A NEW GENERALIZATION OF KUMARASWAMY LINDLEY DISTRIBUTION

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1 ou of Sttt: dv Thoy d ppto Vou 4 Nu 5 Pg 69-5 v t DOI: NW GNRLIZTION OF KUMRSWMY LINDLY DISTRIBUTION M. MHMOUD M. M. NSSR d M.. F Dptt of Mtht Futy of S Sh Uvty Co 566 gypt -:..@g.o tt I th pp w of gzd dtuto d th pottd Kuwy Ldy (KL) dtuto todud. Th w dtuto qut f od yzg potv dt. W povd ophv tht ttt of th ttt popt of th dtuto. So tutu popt of th popod w dtuto dud udg poty dty futo d pt fo fo t uvv futo hzd futo d qut futo. Th thod of u hood ud to tt th od pt d th ovd d ptd foto t dvd. dt t ud to op th w od wth wdy ow dtuto. uto tudy odutd d th d qud o of tt ptd fo dfft p z. Mtht Sut Cfto: Kywod d ph: Kuwy dtuto Ldy dtuto ot gtg futo topy u hood tto foto t d uto. Rvd Spt Stf dv Puh

2 7 M. MHMOUD t.. Itoduto I y fd of ppd uh d gg d f ogt oth odg d yzg ft dt u. Sv ft dtuto hv ud to od uh d of dt fo t th pot Wu g Rygh dtuto d th gzto (.g. Gupt d Kudu [ ]). h dtuto h t ow htt du pfy to th hp of th fu t futo whh y oy ootoy dg o g o ott t hvou w oooto g thtu hpd o v uod. Ldy dtuto pott fo tudyg t-tgth ty odg. Ghty t. [] dud vou popt of th dtuto d howd tht th Ldy dtuto povd tt od th th pot dtuto y wy. S [4] todud th dt Poo- Ldy dtuto y og th Poo d Ldy dtuto. Mzuh d h [5] dud th ppto of Ldy dtuto to optg ft dt. Ghty d -Mut [6] d Ghty t. [7] otd z-d d zo-tutd vo of Poo-Ldy dtuto d dud th vou popt d ppto. Bouh t. [8] otd tdd Ldy dtuto d dud t vou popt d ppto. Ghty t. [9] dvopd two-pt wghtd Ldy dtuto d dud t ppto to uvv dt. Ghty d -Mut [] dud vou tto thod fo th dt Poo-Ldy dtuto. R d Mh [] tudd qu Ldy dtuto. Ghty t. [] tudd pow Ldy dtuto d otd f. Zzdh d Dot [] d t t. [4] hv otd th gzd Ldy dtuto d th Kuwy qu Ldy dtuto ptvy d dud vou popt d ppto. Cyp d Kd [5] hv otd th Kuwy Ldy dtuto.

3 NW GNRLIZTION OF KUMRSWMY 7 Th Kuwy dtuto dfd y Kuwy (98) [6] dtfd ttv to t dtuto u thy oth hv th hp popt (uod utod g dg ooto o ott). Th poty dty futo (pdf) of th Kuwy dtuto gv foow: f ( ) ( ) ; < <. () Th opodg uutv dty futo (df) gv F ( ) ( ) ; < < () wh > hp pt. Th pdf gv quto () do ot vov y opt t futo to d t gdd g tt u of t d g popt. o [7] pod th goud of th dtuto d dud th dvtg d ddvtg of th dtuto opd wth th t dtuto. Codo d Cto [8] hv popod gzto of th poty dtuto y ug () d (). Th dty d dtuto futo of gzd gv y Codo d Cto hv foowg fo: d f ( ) g( ) G( ) ( G( ) ) () F ( ) ( G( ) ) (4) dg( ) wh g ( ) d pt > hp pt. d Codo t. [9] todud th Kuwy Wu dtuto. o Bougugo t. [] todud th Kuwy Pto dtuto. Lt P t. [] d Go t. [] todud th Kuwy gzd Bu d Rygh dtuto foowd y th pp of d Po t. [] wh thy todud th Kuwy gzd g dtuto.

4 7 M. MHMOUD t. I th t w pt w gzto of Ldy dtuto d th pottd Kuwy Ldy (KL) dtuto. Th tudy vou popt of th w od. Th t of th pp ogzd foow: I Sto w df th gzto of th KL dtuto. Th poty dty futo d uutv dtuto gv. Qut futo ot d ot gtg futo dud Sto. Ry d Sho top utd Sto 4. I Sto 5 w du u hood tto d dt th t of th ovd foto t d ptd Fh foto t. Sto 6 povd ppto to dt t d uto tudy ptd. Sto 7 d th pp wth o ouo.. pottd Kuwy Ldy (KL) Dtuto I th to w todu th pottd Kuwy Ldy (KL) dtuto. Th uutv dtuto futo (df) of th Ldy dtuto todud y Ldy [4] gv y [ ] > >. G (5) Ug (5) (4) w ot th df of th Kuwy Ldy (KL) dtuto. Th tudy of pottd dtuto ufu ttt dtd y Mudho d Svtv [5] fo vy pott o: It povd thod of tdg dtuto fo ddd fty fttg dt. Fo df G ( ) th pottd dtuto F ( ) G( ) wh th potto pt δ > gv th fty to oodt oth ooto d o-ooto hzd t futo. N d [6 7] d y utho tudd th vou popt of th pottd dtuto. I th pp w od th KL wth poty dty futo (pdf) gv y δ

5 NW GNRLIZTION OF KUMRSWMY 7 KL f. (6) Ug th gzd o tho f β tu u th gv y. z z β β β (7.) If β th tu u o zo th ovg outy fo < z d dvg fo > z ( Gdhty d Ryzh [8]). I th th wtt ug th foowg ft u:. z z β β (7.) Th KL pdf wtt fo o-tg foow: KL f.

6 M. MHMOUD t. 74 S w gt f KL. Th KL pdf wtt th fo KL w f (8) wh w. (9) o th uutv dtuto futo (df) of th KL gv y. > > > > F KL ()

7 NW GNRLIZTION OF KUMRSWMY 75

8 76 M. MHMOUD t.

9 NW GNRLIZTION OF KUMRSWMY 77 Fgu. Pot of th pdf of th KL fo td vu of th pt. Fgu d th pot of th pdf d df of th KL fo dfft vu of th pt. It ovou fo th dpyd pot Fgu tht pdf of th KL dtuto uod fo vu of th pt wd to th ght d fo < th gph pd wh fo > th gph dh. o fo g vu of th gph h hgh od p.

10 78 M. MHMOUD t.

11 NW GNRLIZTION OF KUMRSWMY 79 Fgu. Pot of th df of KL fo td vu of th pt.

12 8 M. MHMOUD t. Th uvv (ty) futo of th KL dfd SKL ( ) FKL ( ). () Now th hzd t futo of th KL gv y h KL f ( ) ( ) F ( ) KL ( ) w S ( ). () I Fgu th hp of th hzd t futo ovd fo th foowg vou pot of th hzd t futo fo dfft vu of th pt.

13 NW GNRLIZTION OF KUMRSWMY 8 Fgu. Pot of th hzd t of KL fo td vu of th pt.

14 8 M. MHMOUD t. Not tht th KL dtuto h v w-ow od p whh t of dtguh tf pot fo oth dtuto. If th quto (6) du to th Ldy dtuto. If th quto () du to th g dtuto wth pt ( ). If w gt th Kuwy Ldy dtuto. Th gph of th hzd futo fo v oto of th pt pt vou hp udg ootoy g ootoy dg thtu d up-d dow thtu hp. Th tttv fty KL hzd t futo ufu d ut fo oooto p hzd hvou whh o y to outd o ovd f tuto.. B Popt of th Dtuto I th to w tudy o popt of KL dtuto pfy th qut futo ot d ot gtg futo... Qut futo Tho. Th qut futo of KL dtuto t th foowg ppot pot: X / / / ( ( u) ). () Poof. If F ( ) otuou d tty g th th qut futo Q ( ) F ( u) u ( ) tghtfowd oputd y vtg () to ot

15 NW GNRLIZTION OF KUMRSWMY 8 / / / ( ( u) ). (4) Ug th Tyo po gv y t ( t)! ppot qut futo of th KL dtuto gv y (). I ptu th d of th KL dtuto gv y /. / / d( ) ( ) (5) Th do p o y gtd fo (4) y tg U ufo do v ( )... Mot My of th ttg htt d ftu of dtuto tudd though t ot (.g. tdy dpo w d uto). Mot y d pott y ttt y py ppto. Th foowg tho gv th ot of th KL dtuto. Tho. If X~KL th th -th o-t ot of th KL dtuto gv y th foowg: ( ) ( ) µ ( ) ( X ) w ( ) ( ) (6) wh w gv y quto (9). Poof. Ug th pdf gv quto (8) ( ) µ X f d w ( ) d.

16 84 M. MHMOUD t. Thfo ( ) w ( ) ( ) µ d ( ) ( ) w d d. H w ot th ut gv y quto (6)... Mot gtg futo I th uto w dv th ot gtg futo of KL dtuto. Tho. If X ~ KL th ot gtg futo of X th gv wh M ( ) ( ) () t w ( ) ( ) (7) t t w gv y quto (9). Poof. W tt wth th w-ow dfto of th ot gtg futo d ug th pdf fo quto (8) t t M () t ( ) fkl ( )d ( ) [ ( ) t w ] d w ( ) ( ) ( ) ( ) t t whh opt th poof.

17 NW GNRLIZTION OF KUMRSWMY topy topy of do v X u of vto of th utty. popu topy u Réy topy [9]. If X h th pdf (). f th Réy topy dfd y og f R (8) h > d. Suppo. ~ KL X Sttg (9) th o ut [ ] d f d d d

18 M. MHMOUD t. 86. Thfo o ot th Réy topy og R og. () Now Sho topy [] dfd y [ ] f og th ptu of quto (8). Fo quto (6) d (9) [ ] [ ] [ ] X X f og og og og [ ] [ ( )]. og og Ug th po dfd quto (7.) d og < < z z z () [ ] og X

19 NW GNRLIZTION OF KUMRSWMY 87 [ ] t t t og t X t t [ ( )] u u u og t t u X u t t u [ ] [ ] X X f og og og t X t t X. t t u X u t t u () Now X w [ ] X X og w

20 M. MHMOUD t. 88 X w wh w gv y quto (9). 5. tto of th Pt H w od ttg th pt of th KL dtuto y th thod of u hood d povd po fo th otd ptd foto t. Suppo X X X do p fo KL ξ wh. ξ Th og hood futo of th pt gv y og og og og og og L og og og [ ] og () wh dfd quto (9). Th opot of th o vto T ξ otd y dffttg quto () wth pt to th dfft pt. Thu w hv [ ] ; og og og og L [ ] ; og og og L [ ]; og og L

21 NW GNRLIZTION OF KUMRSWMY 89 L og [ ] wh. (4) By qutg h of th fou quto to zo th u hood tt (ML) of th pt th outo of th yt of o quto whh ovd ttvy. Th ovd foto t gv y ξ wh [ ] og og [ ]; og [ ] [ ] ; og og [ ] ; og

22 M. MHMOUD t. 9 [ ] og [ ] [ og og ]; og og [ ] ; og [ ] ; og [ ] [ ] ; og ; [ ] ; [ ] [ ] [ ] [ ] wh.

23 NW GNRLIZTION OF KUMRSWMY 9 ppyg th uu g p ppoto th ML tt ξ ˆ ˆ ˆ ˆ ˆ ttd g ppoty ( ) 4 ξ ξ I N wh [ ] ξ ξ I ud odto tht fufd fo pt th to of th pt p ut ot o th oudy. Th yptot dtuto ( ). ˆ 4 ξ ξ ξ I N ppot two dd α % ofd tv fo d ptvy gv y ˆ ˆ ˆ / / / ξ ± ξ ± ξ ± α α α I z I z I z ˆ / ξ ± α I z wh α z th upp th - α pt of th tdd o dtuto. Th t of th ptd Fh foto t β og L { β β } β β og β L β β β

24 M. MHMOUD t. 9 og β L L og 4 4

25 NW GNRLIZTION OF KUMRSWMY 9 ; og β L og β L L og 4 4

26 M. MHMOUD t. 94 ; og L L og 4 4 ; L og ω τ τ τ

27 NW GNRLIZTION OF KUMRSWMY 95 ω ω τ ω τ ω ω ω wh τ ; 4 4

28 M. MHMOUD t. 96 ω Nu ppto I th to w utt th ufu of th KL dtuto. 6.. Nu p dt t ud to how tht th KL dtuto tt od th th Ldy dtuto. Th dt t gv T pt uod dt t opodg to o t ( oth) of do p of 8 dd ptt potd L d Wg [].

29 NW GNRLIZTION OF KUMRSWMY 97 T. Th o t ( oth) of dd ptt Th u hood thod ppd to tt th pt of th th od Ldy (L) Kuwy Ldy (KL) d pottd Kuwy Ldy (KL) dtuto. Th utg tt wth th gtv of th hood futo ( ). T. Th ML ttd d Log-hood fo dt t Mod ML tt L ˆ ˆ.5 KL KL ˆ ˆ ˆ ˆ ˆ ˆ

30 98 M. MHMOUD t. Th v ov t I ( ξ) of th ML ud th KL ˆ dtuto fo th dt t oputd Thu v ( ˆ ).98 v( ˆ ).587 v( ˆ ).9 d v ( ˆ ).4. Thfo 95% ofd tv fo d [ ] [.7.659] [.88.6] d [.88.59] ptvy. Th LR tt ttt to tt th hypoth H : vu ω 7.8 > 7.85 o H χ.5 w t th u hypoth. Th vu of th pt tt ud to pot th pdf d th df fo th th dtuto L KL d KL Fgu 4 d Fgu 5.

31 NW GNRLIZTION OF KUMRSWMY 99 Fgu 4. ttd dt of th od fo dt t. Fgu 5. ttd uutv dt of th od fo dt t.

32 M. MHMOUD t. T. Ct opo Mod IC ICC BIC Ldy Ku-Ldy pku-ldy I od to op th th dtuto w od t IC ( foto to) ICC (otd foto t) d BIC (By foto to) fo th dt t. how T th tt dtuto opod to IC ICC d BIC vu wh IC K ; ICC IC ( ) ; BIC og( ). H th u of pt th ttt od th p z d th zd vu of og-hood futo ud th odd od. T how th ML ud th th dtuto T how th vu of IC ICC d BIC vu. T dt tht th KL dtuto d to tt ft th th Ldy d KL dtuto. 6.. Suto tudy W odutd Mot Co uto tud to o th ft p hvo of th ML of d. ut w otd fo Mot Co pto uto. Th KL do u gto w pfod ug th vo thod. I h pto do p of z dw fo th KL ( ) dtuto d th u hood tt of th pt w otd. Th d qud o (MS) fo h pt w oputd ud dfft p z d.

33 NW GNRLIZTION OF KUMRSWMY T 4. M tt d qu o of th ML of. Pt M B MS

34 M. MHMOUD t. T 5. M tt d qu o of th ML of 5.5 Pt M B MS Couo W popod w dtuto d th KL dtuto whh td th KL dtuto th y of dt wth uppot. ovou o fo gzg tdd dtuto u th gzd fo povd g fty odg dt. W hv dvd vou popt of th w dtuto udg th ot ot gtg futo d qut futo. Réy d Sho top hv otd. Th tto of th od

35 NW GNRLIZTION OF KUMRSWMY pt ppohd y u hood d th ovd foto t dvd. Fy ppto to dt t how tht th ft of th w od upo to th ft of t uod. W hop tht th popod od y ttt wd ppto ttt. Rf [] R. D. Gupt d D. Kudu Gzd pot dtuto ut d Nw Zd ou of Sttt 4 (999) [] R. D. Gupt d D. Kudu Gzd pot dtuto: tg ut d o t dvopt ou of Sttt Pg d If 7 (7) [] M.. Ghty M.. B. th d S. Ndh Ldy dtuto d t ppto Mtht d Coput Suto 78 (8) [4] M. S Th dt Poo-Ldy dtuto Bot 6 (97) [5]. Mzuh d.. h Th Ldy dtuto ppd to optg ft dt Coput Mthod d Pog Bod 4 () [6] M.. Ghty d D. K. L-Mut Sz-d Poo-Ldy dtuto d t ppto Mto.-It.. Stt. LXVI () (8) 99-. [7] M.. Ghty D. K. -Mut d S. Ndh Zo-tutd Poo-Ldy dtuto d t ppto Mtht d Coput Suto 79 (7) [8] H. S. Bouh B. M. -Zh.. L-Sho V... Mh d F. Louzd tdd Ldy dtuto ou of Ko Sttt Soty 4 () [9] M.. Ghty F. -Qf D. K. -Mut d H.. Hu two pt wghtd Ldy dtuto d t ppto to uvv dt Mtht d Coput Suto 8 () 9-. [] M.. Ghty d D. K. -Mut tto thod fo th dt Poo- Ldy dtuto ou of Sttt Coputto d Suto 79 (9) -9. [] S. R d. Mh qu Ldy dtuto f ou of Mtht d Coput S Rh 6(4) () [] M.. Ghty D. K. -Mut N. Bh d L.. -z Pow Ldy dtuto d otd f Coputto Sttt d Dt y 64 () -.

36 4 M. MHMOUD t. [] H. Zzdh d. Dot Gzd Ldy dtuto ou of Mth. t. () (9) -5. [4] I. t d M. ghy Sttt popt of Kuwy qu Ldy dtuto Itto 4 () [5] S. Cyp d G. O. Kd w uto ft duto dtuto Th Kuwy Ldy dtuto Itto ou of Td oo d F 5 (4) [6] P. Kuwy gzd poty dty futo fo dou oudd do po ou of Hydoogy 46 (98) [7] M. C. o Kuwy dtuto: t-typ dtuto wth o ttty dvtg Sttt Mthodoogy 6 (9) 7-8. [8] G. M. Codo d M. Cto w fy of gzd dtuto ou of Sttt Coputto d Suto 8 () [9] G. M. Codo. M. M. Otg d S. Ndh Th Kuwy Wu dtuto wth ppto to fu dt ou of th F Ittut 47(8) () [] M. Bougugo R. B. Sv L. M. Z d G. M. Codo Th Kuwy Pto dtuto ou of Sttt Thoy d ppto () () [] P. F. P. M. M. Otg G. M. Codo d M.. d Po Th Kuwy Bu XII dtuto: Thoy d pt ou of Sttt Coputto d Suto 8 () 7-4. [].. Go C. Q. d-sv G. M. Codo d. M. M. Otg w ft od: Th Kuwy gzd Rygh dtuto ou of Sttt Coputto d Suto 84 (4) 9-9. [] M.. d Po. M. M. Otg d G. M. Codo Th Kuwy gzd g dtuto wth ppto uvv y Sttt Mthodoogy 8(5) ( ) 4-4. [4] D. V. Ldy Fdu dtuto d By tho ou of Roy Sttt Soty S B (958) -7. [5] G. Mudho d D. Svtv pottd Wu fy fo yzg thtu fu dt I Tto o Rty 4 (99) 99-. [6] M. M. N d F. H. O th pottd Wu dtuto Couto Sttt- Thoy d Mthod () 7-6. [7] M. M. N d F. H. By tto fo th pottd Wu od Couto Sttt - Thoy d Mthod (4) 4-6. [8] I. S. Gdhty d I. M. Ryzh T of Itg S d P 98.

37 NW GNRLIZTION OF KUMRSWMY 5 [9]. Réy O Mu of topy d Ifoto I Podg of th Fouth By Sypou o Mtht Sttt d Poty By Uvty of Cfo P (96) [] C.. Sho Pdto d topy of Ptd gh Th B Syt Th ou (95) []. L. L d. W. Wg Sttt Mthod fo Suvv Dt y d dto Wy Nw Yo. g

CBSE SAMPLE PAPER SOLUTIONS CLASS-XII MATHS SET-2 CBSE , ˆj. cos. SECTION A 1. Given that a 2iˆ ˆj. We need to find

CBSE SAMPLE PAPER SOLUTIONS CLASS-XII MATHS SET-2 CBSE , ˆj. cos. SECTION A 1. Given that a 2iˆ ˆj. We need to find BSE SMLE ER SOLUTONS LSS-X MTHS SET- BSE SETON Gv tht d W d to fd 7 7 Hc, 7 7 7 Lt, W ow tht Thus, osd th vcto quto of th pl z - + z = - + z = Thus th ts quto of th pl s - + z = Lt d th dstc tw th pot,,