Statistical properties and applications of a Weibull- Kumaraswamy distribution

Transcription

1 Itrtol Jourl of Sttstcs d Appld Mthmtcs 208; 3(6): 8090 ISSN: Mths 208; 3(6): Stts & Mths Rcvd: Accptd: Amu M Dprtmt Mths d Sttstcs, Aukr Ttr Al Polytchc, Buch, Ngr Dkko HG Dprtmt of Sttstcs, Ahmdu Bllo Uvrsty Zr, Ngr Yhy A Dprtmt of Sttstcs, Ahmdu Bllo Uvrsty Zr, Ngr Sttstcl proprts d pplctos of Wull Kumrswmy dstruto Amu M, Dkko HG d Yhy A Astrct Th Kumrswmy prolty dstruto ws orglly proposd y Pood Kumrswmy (980) for doul oudd rdom procsss for hydrologcl pplctos. Hr, w proposd w tso of th Kumrswmy dstruto y ddg o shp d o scl prmtr to th Kumrswmy dstruto usg th Wull lk fucto proposd y Thr t l. (205). Ths study hs drvd som prssos for ts sc sttstcl proprts such s momts, momt grtg fucto, th chrctrstcs fucto, rllty lyss, qutl fucto d th dstruto of ordr sttstcs. Som plots of th dstruto d th rllty fucto wr grtd d trprtd pproprtly. Th modl prmtrs hv stmtd usg th mthod of mmum lklhood stmto. Th prformc of th WullKumrswmy dstruto hs lso tstd y som pplctos to two rl dt sts. Kywords: WullKumrswmy dstruto, sttstcl proprts, ordr sttstcs, rllty lyss, stmto, prmtrs, pplcto. Itroducto Th Kumrswmy prolty dstruto ws orglly proposd y Pood Kumrswmy (980) [6] for doul oudd rdom procsss for hydrologcl pplctos. Th Kumrswmy doul oudd dstruto dotd y (, ) dstruto s fmly of cotuous prolty dstrutos dfd o th trvl [0, ] wth cumultv dstruto fucto (cdf) gv y G( Ad th corrspodg prolty dsty fucto (pdf) gv y () g( (2) Corrspodc Amu M Dprtmt Mths d Sttstcs, Aukr Ttr Al Polytchc, Buch, Ngr For 0, whr > 0 d > 0 r th shp prmtrs. W hv so my grlzd fmls of dstrutos proposd y dffrt rsrchrs tht r usd tdg othr dstrutos to produc compoud dstrutos wth ttr prformc. W lso hv som grlztos of th Kumrswmy dstruto rctly proposd th ltrtur such s th trsmutd Kumrswmy dstruto y Kh t l., (206) [5], th pottd Kumrswmy dstruto y Jvshr t l. (205) [3] d th Kumrswmy Kumrswmy dstruto y ElShrpy d Ahmd (204) [2]. I th t scto, w hv otd th cdf d pdf of th WullKumrswmy dstruto (WKD) usg th stps proposd y Thr t l, (205) [7]. Accordg to thm, th formul for drvg th cdf d pdf of y Wullsd dstruto from th ov WullG fmly s dfd for y cotuous dstruto s follows: F( log G( 0 t t dt ~80~ log G( (3)

2 Itrtol Jourl of Sttstcs d Appld Mthmtcs d g( f ( G( log G( log G( (4) Rspctvly, whr g( d G( r th pdf d cdf of y cotuous dstruto to grlzd rspctvly d α>0 d β>0 r th two ddtol w prmtrs rsposl for th scl d shp of th dstruto rspctvly. Th m of ths rtcl s to troduc w cotuous dstruto clld WullKumrswmy dstruto (WKD) from th proposd fmly y Thr t l. (205) [7]. Th rmg prts of ths ppr r prstd sctos s follows: W dfd th w dstruto d gv ts plots scto 2. Scto 3 drvd som proprts of th w dstruto. Th stmto of prmtrs usg mmum lklhood stmto (MLE) s provdd scto 4. I scto 5, w crryout pplcto of th proposd modl wth othrs to two rl lf dt sts. Lstly, scto 6, w gv som cocludg rmrks. 2. Th WullKumrswmy dstruto (WKD) Usg quto () d (2) (3) d (4) d smplfyg, w ot th cdf d pdf of th WullKumrswmy dstruto s follows: F log ;0 (5) d log f log (6) rspctvly. For 0 ;,,, 0, whr > 0 d > 0 r th shp prmtrs whl 0d 0 r scl d shp prmtrs rspctvly. Gv som vlus for th prmtrs α, β, d, w provd som possl shps for th pdf d th cdf of th WKD s show fgur d 2 low: Fg. : PDF of th WKD for dffrt vlus of th prmtrs d whr s dsplyd th ky o th grph. Fgur dcts tht th WKD dstruto hs vrous shps such s lftskwd or rghtskwd shps dpdg o th prmtr vlus. Ths ms tht dstruto c vry usful for dtsts wth dffrt shps. ~8~

3 Itrtol Jourl of Sttstcs d Appld Mthmtcs Fg 2: CDF of th WKD for dffrt vlus of th prmtrs d whr s dsplyd th ky o th grph. From th ov cdf plot, th cdf crss wh X crss, d pprochs wh X coms lrg, s pctd. 3. Proprts I ths scto, w dfd d dscuss som proprts of th WKD dstruto. 3. Momts Lt X dot cotuous rdom vrl, th th momt of X s gv y; ' E X 0 f ( d (7) Cosdrg f( to th pdf of th WullKumrswmy dstruto s gv quto (7). ' Rcll, E X 0 f ( d f ( log log (8) Lt A log Th, usg powr srs pso for A, w c wrt A s: log A log! 0 Susttutg for th pso ov quto (8), w hv; ~82~

4 Itrtol Jourl of Sttstcs d Appld Mthmtcs 0! f ( log log log 0! (9) Also, lt B log th, th followg formul holds for B for ( d th w c wrt th B s follows: jkl k k, l0 j0 Whr for (for j 0) P j,0 = d (for k=,2,..) k k k log j k j l P j, k l P jk, k k m m m j k m P j, km (0) () Comg quto (9) d srtg th ov powr srs quto (0) d smplfyg, w hv: jkl k! k f P j k l k k l jk, 0 k, l 0 j 0 j! k jkl k k l jk, k f P 0 k, l 0 j 0 j k j l (2) Now, f l s postv otgr, w c pd th lst trm (2) s: l m l m0 m Thrfor, f( coms: jklm! m lk k k k f Pjk, m 0 0 k, l 0 j 0 j k m j l m Whr, j, k, l, m W W m jklm! lk k k k P j k l, j, k, l, m j, k m0 0 k, l0 j0 m j (3) ~83~

5 Itrtol Jourl of Sttstcs d Appld Mthmtcs Hc, ' X, j, k, l, m 0 0 m E f ( d W d (4) Rcll th for th Kumrswmy dstruto; r r r r X Hc, ths mpls tht 0 0 E f ( d d B, ' X, j, k, l, m 0 0 m E f ( d W d, j, k, l, m, j, k, l, m 0 0 m f ( d W d W B, m W,,,, B, m j k l m (5) Th M Th m of th WKD c otd from th th momt of th dstruto wh = s follows: ' E X W, j, k, l, mb m, (6) Also th scod momt of th WKD s otd from th th momt of th dstruto wh =2 s ' 2 2 X 2 E W, j, k, l, mb, m (7) Th Vrc Th th ctrl momt or momt out th m of X, sy μ, c otd s E X ' ' ' ( ) 0 (8) Th vrc of X for WKD s otd from th ctrl momt wh =2, tht s, E[ 2 Vr( X ) E[ X ] X 2 ] 2 2 j k l m Vr( X ) W B, m W B, m, j, k, l, m,,,, (9) (20) Th vrto, skwss d kurtoss msurs c lso clcultd from th octrl momts usg som wllkow rltoshps. 3.2 Momt Grtg Fucto Th mgf of rdom vrl X c otd y M t ( t) E f ( d 0 t (2) Usg powr srs pso quto (2) d smplfyg th tgrl (2), thrfor w hv; ~84~

6 Itrtol Jourl of Sttstcs d Appld Mthmtcs t t t ' t ( ), j, k, l, m, 0! 0! 0! M t E W B m (22) whr d t r costts, t s rl umr d μ dots th th ordry momt of X. 3.3 Chrctrstcs Fucto Th chrctrstcs fucto of rdom vrl X s gv y; ( ) t t E E cos( t s( t E cos( t E s( t (23) Smpl lgr d powr srs pso provs tht t 2 2 ' t ' () t ! 0 2! (24) Whr μ 2 d μ 2+ r th momts of X for =2 d =2+ rspctvly d c otd from μ quto (5) 3.4 Qutl fucto for th WKD. Th qutl fucto, sy X=Q(u), of th WKD c otd s th vrs of Equto (5) s; F log X q p l Qu u (25) By usg (25) ov, th md of X from th WKD s smply otd y sttg u=0.5 whl rdom umrs c grtd X Q from WKD y sttg u, whr u s uform vrt o th ut trvl (0,). 3.5 Skwss d kurtoss Th qutl sd msurs of skwss d kurtoss wll mployd du to ostc of th clsscl msurs som css. Th Bowly s msur of skwss (Kdy d Kpg, 962.) sd o qurtls s gv y; SK 3 Q 2Q Q Q Q 4 4 (26) Ad th Moors (998) kurtoss s o octls d s gv y; KT Q Q Q Q Q 8 4 (27) whr Q(. )s y qurtl or octl of trst. 3.6 Rllty lyss of th WKD. Survvl fucto s th lklhood tht systm or dvdul wll ot fl ftr gv tm. Mthmtclly, th survvl fucto s gv y: S X P X F Applyg th WKD (28), th survvl fucto for th WKD s otd s: ~85~ (28)

7 Itrtol Jourl of Sttstcs d Appld Mthmtcs S log (29) Blow s plot of th survvl fucto t chos prmtr vlus fgur 3 Fg 3: Th survvl fucto of th WKD for dffrt vlus of th prmtrs d whr s dsplyd th ky o th grph. Th fgur ov rvld tht th prolty of survvl for y rdom vrl followg WullKumrswmy dstruto rducs s th vlus of th rdom vrl coms lrgr, tht s, s g grows, prolty of lf dcrss. Ths mpls tht th WullKumrswmy dstruto c usd to modl rdom vrls whos survvl rt dcrss s thr g grows whr 0<<. Hzrd fucto s th prolty tht compot wll fl or d for trvl of tm. Th hzrd fucto s dfd s; h f f S F (30) Susttutg for f( d F( d smplfyg gvs log h ( ) (3) Th followg s plot of th hzrd fucto t chos prmtr vlus fgur 4 Fg 4: Th hzrd fucto of th WKD for dffrt vlus of th prmtrs whr s dsplyd th ky o th grph. ~86~ d

8 Itrtol Jourl of Sttstcs d Appld Mthmtcs Itrprtto: th fgur ov rvld tht th prolty of flur for y rdom vrl followg WullKumrswmy dstruto crss s th vlus of th rdom vrl crss, tht s, s tm gos o, prolty of dth crss. Ths mpls tht th WullKumrswmy dstruto c usd to modl rdom vrls whos flur rt crss s thr g grows. 3.7 Ordr Sttstcs I ths scto, w drv closd form prssos for th pdf of th th ordr sttstcs of th WKD. If X, X 2,..., X X smpl from th WKD d lso lt :, X 2:,..., X : th corrspodg ordr sttstc otd from ths smpl. Th pdf, of th th ordr sttstc c dfd s; s rdom f :! f : ( f ( F( F( ( )!( )! whr f( d F( r th pdf d cdf of th proposd dstruto rspctvly. Usg (5) d (6), th pdf of th th ordr sttstcsx :, c prssd from s; (33) f : k l ( ), j, k log k 0 k 0!! ( W * ( )!( )! k (32) (34) Hc, th pdf of th mmum ordr sttstc X () d mmum ordr sttstc X () of th WKD r gv y; f : k l ( ), j, k log k 0 k 0! ( W * k (35) Ad f ( ),, l W : j k 0 rspctvly. log! (36) 4. Estmto of Prmtr Lt X,, X smpl of sz dpdtly d dtclly dstrutd rdom vrls from th WKD wth ukow prmtrs α, β,, d dfd prvously. Th lklhood fucto s gv y; Lt th loglklhood fucto, log L X, X 2,..., X /,,, log l log L X, X 2,..., X /,,,, thrfor l log log log log ( ) log ( ) log log ( ) log log log (37) (38) Dffrttg l prtlly wth rspct to α, β, d rspctvly gvs; ~87~

9 Itrtol Jourl of Sttstcs d Appld Mthmtcs l l l l log ( ) 0 log log l 0 l l l log log ( ) 0 log 0 log l l log log log log log log log l (39) (40) (4) (42) l l dl 0 0 0, Th soluto of th olr systm of qutos;,, d l d 0, wll gv us th mmum lklhood stmts of prmtrs,, d. Howvr, th soluto cot gott lytclly cpt umrclly wth th d of sutl sttstcl softwr lk R, SAS,.t.c wh dt sts r vll. 5. Applctos Ths scto prsts pplctos of th two dtsts to som slctd grlztos of th Kumrswmy dstruto. W hv comprd th prformc of th WullKumrswmy dstruto (WKD) to thos of thr modls such s th trsmutd Kumrswmy dstruto (TKD), th KumrswmyKumrswmy dstruto (KKD) d th Kumrswmy dstruto (KD). Th two dtsts r s follows: Dt st I: Ths dt s flood dt wth 20 osrvtos otd from Dumocu d Atl (973) d t hs usd y Kh t l. (206) [5]. Dt st II: Th scod dt st s o shp msurmts of 48 rock smpls from ptrolum rsrvor. Ths dt ws trctd from BP rsrch, mg lyss y Rot Ktz, u Oford d hs usd for lyss y Jvshr t l. (205) [3]. Th summry of th two dt sts s lso provdd Tl s follows; Tl : Summry Sttstcs for th two dt sts prmtrs Mmum Q Md Q3 M Mmum Vrc Skwss Kurtoss Vlus for dt st I Vlus for dt st II W lso provd som hstogrms d dsts for th two dt sts s show Fgur 5 d 6 low rspctvly. ~88~

10 Itrtol Jourl of Sttstcs d Appld Mthmtcs Fg 5: A hstogrm d dsty plot for th flood dt (Dt st I) Fg 6: A Hstogrm d dsty plot for th rock smpl dt (Dt st II) From th dscrptv sttstcs, hstogrms d dsts show ov for th two dt sts, w osrvd tht oth th frst d scod dt sts r postvly skwd d thrfor sutl for dstrutos tht r skwd to th rght. For us to ssss th modls lstd ov, w md us of som crtr: th AIC (Akk Iformto Crtro), CAIC (Cosstt Akk Iformto Crtro), BIC (Bys Iformto Crtro) d HQIC (H Qu formto crtro). Dcso chmrk: Th modl wth th lowst vlus of ths sttstcs would chos s th st modl to ft th dt. Th stmts d prformc msurs of th ov dstrutos r lstd tl 2 d 3 for dtsts I d II rspctvly s follows: Tl 2: Prformc of th dstruto usg th AIC, CAIC, BIC d HQIC vlus of th modls sd o dt st I. Dstrutos WKD TKD KKD EKD KD Prmtr stmts =0.792 =.2937 α = β = = =9.372 λ = =.4228 = α = β =.844 =.0795 =5.309 γ =9.059 =3.247 = ƖƖ=(loglklhood vlu) AIC CAIC BIC HQIC Rks of modls prformc ~89~

11 Itrtol Jourl of Sttstcs d Appld Mthmtcs From Tl 2, th vlus of th prmtr MLEs d th corrspodg vlus of ƖƖ, AIC, BIC, CAIC d HQIC for ch modl shows tht th WKD hs ttr prformc comprd to th KKD, EKD, TKD d KD. It lso grs wth th fct tht grlzg y cotuous dstruto provds compoud dstruto wth ttr ft th th clsscl dstruto th vw tht th WKD dstruto outprformd th clsscl KD tslf. Tl 3: Prformc of th dstruto usg th AIC, CAIC, BIC d HQIC vlus of th modls sd o dt st II. Dstrutos WKD TKD KKD EKD KD Prmtr stmts =.0850 = α = β = =.4353 = λ = = = α = β = = = γ = =.6673 = ƖƖ=(loglklhood vlu) AIC CAIC BIC HQIC Rks of modls prformc Tl 2 lso shows th prmtr stmts to ch o of th fv fttd dstrutos for th scod dt st (dt st II), th tl lso provd th vlus of ƖƖ, AIC, BIC, CAIC d HQIC of th fttd modls. Th vlus Tl 3 lso dct tht th Wull Kumrswmy dstruto hs ttr prformc wth th lowst vlus of AIC, CAIC, BIC d HQIC. Th WKD prformd ttr th th KKD, TKD, EKD d KD. 6. Cocluso Ths study hs proposd w dstruto. Som mthmtcl d sttstcl proprts of th proposd dstruto hv studd pproprtly. Th drvtos of som prssos for ts momts, momt grtg fucto, chrctrstcs fucto, survvl fucto, hzrd fucto, qutl fucto d ordrd sttstcs hs do pproprtly. Som plots of th dstruto rvld tht t s postvly skwd dstruto. Th modl prmtrs hv stmtd usg th mthod of mmum lklhood stmto. Th mplctos of th plots for th survvl fucto dct tht th WullKumrswmy dstruto could usd to modl tm or gdpdt vts, whr survvl rt dcrss wth tm or g. Th prformc of th w dstruto s llustrtd y som pplctos to two rl dt sts. Th rsults showd tht th w dstruto, WKD prforms ttr th th KumrswmyKumrswmy, Trsmutd Kumrswmy, Epottd Kumrswmy d th Kumrswmy dstrutos s show th lyss for dt st I d II scto Rfrcs. Cordro GM, Ortg EMM, Popovc BV, Pscm RR. Th Lom grtor of dstrutos: Proprts, mfcto procss d rgrsso modl. Appld Mthmtcs d Computto. 204; 247: ElShrpy EE, Ahmd MA. O th KumrswmyKumrswmy dstruto. Itrtol Jourl of Bsc d Appld Sccs. 204; 3(4): Jvshr Z, H Rd A, Arghm NR. EpKumrswmy dstrutos: Som proprts d Applctos. Jourl of Sccs, Islmc Rpulc of Ir. 205; 26(): Ky JF, Kpg ES. Mthmtcs of Sttstcs, 3 d, Chpm & Hll Ltd, Nw Jrsy, Kh MS, Kg R, Hudso IL. Trsmutd kumrswmy dstruto. Sttstcs I Trsto. 206; 7(2): Kumrswmy P. Grlzd prolty dstyfucto for douloudd rdom procsss. Jourl of Hydrology. 980; 46: Thr MH, Zur M, Msoor M, Cordro GM, Alztrh A, Hmd GG. Th w WullG fmly of dstrutos. Commu SttThory Mthods (forthcomg), Moors JJ. A qutl ltrtv for kurtoss. Jourl of th Royl Sttstcl Socty: Srs D. 988; 37:2532. ~90~