as follow : he T 0, for X 0 f ( x) () β β 1 ( X ) for X 0 α ( X ) e Tprobablty Tfucto Ts gve 0, for X 0 Tcumulatve F( x) β (3) ( X ) for X 0 1 e Ad the adopto of desty dstrbuto fucto we get geeralzed Webull dstrbuto ad whe (θ 0 ), we get the Webull dstrbuto fucto wth two parameters ad, desty fucto, are: 0, for X 0 f x X e, for X 0 λ λ κ ( ) κ 1 x κ (4) λ Because the Webull dstrbuto to applcatos, t s used today several felds, cludg: 1. he aalyss of. Idustral relablty aalyss. 3. heoretcal maxmum value. 4. The ad the descrpto of dstrbuto of speed. 5. Commucato Systems Egeerg. 6. he feld of surace. Copyrght 014 ScResPub.

3 two probablty (T), the ad ) Iteratoal Joural of Advacemets Research & Techology, Volume 3, Issue 10, October Webull dstrbuto. Ths s oe of the extreme dstrbutos fucto, ad t s gve as uque for the smallest value for a large umber of radom varables depedet (X), It s also a cotuous dstrbutos fucto called the dstrbuto test, such as the dstrbuto of ( χ ) ad dstrbuto (t) ad dstrbuto (F), due to the mportace mathematcal statstc methods. Those depedg o the probablty dstrbuto of the extreme values, we ca express the geeralzed dstrbuto ad (Geeralzed Webull Dstrbuto), the fucto of the dstrbuto of a radom varable (X). As follows: 1 κ κ GX (, θακ,, ) px ( x) exp 1 ( x α), κ 0 (5) θ x α G( X, θα, ) exp exp, κ 0 (6) θ Where ; (κ ) shape parameter, (θ ) Scale parameter ad ( β ) Locato parameter. As the relatoshp dstrbuto, are: ( T X ) betwee the Webull dstrbuto ad geeralzed that (T) represets extreme value ote radom varable dstrbuto ad : Webull geeralzed, ad we ca wrte : pt ( t) p( X x) 1 G( x; θακ,, ) Where :(t) the value of a radom varable θ ad be specfc from the β, ad order to get the Webull dstrbuto fucto κ wth two ad three parameters, we replace (x) wth (-t ) ( α )wth value (( β ) the two equatos (5) ad (6), t becomes: 1 κ κ Ft (, θβκ,, ) 1 exp 1 + ( t β), κ 0 (7) θ t β Ft (, θ, β) 1 exp exp, κ 0 (8) θ o get desty fucto for the geeralzed Webull dstrbuto wth three ad a parameters, we take a dervatve of both two relatoshps (5) ad (6 ), we get ; κ κ κ κ f ( t; θβκ,, ) 1 ( t β) exp 1 ( t β) + +, κ 0 (9) θ θ θ 1 t β t β f ( t; θ, β) exp( )exp exp( ), κ 0 (10) θ θ θ Copyrght 014 ScResPub.

4 Iteratoal Joural of Advacemets Research & Techology, Volume 3, Issue 10, October Cosequetly we ca wrte a Relablty Fucto as follows R t t β θ ( ) exp exp (11) Ad the (Hazard rate) for ths dstrbuto, as follows: 1 ht ( ) exp t β θ θ 3. Lear Estmato Techques: Let (T) be the radom varable wth Webull dstrbuto fucto wth two parameters, ad accordg to the relatoshp (8) we wrte: t β Ft ( ; θ, β) 1 exp exp ; κ 0 θ Let (t 1, t, t 3,..., t ) the data dstrbuted to the Webull dstrbuto wth two parameters, the we ca estmate ths parameters, as follows: ˆ θ ct, ˆ β 1 1 at As: s (1,,3,..., ) (a, c ) s the lear weghts (Lear Weghtg Factors). Accordgly, we wrte mathematcal relatoshp to the verse fucto of the Webull dstrbuto wth two parameters, as follows: t β + θx (1) x p p l( l(1 p)) p Ad the we ca fd the value of each ( θ, β,cosders ) represet a slop ad the lump the equato of the regresso le (15), through several methods eable us to estmate two parameters of Webull dstrbuto, but we wll lmt ourselves here oly to use a Whte (Whte Method), based o the theory of regresso aalyss, ad so trasformg the dstrbuto fucto of the relatoshp (8) to form a smple lear regresso model as t β Ft (, θ, β) 1 exp exp (13) θ or 1 Ft (,, ) exp exp t β θ β θ Ad we get to take twce the logarthm of the fucto above, o: t β l{ l[ 1 Ft ( ; βθ, )]} (14) θ ( t... t t t ) Assumg that 3 1, s order sample arraged for a radom sample sze (), the t wrte prevous relatoshp after takg the logarthm twce, as follows l { l 1 ( ( ) ( ) ;, ) t β Ft βθ } θ Copyrght 014 ScResPub.

5 Iteratoal Joural of Advacemets Research & Techology, Volume 3, Issue 10, October Ad cludg t( F ) l l ( 1 ˆ β + θ F ) Ad f we put x ad stead of l l ( 1 F ˆ ) ad y stead of t( F ), the become lear regresso equato s as follows: y β + θx (15) Because that s the pot estmate for, whch s created through oe of the followg oparametrc methods? ˆ 1 1) F ˆ 8 ) F ˆ 0.5 3) F But the goal of estmatg the two parameters of geeralzed Webull dstrbuto accordg to( Whte method), usg the method of least squares mmzg estmated the followg: ( ) { ( ˆ ) } 1 G( βθ, ) t( F ) β θ l l 1 F (16) We wll get the two parameters of the regresso equato above, by applyg the followg relatoshp: ( x x )( y y ) ˆ 1, (17) θ 1 ˆ β y ˆ θx ( x x ) x 1. x,. y Where 1 1 y. y t( F), represeted average values are Vews pheomeo studed ad raked ascedg order. ( F ˆ ) 3. x l l 1 4. Fˆ 0.05 We ca also obta t by estmatg the frst order for a smooth Taylor seres raked ascedg order, as follows: Copyrght 014 ScResPub.

6 Iteratoal Joural of Advacemets Research & Techology, Volume 3, Issue 10, October Z l( l(1 p )), t( p ) β + θz t t 1, Z β θ f( a+ h) f( a) + h f ( a) + h f ( a) 1 as f ( a) β0 + θ0z Ε ( t ) β + θz + ( β β ).1 + ( θ θ ). Z β0 + θz + β β0 + θz θ0z β + θz Ad we ca use method (OLS) based o the style of matrces, estmatg parameters of Webull dstrbuto, where the extract estmated values for the followg relatoshp: ˆ β ˆ θ As ( HH) 1 Ht ( ) z (18) H 1 Z 1 1 Z.. 1 Z I order to determe the approprate data of ay pheomeo studed the Webull dstrbuto (GSP), t s possble to subject these data to test goodess of ft, ad based o the table values (Aderso, A ) for the dstrbuto of Webull wth two parameters ad the dstrbuto of the value of the extremst, made by (D, agosto, R, B., Stephes, M, A., 1986, P.86-90), t ca reach a table values (A ) for the Webull dstrbuto wth two parameters, as follows: Let we have a sample of the sze raked (t 1, t, t 3,..., t ) ad estmated values ( ˆ, ˆ) βθ for the Webull dstrbuto wth two parameters, as follows: ˆ ( ; ˆ, ˆ t β Ftθ β) 1 exp exp ; κ 0 (19) ˆ θ Suppose that Z t ˆ β exp ˆ θ Substtutg ths (suppose) hypothess equato (19), we get: [ Z ] Ft ( ; ˆ θ, ˆ β ) 1 exp Copyrght 014 ScResPub.

7 Iteratoal Joural of Advacemets Research & Techology, Volume 3, Issue 10, October Accordgly, to applyg equato (0) we get a values of (A). l( ) ( 1 ) l ( 1 ) (0) A P F + P F 1 Where 0.5 P, 1,,3,..., 4. Applcatos part: After our revew of probablty dstrbutos,spatally the Webull dstrbuto, we ca say that the use of probablty dstrbutos s oe of the mportat statstcal method ad essetal forecastg the pheomea of lfe. I order to reach statstcal models that desged to dagose these pheomea, estmatg ad aalyzg the dfferet of the teracto amog them, by usg mathematcal ad statstcal methods to determe treds, explaed by ts fdg of dcators ad accurate estmates have,that would be the bass for the developmet of possble solutos ad how to deal wth them the forecastg. I our research we focused o study treds the rate of traffc accdets the cty of Sulama by durato ( ) based o the Webull dstrbuto ad estmate parameters for ths dstrbutos by usg the least squares method to fd the probablty of occurrece of the expected rate of traffc accdets. I ths study, we used data o mothly rates for a umber of traffc accdets to the cty of Sulama durg ( ) from the records of the drectorate traffc. Table (1) Ascedg order of the umber of traffc accdets to the cty of Sulama durg ( ) Number of Moth Number of Moth accdets accdets Copyrght 014 ScResPub.

8 Iteratoal Joural of Advacemets Research & Techology, Volume 3, Issue 10, October Accordg to the above revew theatrcal part for "Webull geeralzed dstrbuto", after ascedg order of data of traffc accdets for the cty of Sulama (as show Table (1), ad after estmatg the parameters for Webull wth two parameters( βθ, ) for those data wth applyg equato (17), we have obtaed ad estmate ts features accordg to the least squares method (OLS), usg software "easy ft" to test, ad estmate the values of the parameters, as show the followg tables ( ) ad (3) respectvely : Table () value of A Aderso-Darlg Sample Sze Statstc Rak α Crtcal Value parameter estmate Reject? No ˆβ No No No No θˆ Table (3) Shows the estmated values of the two parameters of Webull dstrbuto Through our use of the (Aderso Darlg) test, wth the equato(19) by usg the "easy ft" software.we acheved the value of ( A ) to esure the approprateess of data traffc accdets, accordg to the estmated values Table (3). for the dstrbuto Webull wth two parameters, for the value of the accdet umber (0.7654) gve table (), whle comparg these values at level of sgfcace (0.05) ad equal to (.50), we accept the data for the umber of accdets, because the calculated value s Copyrght 014 ScResPub.

9 Iteratoal Joural of Advacemets Research & Techology, Volume 3, Issue 10, October less tha the value, so ca we say about these data t s belog to Webull dstrbuto wth two parameters. From equato (15) we ca fd the estmate umber for umber of accdets as show table (4), ad Fgure (1) Shows smlarty betwee real values of umber accdets ad estmate values. Table (4 ) Show the estmate umber for umber of accdets umber of accdets estmate umber umber of accdets estmate umber Copyrght 014 ScResPub.

10 Iteratoal Joural of Advacemets Research & Techology, Volume 3, Issue 10, October estmate umber Number of traffc accdets Fgure (1) Show smlarty betwee real ad estmate values Also we ca fd expected probablty of occurrece of the umber of accdets for (N) years, by usg equato (19) ad estmate value of ( βθ, ). G 1 [ 1 Ft ()] N For example, f the rate of accdets for the December s equal (44),F(t) wll be : F(t)0.01 Smlarty for two years ad four year equal ( 0. 9 ),( 0.06 ) respectvely,ad so o. 4. Cocluso: Through ths study, usg Webull dstrbuto to aalyss the umber of accdets we coclude these pots: 1) Ca be usg mathematcal form of Webull dstrbuto to forecastg. ) Ca be usg regresso techque to estmate Webull parameters. 3) From table ( 4) for average umber of accdets are creasg. At the ed we recommed for future work ca be usg these theorems of probablty dstrbuto realzed process. Referece : Copyrght 014 ScResPub.

11 Iteratoal Joural of Advacemets Research & Techology, Volume 3, Issue 10, October AI-Baddha,F.A.(1990),Relablty Theory operatoal Research,Uversty of St. Adrews,U.K..D,agosto,R.B.,ad Stephes,M.A.,(1986),Goodess of Ft Techques,Marcel Dekker, c, New York. 3. Grosh,D.L,(1989)" A Prmer of Relablty Theory" Joh Wley & Sos, c, Caada. 4.Hug,W.L,(004),Estmato of Webull parameters usg a Fuzzy,Least squares method, Iteratoal Joural of Uceato,vol.1,No.,pp: Kahadawala,C.,(006),Estmato of Webull Dstrbuto :the add Webull famly,statstcal modelg,6;pp: Zacks,S.,(199),Itroducto to Relablty Aalyss Probablty Models aalyss Statstcal Methods, New York. 7.htpp:/ -06.htm.Retrved. 8.Ch-Dew La,(014)"Geeralzed Webull Dstrbutos Sprger,Hedelberg, New York, Dordrecht,Lodo. 9. Horst Re,(009)"The Webull Dstrbuto A Hadbook", Justus-Lebg-Uversty Gesse, Germay, Chapma & Hall/CRC, Taylor & Fracs Group. Copyrght 014 ScResPub.