KUMARASWAMY WEIBULL-GENERATED FAMILY OF DISTRIBUTIONS WITH APPLICATIONS

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1 Advs d Aos Sss 26 Ps Ps Hos Ad Id Psd O: My 26 ://d.do.o/.7654/as48325 Vo 48 N 3 26 Ps ISSN: KUMARASWAMY WEIBULL-ENERATED FAMILY OF DISTRIBUTIONS WITH APPLICATIONS A S. Hss d M. Ey 2 Is of Ss Sds d Rs (ISSR) D of M Sss Co Uvsy Ey -: 2 D of M Sss Byd Cos Qss Sd A -: As I s w od d sdy w fy of oos dsos d Kswy W-d ( KwW - ) fy of dsos w s so of W- fy of dsos oosd y Boo [3. T w fy ds sv ow ods. W o ssos fo fo os oy wd os fo dvo d od sss. M s of ood od w dsss od Rvd: Fy 4 26 Rvsd: M 26 Ad: M Ms S Cssfo: 62-XX. Kywods d ss: Kswy W dso W- fy os od sss ood so. Cod y K. K. Ad

2 26 A S. Hss d M. Ey s. Fo sds sow sfss of o oosd fy of dsos y osd d s vd of s fy o ofo o sss of f ods s od.. Iodo Ov s dds sv ss dsos v svy sd d d sv s s s d ss doy. Howv y sos ss dsos o s fo ds d d wod o. Ry s sd s df w os o d sss of v oos dsos y od ddo s (s) o s od. T w fs dd w-ow dsos d s ovd fy od d. E. [8 odd w ss of d dsos sd o dso d -d fy of dsos. Zofos d Bs [4 oosd d fy. Codo d d Cso [4 odd Kswy d dsos. Fo y v dso fo (df) Rs d Bs [3 oosd v o fo y dso. Ps. [ oosd K -d fy of dsos fo y df. Codo. [5 oosd od d ss of dsos. A. [ dd -d fy of dsos y s y o-v oos do v T s o of do v. Ty dfd s ss s foows: W ( ) F ( ) d () w () s oy dsy fo (df) of o-v

3 Kswy W-d Fy of Dsos 27 oos do v T. I s w ss dso of do v T s o d s sfoo W ( ) o[. Ts d fy of dsos s d T-X dso. I s o df dff fo dff ys of T-X dsos. T sfoo W ( ) ssfs foow wo odos: W ( ) [ ) d s ooo o ds fo. Ry Boo. [3 oosd W- ( W - ) fy of dsos. Ty osdd oos dso ( ) w dsy fo () d W df R() ( fo > ) w osv s d. Bsd o s dsy y W ( ) () w ( ) ( ) [ ( ) ( ) y dfd df of fy y ( ) ( ) F d ( ) D R > w ( ) s s df w dds o vo. I s w so of W - fy of dsos d Kswy W- fy of dsos s oosd. T s of s s od s foows: I So 2 Kswy W- fy of dsos s dfd. So os of fy dsssd So 3. So s of oosd fy dsssd So 4. I So 5 so of od s s fod y od of ood. A sv o sd o d s d So 6. Fy od s sd So 7.

4 28 A S. Hss d M. Ey 2. T Kswy W-d Fy I s so w ss of dsos d dsos s odd. KwW - fy of Codo. [6 sd KwW dso w foow df d df: d R() [ [ > > () [ [ [ > >. (2) W o w fy y o ( ) dfd () y df o dfd (2) s foows: ( ) F [ [ [ T dso fo of d. KwW - fy s foow fo: F [ ( ( ) ) > > (3) w > s s d > s s. T df (3) ovds wd fy of oos dsos. T df osod o (3) s v y f ( ) ( ) ( ) [ ( ) ( ( ) ) ( ) ( ( ) ) [ ( ( ) ) > >. (4) Hf do v X w df (4) s dod y X ~ KwW -.

5 Kswy W-d Fy of Dsos 29 No. () Fo KwW - dso ds o w fy d od W-d ( EW - ) fy of dsos. () Fo KwW - dso ds o W - fy of dsos w s od y Boo. [3. T svv d vsd d d v d fos od svy s foows: F [ ( ( ) ) > > τ ( ) ( ) ( ( ) ) [ ( ( ) ( ) [ ( ( ) ( ) ( ) [ ( ) ( ) ) ( ) ( ) ( ) [ ( ) ( ) ( ) ) ( ) ( ) { [ ( ) } d H F [ ( 3. Ss Pos I s so so os of s foows: 3.. Q d d ( ) ). KwW - fy w od Q fos sd o ss of oy oy

6 2 A S. Hss d M. Ey ss os d sos. So ods fo o od sd do vs fo ss d w oos dsos. T fo sy Q F of X s v y [ ( ( Q ) Q ( ) ) f so sfos ds o foow fo: { [ [ } Q (5) { [ [ ( ) } w s osdd s fo do v o v ( ) d () s vs fo of ( ). I d dvd fo (5) y s.5.. d s v y { [ [ (.5) } Md. { [ [ (.5) } 3.2. Esos fo dso d dsy fos I s so so sos of df d df fo Kswy W- fy of dsos w sd. T o v ow w sf s so. I s w-ow f > s o d < d o o s w s foows: ( ) ( ). (6) T y y o o (6) (4) oy dsy fo of KwW - fy w s o os

7 Kswy W-d Fy of Dsos 2 f ( ) ( ) ( ( ) ) ( ) ( ) [ ( ) ( ) ( ). T s o so vos o w s o ds o: f ( ) ( ) ( ( ) ) ( ) (7) y s ow ss fo o fo w o. (8)! ( ) Is so (8) (7) w v f ( ) ( ) ( )! [ ( ) ( ) [ ( ) [ ( ). (9) Now s d o o w w [ ( ) ( ) ( ( ) ). () Is () (9) oy dsy fo of KwW

8 A S. Hss d M. Ey 22 ssd s f oo of od- (- fo so) dsy fos.. f () W f (2) w! W d. W If d s d sos sos d sos. Ao fo ydd w s o y dd d s o o () s foows:. f Us o so yds:. f Us o so o yds:

9 Kswy W-d Fy of Dsos 23. f So so fo of df s s foows: f (3) w. Fo so fo v fo s dvd s foows: Us o so fo F w s ds o: [ ( ) F. Us o so o w s o ds o ( ) F. Us o so w s o ds o F y s ow ss fo o fo (8) vos o w o

10 A S. Hss d M. Ey 24 F! y s o () vos o w s o w o F! F.! Us o so ds o F.! Us o so w s ds o F!. R w yd F!. Fy

11 Kswy W-d Fy of Dsos 25 s F (4) w s.! 3.3. Mos If X s df () s o od o foow o:. d f X E (5) Fs w s ss () o (5) yds:. d X E T τ w τ s oy wd os (PWMs) of s dso. Sod w s o ss (3) o (5) yds:. d X E T τ.

12 26 A S. Hss d M. Ey Ao fo sd o fo: Fs w s yds ( ) ( ) Q d. Sod w s o yds ( Q ) d T oy wd os A oy fo PWMs ovs so d dso of o oy dsos so of s s of oy dsos d yoss s fo oy dsos. T PWMs od y sd fo s s of dso wos vs fo o ssd y. T oy wd os od fo foow o: s τ s E[ X F f ( F ) d. (6) Fs w s y ss () d (4) o (6) w s ds o: ( ) ( ) τ s s d. T τ s s τ ( ). s

13 Kswy W-d Fy of Dsos 27 Sod w s o y ss (3) d (4) o (6) w s ds o: τ. s d s T τ τ. s s Ao fo sd o fo: Fs w s yds τ. s d Q s Sod w s o yds τ. s d Q s 3.5. T o fo y o fo s () X X E M y s o so w s ().! X X E M Ao fo sd o fo: fs w s yds (). Q d M

14 28 A S. Hss d M. Ey Sod w s o yds M () ( Q ) d T dvo Bsy dvo s s fo o of s X. y dvo s ssd y: δ ( X ) 2F ( ) 2T ( ) d δ 2( X ) 2T ( M ) w T ( ) f d w s fs o o. Ao fo sd o fo: Fs w s yds ( T Q ) Sod w s o yds d. T ( ) Q d Od sss L X () < X( 2) < < X( ) od sss of do s of s foow Kswy W- fy of dsos w s d. T df of od ss s od Dvd [7 w s foows: f X ( ( )) f ( ( ) ) B( ) ν ν ( ) F( ( ) ) ν ν. (7)

15 Kswy W-d Fy of Dsos 29 Fs w s ss () d (4) (7) w ν ds o f ( ( )) X ( ( ) ) B( ) ( ) v ν ( ) (8) w ν s. ν ν Sod w s o ss (3) d (4) (7) w ν ds o f X ( ( )) ( ( ) ) B ν v ( ) (9) w () d () dsy d v fos of KwW - dsos svy. Moov o of od sss s od s foows: E ( X ) f ( ( ) ) d( ). (2) Fs w s ss (8) (2) ds o ( ) E X τ ( ) B v ( ) ν.

16 22 A S. Hss d M. Ey E Sod w s o ss (9) (2) ds o ( ) X B ν 4. So S S-ods H w dsss fw s of 4.. KwW-fo dso vτ. KwW - fy of dsos. As fs sos dso s fo v < < < d ( ). Tfo KwW-fo (KwWU) dso s foow df df y d ss ( ) (3) d (4) s foows: F [ ( ) > < < f ( ) [ [ [ > < <. Fo svv d d vsd d fos od svy s foows: F [ ( ) [ [ [

17 Kswy W-d Fy of Dsos 22 d [ [ [. τ 4.2. KwW-BXII dso L s osd BXII dso w df d df v y > d svy. T KwW-BXII (KwWBXII) dso s foow df df svv d d vsd d fos: [ > > F [ f [ F

18 A S. Hss d M. Ey 222 [ d [ [ [ τ w > > s s > s s KwW-W dso If do v X foows W dso w df v y > w d s d s s. T df d df of KwW-W (KwWW) s dvd fo KwW- fy of dsos s foows: [ ( ) > > F { } ( ) f [ ( ).

19 Kswy W-d Fy of Dsos 223 Addoy svv d d vsd d fos of KwWW foow fos: [ ( ) F { } ( ) ( ) d { } ( ) [ ( ) [ ( ) τ svy. No fo KwW-W dso ds o KwW-o dso KwW-s Ldy dso Qs Ldy dso s odd y S d Ms [2. Ty dfd df d df of s Ldy w wo s s foows:. T df d df fo Kswy W s Ldy (KwWQL) dso dvd fo (3) d (4) svy s foows: [ > > > F

20 A S. Hss d M. Ey 224 [ ( ) 2 f [. Fo svv d d vsd d fos od s foows: [ ( ) F ( ) [ ( ) 2 d [ [ ) ( ) [ [ ( ) τ 2 [ svy. Fo KwW-Ldy dso w od.

21 Kswy W-d Fy of Dsos 225 Pos of df d d fo fo w fo s dff dsos ddd fo KwW- fy fo so vs dsyd Fs d 2 svy. () () () (d) F. Dsy os () KwWU () KwWBXII () KwWW (d) KwWQL.

22 226 A S. Hss d M. Ey () () () (d) F 2. Hd os () KwWU () KwWBXII () KwWW (d) KwWQL.

23 Kswy W-d Fy of Dsos M Lood Eso T ood sos of ow s of KwW- fy fo o ss dd. L X... X osvd vs fo KwW- dso w s d. L Θ ( ) T vo. T o o-ood fo fo vo of s Θ ssd s L( Θ) ( ) [ ( ) [ ( ) [ ( ) [ ( ) H ( ) [ [ H ( ) [ ( [ H ( ) ) ( ) w H ( ) ( ). T s of so fo U ( Θ) ( U U U U U ) v y U [ ( [ ) ( [ H H H ( ) ( [ H ) ) U [ ( [ H ( ) ) U [ H ( ) ( ) [ H ( ) [ ( ) H [ H ( ) ( ) [ H [ [ H H [ [ H

24 228 A S. Hss d M. Ey U [ H ( ) [ H ( ) [ H ( ) ( ) ( ) [ H ( ) [ H ( ) [ H ( ) [ ( ) H [ [ H [ [ H H ( ) [ H ( ) [ H ( ) d U ( ) ( ) ( ) ( ) ( ) ( ) ( ) H ( H ) ( ) [ H ( ) [ H ( ) H ( ) [ ( ) H ( ) [ H ( ) [ H ( ) H ( ) [ [ H H ( ) [. S U U U U d U o o d sov os sosy yds ood so Θ T ( ) of Θ ( ). Ts os o sovd yy d ss sofw sd o sov y. Fo v so of s 5 5 osvd foo I { } fo ( v ) wos s Θ I ν

25 Kswy W-d Fy of Dsos 229 sod dvvs of o-ood ( L) fo. Ud y odos ow syo os of ood d od s : ( Θ Θ) N( I ( Θ)) s w d s ov dso w o d ov I ( Θ) ( ω)% ofd v fo Θ ( ) s v s foows: Θ ± Z ω 2 v( Θ) w Z ω 2 s ( ω 2) of sdd o dso d v () do do s of I ( Θ) osod o od s. 6. Aos I s so w fy d soy of w KwWE s s dso fo Kswy W- fy oosd. W d ss o o fs of KwWE w od W o (EWE) dso d W o (WE) dso w foow osod dss: d f f KwWE EWE [ { } ( ) [ ( ) > > [ { } ( ) > > f WE { ( ) } > >.

26 23 A S. Hss d M. Ey I d s s sd y ood od. A foo o (AIC) o A foo o (CAIC) Bys foo o (BIC) H foo o (HQIC) 2 L Koooov-Sov ( K - S ) d -v sss osdd o o ods. T fo fo s s s foows: AIC 2 2 L CAIC AIC 2( ) BIC ( ) 2 L d HQIC 2 [ ( ) 2 L w s of s ss od s s s d L s d v of o-ood fo d osdd od. Aso s s y[ F ( y) F( y) w F ( y) ( of osvo y) d F ( y) dos df. I s dso osods o ss vs of 2 L AIC BIC CAIC HQIC K-S d s v of -v. 6.. T fs d s T fs d s ws fo Hy [9 w osss of y sssv vs of M o ( s) Mos/S P. T d s s s foows: T ovds ood ss (MLEs) of od s. Aso vs of sss AIC BIC CAIC HQIC d Koooov-Sov vs s w s -v sss sd T. Fs 3 d 4 ovd os of sd v d sd dss of fd KwWE EWE d WE ods fo fs d s.

27 Kswy W-d Fy of Dsos 23 T. T MLEs AIC CAIC BIC HQIC K-S d -v of ods sd o fs d s Mod KwWE EWE WE P Ess ˆ 3.9 ˆ ˆ ˆ ˆ ˆ ˆ.578 ˆ. 58 ˆ. 572 ˆ ˆ ˆ. 2 2 L AIC CAIC BIC HQIC K-S v Bsd o vs of AIC CAIC BIC HQIC K-S d -v T KwWE od ovds f EWE d WE ods.

28 232 A S. Hss d M. Ey F 3. Esd v dss fo fs d. F 4. Esd dss of ods fo fs d.

29 Kswy W-d Fy of Dsos 233 I s fo F 4 fd dsy fo KwWE od s os o so EWE d WE ods T sod d s T sod d s ws fo My. [. T osod d s f o w fs fo. T d s s s foows: T foow T 2 ovds MLEs AIC BIC CAIC HQIC d K-S vs s w s -v sss fo sod d s. T 2. T MLEs AIC CAIC BIC HQIC K-S d -v of ods sd o sod d s Mod KwWE EWE WE P ss ˆ ˆ ˆ ˆ 6.52 ˆ ˆ ˆ.593 ˆ. 58 ˆ. 38 ˆ ˆ ˆ. 5 2 L AIC CAIC BIC HQIC K-S v

30 234 A S. Hss d M. Ey I od o ssss w KwWE od s o Fs 5 d 6 ovd so os of sd df s w s sd oy dss of fd KwWE EWE d WE ods fo sod d s. F 5. Esd v dss fo sod d. F 6. Esd dss of ods fo sod d.

31 Kswy W-d Fy of Dsos 235 Fo F 6 s odd KwWE dso s s fo s d s T d d s T foow d s svv s ( dys) of 72 s fd w v osvd d od y Bd [2. T d s s foows: T foow T 3 vs MLEs AIC BIC CAIC HQIC d K-S vs s w s -v sss fo d d s. T 3. T MLEs AIC CAIC BIC HQIC K-S d -v of ods sd o d d s Mod KwWE EWE WE P Ess ˆ ˆ ˆ ˆ ˆ ˆ ˆ.556 ˆ. 6 ˆ. 584 ˆ ˆ. ˆ. 2 2 L AIC

32 236 A S. Hss d M. Ey CAIC BIC HQIC K-S v T vs T 3 d KwWE od s so oo o o dsos ws sd o f s d s. Fs 7 d 8 ovd so os of sd df d df of fd KwWE EWE d WE ods fo d d s. F 7. Esd v dss fo d d.

33 Kswy W-d Fy of Dsos 237 F 8. Esd dss of ods fo d d. Bsd o os of F 8 o o fd dsy of KwWE od s os o so osod dss of EWE d WE ods. 7. Coso I vw of d of fy of W- dsos d Kswy dso w od w fy of dsos d Kswy W-. T Kswy W- fy ds sv wdy ow dsos. A so fo dsy fo d ssos fo os oy wd os fo dvo fo d od sss od. T so of s s od y od of ood. So os of w fy o d v o dos o d sfss of ssd fy. I s sdy w ws s w o w

34 238 A S. Hss d M. Ey ody os sv s s s svv d f d ydooy. Aowd T os f o do d vw wos os d o ov of. Rfs [ A. A C. L d F. Foy A w od fo fs of oos dsos Mo 7 (23) [2 T. Bd Aso of ss s fd w dff doss of v A Jo of Hy 72 (96) [3 M. Boo R. B. Sv d. M. Codo T W- fy of oy dsos Jo of D S 2 (24) [4. M. Codo d M. d Cso A w fy of d dso J. Ss. Co. S. 8 (2) [5. M. Codo E. M. M. O d D. C. C. d C T od d ss of dsos Jo of D S (23) -27. [6. M. Codo E. M. M. O d S. Nd T Kswy W dso w o o f d Jo of F Is 347 (2) [7 H. A. Dvd Od Sss 2d d. Wy Nw Yo 98. [8 N. E C. L d F. Foy T -o dso d s os Co. Ss. Toy Mods 3 (22) [9 D. Hy O o of ow sfoos Jo of Roy Ss Soy Ss () Ad Sss 26 (977) [ D. N. P. My M. X d R. J W Mods Jo Wy Nw Jsy 24. [ R. R. Ps. M. Codo C.. B. Do E. M. M. O d S. Nd T w ss of K d dsos Sss d Oos Rs Tsos 36 (22) 53-8.

35 Kswy W-d Fy of Dsos 239 [2 R. S d A. Ms A s Ldy dso Af J. M. Co. S. Rs. 6 (23) [3 M. M. Rs d N. Bs T -od o dso J. Ss. Co. S. 82 (22) [4 K. Zofos d N. Bs O fs of - d d d dsos d ssod f Ss. Mod. 6 (29)

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