Moments of Order Statistics from Nonidentically Distributed Three Parameters Beta typei and Erlang Truncated Exponential Variables

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1 Joural of Mahemacs ad Sascs 6 (4): , 200 SSN Scece Publcaos Momes of Order Sascs from Nodecally Dsrbued Three Parameers Bea ype ad Erlag Trucaed Expoeal Varables A.A. Jamoom ad Z.A. AL-Saary Deparme of Sascs, Grls College of Scece, Kg Abdul-Azz Uversy, Saud Araba Absrac: Problem saeme: Momes of order sascs of depede o-decally dsrbued (ND) radom varables s o a easy subec o deal wh for couous dsrbuos. Oe s forced o use messy algebrac calculaos (wheher oe uses permaes or o). Ths was he movao behd hs sudy. hs sudy he momes of order sascs arsg from depede odecally dsrbued hree parameers Bea ype dsrbuo ad Erlag Trucaed Expoeal dsrbuo were derved. Approach: We employed a easer echque esablshed by Baraa & Abdelader wll be referred o as (BAT). Resuls: The mea, he secod mome ad he varace of he meda ad he smalles order sascs for he frs dsrbuo were gve for dffere values of he shape parameer ad dffere sample szes. Cocluso: The resuls ca be used o mae some fereces ad used he BAT echque o derve momes of order sascs arsg from depede odecally dsrbued for ay oher couous dsrbuo wh dsrbuo fuco (cdf) he form: F(x) = -λ (x). Key words: Momes, o-decally dsrbued order sascs, permaes, hree parameers bea ype dsrbuo, erlag rucaed expoeal dsrbuo NTRODUCTON The momes of order sascs (os) of depede No-decally Dsrbued (ND) radom varables (rvs) have bee esablshed leraure wo drecos. The frs dreco was aed by (Balarsha, 994). requres a basc relao bewee he probably desy fuco (pdf) ad he cumulave dsrbuo fuco (cdf). Ths echque s referred o as Dffereal Equao Techque (DET). eables oe o compue all he sglead produc momes of all order sascs a smple recursve maer ad he dervao of he momes depeds maly o egrao by pars. The secod echque was esablshed by Baraa ad Abdelader(2003) wll be referred o as (BAT). s a easer maer o evaluae he momes of ND os bu ca be appled o dsrbuos wh cdf he form: F( x) = -λ(x) or by usg he survval fuco of he dsrbuo uder sudy. BAT cao be used o evaluae he produc momes. hs sudy was our eres o use he BAT echque o ge he momes of os arsg from ND dsrbued hree parameers Bea ype (Johso e al., 994) ad Erlag Trucaed Expoeal (El- Alosey, 2007) ad (Mohs,2009) radom varables. The subec of os from ND rvs s dscussed wdely he leraure (Davd, 98; Bapa ad Beg, 989; Davdad Nagaraa, 2003). (Baraa, 2002). foud he lm behavor of bvarae order sascs from ND dsrbued radom varables. (Gügor, Goha ad Bulu 2005) expressed he mulvarae order sascs by margal Orderg of ND radom vecors uder dscouous df`s. Applcaos of he prevous wo mehods are also foud leraure for several couous dsrbuos. The DET was used by (Balarsha, 994) o derve recurrece relaos sasfed by sgle ad produc momes of os from ND rvsfor he Expoeal ad rgh rucaed dsrbuos. (Chlds ad Balarsha, 2006) appled DET o derve he momes of os from ND rvs for Logsc radom varables. The frs applcao of BAT was by (Baraa ad Abdelader, 2000) o webull dsrbuo ad he he mehod was geeralzed by (Baraa ad Abdelader, 2003) ad was Correspodg Auhors: A.A. Jamoom, Deparme of Sascs, Grls College of Scece, Kg Abdul-Azz Uversy, P.O. Box 45, Jeddah, Saud Araba 24 Tel: (02) / Fax: (02)694204, (02)

2 J. Mah. & Sa., 6 (4): , 200 appled o Erlag, Posve Expoeal, Pareo ad Laplace dsrbuos. Abdelader(2004) ad (Abdelader, 2008) used a closed expresso for he survval fuco of Gamma ad Bea dsrbuos o compue he momes of os from ND rvs usg BAT. Jamoom(2006) appled o Burr X rvs. Le X, X 2,,X be depede radom varables havg cdfs F (x), F 2 (x),, F (x) ad pdfs,ƒ (x), ƒ 2 (x),, ƒ (x) respecvely. Le X :, X 2: X : deoe he os obaed by arragg he X 's creasg order of magude. The he pdf of he rh os.x r: ( r ) ca be wre as: MATERALS AND METHODS We cosder he hree parameers Be ype dsrbuo (Johso e al., 994) wh cdf: x F(x) =,w x w P (5) ad Erlag rucaed expoeal (El-Alosey, 2007) ad (Mohs, 2009) wh cdf: βx λ F(x) e ( e ),0 x,, 0 = β λ > (6) r f (x) = F (x)f (x) { F (x)} r: α r c (r )!( r)! p α= c= r + () Now, we cosder he case whe he varables X, s be ND rvs, he he above cdfs ca be wre as: Where, deoes he summao over all p permuaos[(, 2,, ) of (, 2,.,). (Bapa ad Beg, 989) pu he prevous pdf of he rh os X r: he form of permae as: f r: (x) = (r )!( r)! per F(x) f (x) { F(x)} r r (2) To derve he momes of os from ND rvs arsg from hs dsrbuo we eed he followg heorem whch s esablshed by (Baraa ad Abdelader, 2003). Theorem : Le X, X 2,,X be depede odecally dsrbued rvs. The h mome of all order sascs,, for r ad =, 2, s gve by: P x F (x) =,w x, p > 0,,w > 0 w βx λ F (x) e ( e ),0 x,, 0 (7) = β λ > (8) Applcaos of heorem for he above dsrbuos wll be saed as heorem 2 d 3. Momes of os from ND hree parameers bea ype rvs: Theorem 2: For ay real umbers, w > 0, r ad =, 2, : P = + () =... β w P +, P < 2 < = = ( w) (9) Proof: O applyg heorem ad usg (4), we ge: ( ) () (3) ( r+ ) () r: = = r+ r Where: (4) < 2 < 0 = () =... x G (x)dx, = 2,..., o: P K x = () =... x dx < 2 < w w x Subsug y = he above equao reduces w G (x) = -F (x) wh (, 2, ) s a permuao of (, 2,, ) for whch < 2 <,, <. < 2 <... < 0 P K w = () =... ( w)( ) y y dy Proof: The proof of hs heorem ca be foud Baraa ad Abdelader (2003). 443 x By subsug z = y we ge:

3 J. Mah. & Sa., 6 (4): , 200 J P = + w P + = () =... z z dz < P 2 <... < 0 = ( w) P = [ ] + () =... β ( P +,) w < P 2 <... < = = ( w) where, β z (a,b) s he complee bea fuco defed by: z a b x ( x) dx = βz(a, b) 0 Momes of os from ND erlag rucaed expoeal rvs: Theorem 3: For r ad =,2, : () =... Γ() ( ) e λ β = < 2 <... < Proof: O applyg heorem ad usg (8), we ge: We ge: < 2 <... < 0 < 2 <... < 0 λ βx( e ) () =... x e dx, = = λ βx (e ) =... x e dx O egrag erm by erm usg he egral: 0 x Γ() e dx = a ax () =... Γ() ( β) ( e ) = Γ() < 2 <... < λ =... ad he proof s compleed. ( β) e = < 2 <... < λ (0) 444 RESULTS Resul : Subsug (9) (3) he h momes of he rh os from ND hree parameers bea ype ca be fally wre as: ( r + ) () r: = = r + r < 2 <... < + P = ( w) ( )... w P = = β ( P +, ) () Remar : The h mome of he larges os X : from ND hree Bea ype rvs ca be wre as: () : = = ( ) () (2) where, () s defed (9). Remar 2: The h mome of he frs os X : from ND hree parameers Bea ype rvs ca be wre as: = () (3) () : Where: + P = w P = = () = β P +,,,w > 0 (4) ( w) Remar 3: D case The depede decally Dsrbued (D) case ca be deduced from heorem (2). () ca be wre as: + p ( ) () = β p w p +, (5) ( w) where, βz (a, b) s he complee bea fuco defed before. Resul 2: Subsug (0) (3) he h momes of he rh os from ND Erlag rucaed xpoeal ca be fally wre as: ( r + ) () r: = = r + r < 2 <... < λ ( β) e = ( )... Γ() (6)

4 J. Mah. & Sa., 6 (4): , 200 Remar : The h mome of he larges osx : from ND Erlag rucaed expoeal rvs ca be wre as: () : = = ( ) () (7) where, () s defed (0) Remar 2: The h mome of he frs os X : from ND Erlag rucaed expoeal rvs. ca be wre as: () : = () Where: () = Γ() K λ ( β) [ e ] = (8) (9) Remar 3: The depede decally Dsrbued (D) case ca be deduced from heorem (2). () s wre as: Γ() () = ( β) [ e ] K λ (20) Numercal calculaos for hree parameer bea ype : Smple programs wre by Mahemaca 7 were used o calculae he momes gve ables, 2, 3, 4, 5. Example (): Mulple oulers sample: Le: =3, X Bea yp ( =, w =0.5, P = 0.), X 2 Bea yp ( =, w = 0.5, p 2 = (0,(0.5)2)), X 3 Bea yp ( =, w = 0.5, p 3 = 0,(0.5)2). () = P = 2 < 2 <... < 3 2 β ( P +, ) w = ( w) P = + p + p2 = β (p + p +, ) p + p2 ( w) w 2 + p + p2 + β (p + p +, ) p + p3 ( w) w 3 + p2 + p3 + β (p + p +, ) p2 + p3 ( w) () =... w P = 3 3 < 2 <. 3.. < 3 w ( w) P = (22) 3 (23) = β ( P +, ) + p + p2 + p3 = β (p + p + p +, ) p + p2 + p3 ( w) w 2 3 () Subsug (22) ad (23) (2), 2:3 s he ca be obaed. Table represes he mea, he secod mome ad he varace of he medaof he sample sze = 3 arsg from hree parameer Bea ype dsrbuo. These compuaos are doe whe ( =, w = 0.5, p = 0., p 2 =(0,(0.5)2)). Example (2): Sgle ouler sample: Whe w = 0, =, he hree parameer Bea ype dsrbuo reduces o F(x ) = [ x] p, 0 x ad Eq (9) ca be wre as: (24) < 2... < = () =... β ( P +, ) The from (3) he h of he meda X 2:3 s gve by: 3 () 2 2:3 = ( ) () = 2 = () 2 () From (9): 2 3 (2) Le: X, X 2,, X 9 Bea yp ( =, w =0, P = (0,(0.5)2)), X Bea yp ( =, w = 0.5, p 0 = (0,(0.5)2))0, 20, 50, 00, 0000) From (3, 4) he h mome of he smalles os whe he sample sze = 0 s: 445

5 J. Mah. & Sa., 6 (4): , 200 = () () : = () () :0 0 0 =... β P +, = < 2 <... < = β ( () + p +, ) = 0 0 = β (0 + p,) (25) Table 2 ad 3 represe he mea, he secod mome ad he varace of hesmalles os of he sample sze =0 arsg from hree parameer Bea ype dsrbuos. These compuaos are doe whe: ( =, w = 0, p = (0,(0.5)2)), p 0 = (0,(0.5)2))0, 20, 50, 00, 0000) Theorem 2 of Abdelader (2008) wh α = s he same as Eq. 25. Example 3: Sgle ouler sample: Le: X, X 2,, X 9 Bea yp ( =, w = 0.5, P = (0,(0.5)2)), X Bea yp ( =, w = 0.5, p 0 = (0,(0.5)2))0, 20, 50, 00, 0000) From (3, 4) he h mome of he smalles os whe he sample sze = 0 s: = () () :0 0 0 =... β P +, < 2 <... < P P + p 0 = = = 0 = 9 = β ( () + p +,) (26) (0.5) = β 9 p 0.5(0 + p 0,) + 0 (0.5) Table : = 3, ( =, w = 5P = 0., P2 = (0(0.5)2),P3 = 0,(0.5)2) P2/P The mea, he secod mome ad he varace of he meda of he sample sze = 3 he presece of mulple oulers usg Eq 2 Table 2: = 0, =, w = 0, p = p 2 = = p 9 = p = (0,(0.5),2), p 0 = (0,(0.5),2 p/p The mea, he secod mome ad he varace of he smalles os of he sample sze = 0 he presece of sgle ouler sg Eq

6 J. Mah. & Sa., 6 (4): , 200 Table (3): = 0, =, w = 0, p = p 2 = = p 9 = p = (0,(0.5),2), p 0 = (0,20, 50, 00, 000) p/p The mea, he secod mome ad he varace of he smalles osof he sample sze = 0 he presece of sgle ouler Eq. 25 Table 4: = 0, =, w = 0.5, p = p 2 = = p 9 = p = (0,(0.5),2), p 0 = (0,(0.5),2 p/p E(X 2 :0) The mea, he secod mome ad he varace of he smalles os of he sample sze = 0 he presece of sgle ouler Eq. 25 Table 5: = 0, =, w = 0.5, p = p 2 = = p 9 = p = (0,(0.5),2), p 0 = (0, 20, 50, 00, 000) p/p The mea, he secod mome ad he varace of he smalles os of he sample sze = 0 he presece of sgle ouler Eq

7 J. Mah. & Sa., 6 (4): , 200 Table 4 ad 5 represe he mea, he secod mome ad he varace of he smalles o.s. of he sample sze =0 arsg from hree parameer Bea ype dsrbuos. These compuaos are doe whe: ( =, w = 0.5, p = (0(0.5), 2), p 0 = (0,(0.5)2))0, 20, 50, 00, 0000). DSCUSSON Usg he BAT Techque beauful ad elega recurrece rlaos were obaed for he momes of rder sascs from odecally bu depedely dsrbued varables for boh dsrbuos uder sudy whch ca be cosdered a addo o ND. os. CONCLUSON BAT echque s srogly recommeded for ay oher couous dsrbuo wh cdf he form: F( x ) = - λ (x). Comparso bewee he DET ad BAT echque s also recommeded for fuure sudy. REFRENCES Abdelader, Y., Compug he momes of order sascs from odecally dsrbued Gamma varables wh applcaos.. J. Mah. Game Theo. Algebra., 4: -8, SSN: Abdelader, Y., Compug he momes of order sascs from depede odecally dsrbued Bea radom varables. Sa. Pap., 49: DO: 0.007/SOO Balarsha, N., 994. Order sascs from odecally expoeal radom varables ad some applcaos. Compu. Sa. Daa Aal., 8: DO: 0.06/ (94) Bapa, R.B. ad M.. BEG, 989. Order sascs from o-decally dsrbued varables ad permaes. Sahya, A., 5: MR Baraa, H. ad Y. Abdelader, Compug he momes of order sascs from odecal radom varables. Sa. Meh. Appl., 3: DO: 0.007/s Baraa, H.M. ad Y.H. Abdelader, Compug he momes of order sascs from odecally dsrbued Webull varables. J. Comp. Appled Mah.,7: DO: 0.06/S (99)00343-X Baraa, H.M., The lm behavor of bvarae order sascs from No-decal dsrbued radom varables. J. Appl. Mah, Polsh Academy Sc., 29: SSN: Chlds, A. ad N. Balarsha, Relaos for order sascs from o-decal logsc radom varables ad assessme of he effec of mulple oulers o bas of lear esmaors. J. Sa. Pla. ferece., 36: DO: 06/.sp Davd, H.A, 98. Order Sascs. 2d Ed., Wley, New Yor, pp: 360. SBN: Davd, H.A. ad H.N. Nagaraa, Order Sascs. 3rd Ed., Wley, New Yor, pp: 458. SBN: El-Alosey, A.R., Radom sum of ew ype of mxure of dsrbuo. JSS, 2: SSN: Gügor, M., A. Goha ad Y. Bulu, Mulvarae order sascs by margal orderg of (ND) radom vecors uder dscouous df`s. J. Mah. Sa., : SSN: Jamoom, A.A., Compug he momes of order sascs from depede odecally dsrbued Burr ype X radom varables. J. Mah. Sa., 2: SSN: Johso, N.L., S. Koz ad N. Balarsha, 994. Couous Uvarae Dsrbuos. 2d Ed., Joh Wley ad Sos, c., New Yor, pp: 756. SBN: Mohs, M., Recurrece relao for sgle ad produc momes of record values from Erlagrucaed expoeal dsrbuo. World Appled Sc. J., 6: SSN: