LIFE LENGTH OF COMPONENTS ESTIMATES WITH BETA-WEIGHTED WEIBULL DISTRIBUTION

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1 Jourl of Sisics: Advces i Theory d Applicios Volume Numer 2 24 Pges 9-7 LIFE LENGTH OF COMPONENTS ESTIMATES WITH BETA-WEIGHTED WEIBULL DISTRIBUTION N. IDOWU BADMUS d T. ADEBAYO BAMIDURO Deprme of Sisics Arhm Adesy Polyechic Ieu-Igo Nigeri e-mil: idowuolsukmi869@yhoo.com Deprme of Mhemicl Scieces Redeemer s Uiversiy Mowe Ogu Se Nigeri Asrc Two mi disriuios re comiig y usig he logi of e fucio y Joes []. The weighed Weiull disriuio proposed y Shhz e l. [9] d e disriuio i order o hve eer disriuio (e-weighed Weiull disriuio) h ech of hem idividully i erms of he esime of heir chrcerisics i heir prmeers. We sudy d provide comprehesive reme of he mhemicl properies of he e weighed Weiull disriuio d derive epressios for is momes d mome geerig fucio survivl re fucio hzrd re fucio skewess d kurosis coefficie of vriio d sympoic ehviours. We lso discuss mimum likelihood esimio d provide formule for he elemes of he Fisher iformio mri. The ew disriuio is pply o lifeime d se d clerly shows h i is much more fleile d hs eer represeio 2 Mhemics Suec Clssificio: A69 A7 3C98. Keywords d phrses: e-weighed Weiull Weiull mome geerig fucio kurosis skewess. Received Mrch Scieific Advces Pulishers

2 92 N. IDOWU BADMUS d T. ADEBAYO BAMIDURO of d h weighed Weiull disriuio. We hope h his model my rc wider pplicio i iology iomedicl eviromeric d lifeime d lysis.. Iroducio The Weiull disriuio hs ee powerful proiliy disriuio i reliiliy lysis; he weighed Weiull disriuio is used o dus he proiliies of he eves s oserved d recorded; while he e disriuio is oe of he skewed disriuios used o descriig uceriy or rdom vriio o sysem. Pil d Ro [8] ivesiged how for isce ruced disriuios d dmged oservios c give rise o weighed disriuio d Azzlii [3] proposed model h c e used s lerive o Gmm d Weiull disriuio; d Shhz e l. [9] followed he sme ide of Azzlii s mehod o proposed model h is slighly modifyig wih ddiiol prmeer clled weighed prmeer. The proiliy desiy fucio of he ww ( λ ) disriuio is give y λ ep ( λ ) [ ep ( λ )] λ > f () d he ssociig cumulive disriuio fucio is give y F [{ ep } { ep ( ( ) X )}]. (2) Sudies o geerlized forms of weighed Weiull disriuio re scy. This pper is rrged s follows: We iroduce he ew proposed e weighed Weiull disriuio (BWW) icludig he desiy d disriuio fucio he sympoic ehviours survivl re fucio hzrd re fucio ec. d specil models hese were sudies i Secio 2. I Secio 3 we discussed mome d mome geerig fucio. Secio 4 cois he prmeer esimio i Secio 5 empiricl pplicio o lifeime d se d Secio 6 cocludes he sudy.

3 LIFE LENGTH OF COMPONENTS ESTIMATES Mehods 2.. The ew proposed e weighed Weiull disriuio Numerous works hve ee doe cocerig e disriuio comied wih oher disriuios i priculr fer rece works of Eugee d Fmoye [6] d Joes [] e geerlized logisic (Moris e l. [4]) e log-logisic (Lemoe []) e-hyperolic sec (Fischer d Vugh [8]) e-gumel (Ndrh d Koz [5]) momes of he e Weiull (Cordeiro e l. [4]) e Weiull (Fmoye e l. [7]) e epoeil (Ndrh d Koz [6]) e Preo (Akisee e l. []) e Ryleigh (Akisee d Lowe [2]) e modified Weiull (Cordeiro e l. [5]) e epoeied Preo (Ze e l. [2]) e Nkgmi (Shiu d Adepou [2]) e-e mog ohers. Now le X e rdom vrile form he disriuio wih prmeers d defied i () usig he logi of e fucio y Joes [] g B( ) [ G ] [ G ] g. (3) The e weighed Weiull BWW ( λ ) disriuio is oied s follows: g BWWD B( ) ( ) ( ) λ λ λ (4) where G F d g f λ d > ~ BWWD( λ ).

4 N. IDOWU BADMUS d T. ADEBAYO BAMIDURO 94 I (4) ove whe i ecomes epoeied weighed Weiull whe i ecomes Lehm ype II weighed Weiull (Bdmus e l. [7]) d whe he disriuio lso leds o e weighed disriuio (ll re ew specil su-models). The se i ecomes weighed Weiull disriuio (pre disriuio). Agi ssume we se λ i (4) we hve BWWD B g. (5) Such h. ~ BWWD X From (5) se (6) i.e. d d puig d io (5) we ge BWWD B g d (7)

5 LIFE LENGTH OF COMPONENTS ESTIMATES 95 h is ( ). λ λ λ y e y e y y Equio (5) ecomes he proiliy desiy fucio of BWW disriuio d c e rewrie s [] [ ]. B g BWW (8) Figure. Figure shows he grph of pdf of he BWWD disriuio d i is cler h ideed i is righly skewed d where c d. d 2... Cumulive desiy fucio The cumulive disriuio fucio (cdf) is derivig d he epressio i (7) is give s d f X P G BWWD B d (9)

6 96 N. IDOWU BADMUS d T. ADEBAYO BAMIDURO susiuig (5) i (8) we oi GBWWD P( X ) G T B( ) ( T ) d; B( ; ) T ( T ) d () B( ) B( ) BWWD where B ( ; ) is clled icomplee e fucio. Figure 2. This figure lso shows he grph of he CDF of he BWW disriuio for 2 3 c 4 d d Survivl re fucio The survivl re fucio of rdom vrile y wih cumulive disriuio fucio G ( y) is give y SBWW G where G BWW equl o () herefore S BWW BWW B( ) B( ; ). () B( )

7 LIFE LENGTH OF COMPONENTS ESTIMATES Hzrd re fucio hbww gbww G BWW BWW where g B( ) ( ) d G BWW s i (). Susiuig g BWW d G BWW i he ove epressio of hzrd fucio we oied he hzrd re fucio of he BWW disriuio s give elow: hbww B( ) ( ) B( ) B( ; ) (2) where is he disriuio i (6). Figure 3. The grph of he hzrd re of he e weighed Weiull disriuio for 2.5 d 2 is he show The sympoic ehviour The sympoic properies of he BWW disriuio re emied y cosiderig he ehviour of lim g BWW d lim g BWW s follows: lim gbwwd lim B( ) ( )

8 98 N. IDOWU BADMUS d T. ADEBAYO BAMIDURO ( ). For he ske of simpliciy we ke he limi of lim lso lim. This hs show h les oe mode eiss. Accordig o lierure wheever d he he PDF lso eds o zero hece he BWW disriuio hs mode Specil models Specil su-models The desiy (4) is impor sice i lso icludes s specil sumodels some disriuios o previously focused o or cosidered i he eisig lierure. The ew specil su-models re give elow: () Whe i (4) he disriuio ecomes e weighed epoeil disriuio (ew). () Seig i (4) reduces o weighed epoeil (WE) (Gup d Kudu [9]) disriuio d weighed Weiull (WW) whe d (Mhdy Rmd [3]). (c) Whe desiy (5) ecomes Lehm ype II weighed Weiull (LWW) (Bdmus e l. [7]) disriuio (ew).

9 LIFE LENGTH OF COMPONENTS ESTIMATES 99 (d) The desiy (5) c lso e simplifies o ew epoeied weighed Weiull disriuio whe (ew) d he WW disriuio eig he pre disriuio (s eemplr) whe (Shhz e l. [9]). 3. Momes d Mome Geerig Fucio Hoskig [2] descried d used i Bdmus e l. [7] h whe rdom vrile followig geerlized e geered disriuio h is ~ GBG( f c) r he µ E[ F c ] where ~ B ( ) c r is cos d F is he iverse of CDF of he weighed Weiull disriuio sice BWW ( ) disriuio is specil form whe c. We he derive he mome geerig fucio (mgf) of he proposed disriuio m () E( e ) d he geerl r-h mome of e geered disriuio is defied y r µ r [ F ] [ ] dy. (3) B( ) Cordeiro e l. [5] discussed oher mgf of y for geered e disriuio () ρ( ; ) ( ) i M k r B (4) where m ρ ( k r) e [ F ] f d. Therefore () [ ] ( e F ) M f d B (5)

10 N. IDOWU BADMUS d T. ADEBAYO BAMIDURO y susiuig oh pdf d cdf F f & of he weighed Weiull disriuio io (5) we hve () BWWD B M (6) seig i (6) gives he mome geerig fucio of he pre disriuio. Now o oi he r-h mome of he e-weighed Weiull disriuio we hve he followig: Sice he mome geerig fucio of weighed Weiull disriuio y Shhz e l. [9] is give y () d M { }.! r (7) Equio (5) c e wrie s () i BWWD i B M { } k i

11 LIFE LENGTH OF COMPONENTS ESTIMATES { } () ( ) i M BWWD B i! i The r-h mome of BWW disriuio c e wrie from (8) s µ E X BWWD r B r i i ( i ). (8) ( ) ( i ) r r { ( ) } r. r! (9) Puig i (9) leds o he r-h mome of he pre disriuio y Shhz e l. [9] d is give y µ r r r r { ( ) } r. E X I is he esy o oi he momes d oher mesures like he coefficie of vriio ( V BWWD ) of BWWD ( ) he skewess S BWWD ( ) d kurosis KR BWWD ( ) c lso e esily oied i eplici forms from (9). 4. Prmeer Esimio A emp is mde o oi he mimum likelihood esime MLEs of he prmeers of he BWW disriuio. Now le θ e vecor of prmeers Cordeiro e l. [5] gve he log-likelihood fucio for θ ( c τ) where τ ( ) d eve ws used y Bdmus e l. [7].

12 2 N. IDOWU BADMUS d T. ADEBAYO BAMIDURO L( θ) log C log[ B( ) ] log[ f ( X τ) ] i i i c ( ) log F ( X τ) ( ) log[ F ( X τ)] (2) seig c reduces he clss of geerlized e disriuio o he clss of e geered disriuio. The we oi θ ( τ) give s L( θ) log[ B( ) ] log[ f ( X τ) ] ( ) log F ( X τ) i i c ( ) log[ F ( X τ) ] i where f ( ; τ) d F ( ; τ) s i () d (2) ove. ( θ) LBWWD log[ B( ) ] i log ( ) log i log ( ) i. (2) Usig differeil equio i (2) wih respec o ( ) d recll h B( ) ( ) ( ) ( )

13 LIFE LENGTH OF COMPONENTS ESTIMATES 3 θ log L. (22) log θ L. (23) θ L. (24) θ L. (25)

14 4 N. IDOWU BADMUS d T. ADEBAYO BAMIDURO Equios (22)-(25) c e solved y usig Newo Rlphso mehod o oi he ˆ ˆ ˆ ˆ he MLE of ( ) respecively. Tkig secod derivives of equios (22) (23) (24) d (25) wih respec o he prmeers ove we c derive he iervl esime d hypohesis ess o he model prmeer d iverse of Fisher s iformio mri eeded. 5. Resuls d Discussio 3.. Applicio o lifeime d This secio we pply he d se ivesiged y Shhz e l. [9] o life compoes i yers o compre he BWW d WW disriuio. The d cois life (grouped d sy (.. 2. > 5.) d he d se (secodry d) cosiss of frequecy ll ogeher 2369 o compre ewee he resuls of he proposed BWW LWW EWW d WW. Usig R sofwre (codes) o deermie some descripive sisics d he mimum likelihood esimes d he mimized loglikelihood for he e weighed Weiull (BWW) d weighed Weiull (WW) disriuios (wih correspodig sdrd errors i preheses) re show i he Tles d 2 elow: Tle. Descripive sisics for he life legh of compoes d i yers Mi Q Medi Me Q 3 M Skewess Kurosis

15 LIFE LENGTH OF COMPONENTS ESTIMATES 5 Tle 2. MLEs of he model prmeers d he correspodig sdrd error Esimes d sdrd errors i preheses Model â ˆ ˆ ˆ BWW (.89564) (.889) (.843) (.7843) LWW -: whe (.68954) (.733) (.75332) EWW : whe (.68954) (.733) (.75332) WW -:- -: Whe (.654) (.5777) Sice he vlues of he esimes re smller for he BWW disriuio compred o oher models herefore he ew model is eer represeive model o hese d. The sympoic covrice mri of he mimum likelihood esimes for he e weighed Weiull disriuio which is geered from he iverse of Fisher s iformio mri d is give y Coclusio The hree prmeer weighed Weiull disriuio ler reduced o wo prmeer y seig λ for he ske of simpliciy pioeered y Shhz e l. [9] is eeded y iroducig wo ddiiol shpe prmeers clled he e weighed Weiull (BWW) disriuio hvig

16 6 N. IDOWU BADMUS d T. ADEBAYO BAMIDURO roder clss of desiy fucios d hzrd re. This is oied y kig (2) s selie cumulive desiy fucio of he logi of e fucio defied y Joes []. We prese deiled sudy o he mhemicl properies of he ew propose disriuio; d he ew model icludes s specil su-models he Lehm weighed Weiull (LWW) (Bdmus e l. [7]) weighed epoeil (WE) (Gup d Kudu [9]) weighed Weiull (WW) (Mhdy Rmd [3]) weighed Weiull (WW) (Shhz e l. [9]) d oher disriuio. We lso derive he desiy d disriuio fucio survivl re hzrd re sympoic ehviours momes d mome geerig fucio. The prmeers of he propose disriuio were esimed d iverse of fisher iformio mri is derived. Applicio o lifeime d se idices h pr from he e weighed Weiull is more fleile i is lso hs eer represeio of d d superior o he fi of is pricipl su-model. Furhermore we hope h he ew model my e pplicle o my res such s survivl lysis ecoomics egieerig eviromel ec.. Refereces [] A. Akisee F. Fmoye d C. Lee The e Preo disriuio Sisics 42(6) (28) hp//d.doi.org/.8/ [2] A. Akisee d C. Lowe The e-ryleigh disriuio i reliiliy mesure Secio o physicl d egieerig scieces Proceedigs of Americ Sisicl Associio () (29) [3] A. Azzlii A clss of disriuios which iclude he orml oes Scd. J. S. 2 (985) [4] G. M. Cordeiro A. B. Sims d D. B. Sosic Eplici epressios for momes of he e Weiull disriuio (28) -7. rxiv:89.86vl[s.me] hp:/fucios.wolfr.com/gmmeerf/beregulrized/3//pp-4 [5] G. M. Cordeiro C. Aledr Edwi M. M. Oreg d J. M. Sri Geerlized e geered disriuios ICMA Cere Discussio Ppers i Fice DP 2-5 (2) -29. [6] N. C. Eugee d F. Fmoye Be-orml disriuio d is pplicios Commuicios i Sisics Theory d Mehods 3 (22)

17 LIFE LENGTH OF COMPONENTS ESTIMATES 7 [7] F. Fmoye C. Lee d O. Olugeg The e-weiull disriuio Jourl of Sisicl Theory d Applicios 4(2) (25) [8] J. M. Fischer d D. Vugh The e-hyperolic sec disriuio Ausri Jourl of Sisics 39(3) (2) [9] R. D. Gup d D. Kudu A ew clss of weighed epoeil disriuios Sisics 43(6) (29) [] M. C. Joes Fmilies of disriuios risig from disriuios of order sisics es 3 (24) -43. [] Arur J. Lemoe The e log-logisic disriuio BJPS Acceped Muscrip (24) -2. [2] Luz M. Ze Rodrigo B. Silv Mrcelo Bourguigo Adre M. Sos d Guss M. Corderio The e epoeied Preo disriuio wih pplicio o ldder ccer suscepiiliy Ieriol Jourl of Sisics d Proiliy (2) (22) 8-9. [3] M. Rmd Mhdy A clss of weighed Weiull disriuios d is properies Sudies i Mhemics Scieces (Cd) VI() (23) DOI..3968/.sms [4] A. L. Moris G. M. Cordeiro d Audrey H. M. A. Cyeiro The e geerlized logisic disriuio Sumied o he Brzili Jourl of Proiliy d Sisics (BJPS Acceped Muscrip) (2) -3. [5] S. Ndrh d S. Koz The e Gumel disriuio Mhemicl Prolems i Egieerig 4 (24) [6] S. Ndrh d S. Koz The e epoeil disriuio Reliiliy Egieerig d Sysem Sfey 9 (26) [7] N. I. Bdmus T. A. Bmiduro d S. G. Oguoi Lehm ype II weighed Weiull disriuio Ieriol Jourl of Physicl Scieces 9(4) (24) DOI:.5897/IJPS23.4. [8] G. P. Pil d C. R. Ro Weighed disriuio survey of heir pplicio; I P. R. Krishih (Ed) Applicios of Sisics (Norh Holld Pulishig Compy) (977) [9] S. Shhz M. Q. Shhz d N. Z. Bu A clss of weighed Weiull disriuio Pkis J. Sisics Operio Res. VI() (2) [2] O. I. Shiu d A. K. Adepou O he e-nkgmi disriuio Progress i Applied Mhemics 5() (23) [2] J. M. R. Hoskig L-Momes lysis d esimio of disriuios usig lier comiios of order sisics Jourl of he Royl Sisicl Sociey B 52 (99) g