Testing EBUASI Class of Life Distribution Based on Goodness of Fit Approach

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1 Tsng BUAS Class f Lf Dsrbn Basd n Gdnss f Apprach.A.Rad a,..lashn b and.a.abass b a ns f Sascal Sds and Rsarch, Car Unrs, gp b Mahmacs Dp., acl f ngnrng, Tana Unrs, gp Absrac: Ts sascs fr sng pnnal agans pnnal Br Wrs han Usd n Arag n Spcfc nral BUAS WUAS s prpsd basd n h gdnss f f apprach. Th crcal als f hs s ar calclad and ablad fr sampl sz n50, 5, 50. Th Pman asmpc ffcnc PA s dscssd and h pwr f h s fr sm cmmnl sd dsrbns n rlabl s calclad. nall ral ampl s prsnd llsra h hrcal rsls. K Wrds: BUAS, WUAS, pnnal dsrbn, gdnss f f, Pman's asmpc ffcnc.. NTRODUCTON Cncps f agng dscrb hw a pplan f ns r ssms mprs r drras wh ag. Man classs f lf dsrbns ar cagrzd and dfnd n lrar accrdng hr agng s mpran n an rlabl analss. Tsng pnnal agans h classs f lf dsrbns ha rcd a gd dal f ann. r sng agans nw br han sd NBU, w rfr Hllandr and Prschan 97, Kl 977, Kmazawa 98 and Ahmad 99 amng hrs. r sng agans dcrasng man rsdal lf DMRL w rfr Hllandr and Prschan 975, Langnbrg and Srnasan 979 and Ahmad 99 amng hrs. r nw br han sd n pcan NBU, w rfr Kl 978 and Ahmad al. 999 amng hrs. r harmnc nw br han sd n pcan HNBU, w rfr Klfsj 98, Abammh and Khalq 987, Ahmad 995 and Hnd al. 998 amng hrs. L b a nnnga randm arabl wh dsrbn fncn df and a sral fncn sf. Assm als ha s absll cnns wh prbabl dns fncn f and has fn man and aranc σ, bh assmd fn. Asscad wh s h nn f "randm rmanng lf" a ag, dnd b whr has sf,,, 0. N ha f and nl f s an pnnal dsrbn. Cmparng and n ars frms and ps cra classs f agng. Amng hs classs ar h fllwng: R ncrasng alr Ra RA ncrasng alr Ra Arag NBU Nw Br han Usd NBU Nw Br han Usd n pcan HNBU Harmnc Nw Br han Usd n pcan r dfnns f hs classs and frhr dals s,.g. Barlw and Prschan 98. Crrspndnc fa_abass@ah.cm

2 Dfnn.: L and Y b w randm arabls wh margnal df's and G, rspcl. W sa ha s lss arabl han Y r s ncrasng rdrng n cnca c han Y, and w wr Y ff c 0 d G d. And w sa ha s ncrasng rdrng n cn c han Y, and w wr 0 Y ff c and 0 d. d G d, fr all 0 Th ncrasng rdrng n cnca c s rlad ncrasng rdrng n cn c as gn n h fllwng hrm. Thrm.: L and Y b w randm arabls wh dsrbn and G, rspcl wh 0- G 0-0 and d G d.. and G ha h sam man, hn, Y Y r G G c c c c r prf f hs hrm s Hnd al Dfnn.: Th nn-nga randm arabl wh dsrbn s sad b pnnal br han sd rdrng BU f c fr all 0. n sch a cas w wr BU r BU, ff r S lbaal 00. Dfnn.: Th nn-nga randm arabl wh dsrbn s sad b pnnal br han sd n arag rdrng BUA f c fr all 0. n sch a cas w wr BUA r BUA. s dal class s pnnal wrs han sd n arag rdrng WUA whch s dfnd b c fr all 0. Th ab nqals ar qaln d ; 0, >0 r S Aa al d d ; 0, >0

3 Dfnn.: A dsrbn fncn wh sppr [0, and fn man d 0, s calld pnnal Br Wrs han Usd n Arag a Spcfc nral BUAS WUAS f d d Ths class f lf dsrbn s prsnd b Abdl-Mnm 007, fr h rlanshp bwn hs class and hr classs lk BU, BUC, BUA and NBU s Abdl-Mnm 007. Ths papr s rganzd as fllws, n scn, w prsn a s sasc basd n gdnss f f apprach fr sng H : s pnnal agans H : s BUAS and n pnnal and larg sampl prprs als prsnd, n scn Pman's asmpc ffcnc PA f h s fr sm cmmn dsrbn ar ablad, n scn Mn Carl Crcal pns ar band fr sampl szs n 50, 5, 50, and pwr f h s s smad n scn 5. nall applcans sng ral daa ar als nrdcd n scn 6. -Tsng pnnal agans BUAS class n hs scn a s sasc basd n h gdnss f f apprach s dlpd s H : s pnnal agans H : s BUAS and n pnnal. Lmma. f s BUAS hn a masr f dparr frm H s whr,. Prf: Th BUAS WUAS class f lf dsrbn s dfnd as, d d Thn d d Dfn d and d Thn

4 Dfn a masr f dparr frm H as [ ] 0 d d d Whr d d, d d, d d L ddd [ ] d d > [ ] 0 > dd dd ddd > ddd ddd ddd rm,, and Thn h prf s cmpld. Basd n a randm sampl,, n wh man frm a dsrbn, a drc mprcal sma f h masr n. s: n n ˆ.

5 Th fllwng hrm gs h larg sampl prprs f h s sasc. Thrm. As n, n ˆ s asmpcall nrmal wh man 0 and aranc σ, whr σ ndr H s gn b σ 0.0 Prf: s sraghfrward, b nng ha ˆ s js an arag f, hn sng h cnral lm hrm h rsl fllws. Th aranc wll b σ Var ndr H h aranc bcms, σ T prfrm h ab s, rjc H f h al f n 0.0 ˆ wll cds Z α whch s h ppr prcnl f h sandard nrmal ar. - Pman Asmpc ffcnc PA Th pman asmpc ffcnc f h class BUAS was calclad sng h Lnar alr Ra LR, Makham, and Wbll dsrbns. Th pman ffcnc s dfnd as: PA σ σ Th fllwng hr famls f alrnas ar fn sd fr ffcnc calclan Lnar alr Ra : Makham : Wbll : Th nll pnnal s aand a 0, 0 and rspcl. Th ffcnc calclan fr h ab hr alrnas ar ablad n abl. Tabl- Pman Asmpc ffcnc Dsrbn ffcnc LR 0.0 Makham 0. Wbll Mn Carl Nll Dsrbn Crcal Pns n hs scn a smlan fr h nll dsrbn crcal pns fr ˆ wll b mad fr sampl szs n50, 5, 50 frm h sandard pnnal dsrbn. Tabl- gs h ppr prcnl f h sasc ˆ ; h fllwd fgr shws h rlan bwn h crcal als and h sampl sz. 5

6 Tabl : Crcal als f ˆ n 90% 95% 98% 99% Rlan bwn crcal als and sampl sz fr ˆ % crcal als 95% crcal als 98% crcal als 99% crcal als Crcal Vals Sampl Sz 6

7 5- Pwr f h s n hs scn an sman f h pwr fr sng pnnal rss BUAS wll b mad sng sgnfcanc ll 95% wh sabl paramrs als f a n0,0,and 0, and fr cmmnl sd dsrbns n rlabl sch as LR, Makham, and Wbll alrnas. Tabl- shws h pwr f h s. Dsrbn LR Makham Wbll Tabl-: Pwr smas n Applcans Th fllwng daa rprsn 9 lr cancr's pans akn frm l Mna Cancr Cnr f Mnsr f Halh f gp, n 999. Th rdrd lf ms n das ar: Th daa ar: 0,,,,,, 5, 7, 8, 0, 0, 0, 0, 0,,,, 6, 0, 0,, 0, 9, 5, 5, 60, 6, 67, 7, 7, 75, 87, 96, 05, 07, 07, 07, 6, and 50. s fnd ha h s sasc fr h s f daa b sng qan. s ˆ whch s lss han h crcal al f h abl, hn w accp H whch sas ha h s f daa dn n ha BUAS prpr a 95% prcnl. Acknwldgmn Th ahrs wld lk hank prf. M.. Hnd fr hs hlpfl cmmns, whch mprd h prsnan f h papr. Rfrncs Abdl-Mnm, A nw class f agng dsrbns. J. gp. Sas. Sc.,, -. Abammh, A.M. and Khalq, A Sm ss fr man rsdal lf crra basd n al m n s ransfrm. Rlabl ngnrng, 9, Ahmad,. A., Hnd, M.., Al-Nachawa, H Tsng nw br han sd class f lf dsrbn drd frm cn rdrng sng krnl mhd. J. Nnparamrc Sas.,, 9-. Ahmad,. A Nnparamrc sng f class f lf dsrbns drd frm a arabl rdrng. Parsankhan Samkka,, -8. 7

8 Ahmad,. A. 99. A class f sascs sfl n sng ncrasng falr ra arag and nw br han sd lf dsrbns. J. Sas. Plan. nf.,, -9. Ahmad,. A. 99. A nw s fr man rsdal lf m. Bmrka, 79, 6-9. Aa, A.., Mahmd, M. A. W. and Abdl-Mnm, "Th class f pnnal br han sd n arag". Th 0 h Annal Cnfrnc n sascs, Cmpr scnc and Oprans Rsarch, Car Un. Barlw, R.. and Prschan,. 98. "Sascal hr f rlabl and lf sng". T Bgn Wh, Slr-Sprng, MD. lbaal, "Th BU and WU classs f lf dsrbn". J. gp. Sas. Sc., 8, Hnd, M.., Al-Nachwa, H. Mnassr, M. and Alwasl,. A An ac s fr HNBU class f lf dsrbn. J. Sas. Cmp. Sml., 60, Hnd, M.., Al-Nachwa, H. and Al-Rzaza, A. S A s fr pnnal agans Nw Br han Usd Arag. J. Kng Sad Un., 07-. Hllandr, M. and Prschan,. 97. Tsng whhr nw br han sd. Ann. Mah. Sas..,, 6-6. Hllandr, M. and Prschan, Ts fr man rsdal lf. Bmrka, 6, Kl, H A nw s fr nw br han sd. Cmm. Sas. Thr. Mh., 6, Kl, H Tsng fr nw br han sd n pcan. Cmm. Sas. Thr. Mh., 7, Kmazawa, Y 98. Tsng fr nw s br han sd. Cmm. Sas. Thr. Mh.,, -. Klfsj, B. 98. Th HNBU and HNWU class f lf dsrbns. Naal Rs. Lgscs Qarrl,, -. Langnbrg, P. and Srnasan, R Nll dsrbn f h Hllandr and Prschan sasc fr dcrasng man rsdal lf. Bmrka, 66,

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